Surveys in Differential Geometry, Vol. 11: Metric and Comparison Geometry

Volume XI

Surveys in Differential Geometry Metric and Comparison Geometry

edited by Jeffrey Cheeger and Karsten Grove

iii

International Press

Surveys in Differential Geometry, Vol. 11

International Press P.O. Box 43502 Sommerville, MA 02143 [email protected] www.intlpress.com Copyright © 2007 by International Press

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Jeffrey Cheeger and Karsten Grove, editors

ISBN: 978-1-57146-117-9 Typeset using the LaTeX system. Printed in the USA on acid-free paper.

Surveys in Differential Geometry XI

Metric and comparison geometry Jeff Cheeger and Karsten Grove

The present volume surveys some of the important recent developments in metric geometry and comparison geometry. These areas represent a vital and expanding part of modern geometry. We begin with an indication of their scope and perspective. While metric geometry could be taken as refering simply to the geometry of metric spaces with no additional structure, our interest here is in those metric spaces which are smooth riemannian manifolds, or more generally, in metric spaces which are either constructed from smooth riemannian manifolds via natural geometric operations such as the process of taking weak geometric limits, or which share properties of riemannian manifolds when these properties are formulated in a suitably weak sense. Also characteristic of the subject is a certain synthetic mode of argument driven by an associated sequence of mental pictures. In establishing the geometric tools on which such synthetic arguments are based, analysis (calculus in some form) intervenes to an extent which varies from instance to instance. By comparison geometry we mean the study of manifolds (or more general metric spaces) whose curvature satisfies definite bounds, for instance, the classification problem for manifolds of positive curvature, which is still far from solved. Comparison theorems are a key tool in comparison geometry. These are theorems which assert that if a particular inequality on curvature is satisfied, then some associated geometric property holds at least to the extent that it does in a corresponding model case, often one in which the curvature is constant. Classical comparison theorems pertain to the behavior of geodesics, and related objects such as the Jacobi equation, the formula for the second variation of arc length and the index form in Morse theory. Date: May 10, 2007. v

vi

PREFACE

Increasingly however, the solution of a problem in comparison geometry is likely to call for a combination of synthetic, analytic and topological arguments. Scalar curvature, for example, is too weak an invariant to control the behavior of geodesics. At the infinitesimal level scalar curvature controls the volume of balls, but it fails to do so for balls of any definite positive radius. Nonetheless, although manifolds with positive scalar curvature are much less constrained than those with positive sectional curvature, there are analytically based results for positive scalar curvature which do not (at present) have synthetic proofs, even under the assumption that the sectional curvature is positive e.g. the vanishing of the A-genus for spin manifolds with positive scalar curvature. The A-genus is actually a concept from topology, and topological techniques, such as surgery theory, also play an important role in the subject. In the study of manifolds with a lower bound on Ricci curvature, estimates on geodesics can be used to control volume. They are also important in situations which are highly constrained (in fact almost rigid). However, some control over geodesics is lost when one passes to weak geometric limits; for example, one can have infinitely many distinct geodesics with the same tangent vector. So in addition to comparison theorems like the Bishop-Gromov inquality, ideas from analysis such as Bochner's formula, the maximum principle, gradient estimates for harmonic functions are required. The theories of Einstein manifolds and Ricci flow involve geometry as well as analysis. Even in the world of more general metric spaces, the connection with analysis is seen in results where the existence of a measure satisfying a suitable compatibility condition with the metric, say a doubling condition, or Poincare inequality, leads to metric or topological conclusions, in whose statement the measure does not appear. In metric riemannian geometry, this is particularly relevant in the context of lower (and two-sided) Ricci curvature bounds. Thus, it seems that distinctions such as "metric geometry" versus "geometric analysis" are to some extent artificial and if pressed too far, are genuinely destructive. To reiterate, increasingly, the solution of specific geometric problems requires a mixture of synthetic, analytic and topological arguments - the work of Perelman (on the program originated by Hamilton) being just one, albeit spectacular, example. This circumstance can only make the subject more interesting. This having'been said, our purpose here is to focus primarily, but not exclusively, on techniques from metric geometry and their use in the study of comparison geometry.

PREFACE

vii

What then does the metric point of view have to offer? Here are few of the possible answers. The metric space structure does not presuppose any assumption of smoothness. Attempting to extend notions of curvature to objects with minimal smoothness can provide fundamental insight. For instance, one can try to understand which notions of curvature are defined for piecewise flat spaces, or for convex hypersurfaces, or what Ricci curvature bounds might mean for a space equipped with a metric and a measure. At a more practical level, certain results which have arisen in metric riemannian geometry have subsequently been realized to have natural extensions to much more general contexts and these have turned out to be of fundamental importance. A prime example is furnished by the work of Gromov in geometric group theory. Finally, even if one were only interested in riemannian manifolds per se, the study of more general metric measure spaces provides useful information, for example, via the following route. Compactness theorems, such as that of Gromov, (whose hypothesis requires a definite lower bound on Ricci curvature) enable one to take weak geometric limits of sequences of riemannian manifolds M? The resulting limit spaces, Y, may be viewed as playing a role in riemannian geometry which is analogous to that played by distributions or Sobolev functions in analysis. Information on the regularity and singularity structure of such Y (the analog of Sobolev embedding theorems) provides information on the sequence, M?, in some cases enough information to show that in actuality, it could not have existed in the first place, and in other cases, information on structure of the M?, for i large. On the face of it, this sort of reasoning might appear circular, since it would seem that the only possibility for obtaining nontrivial information on the limit space would have to be via uniform estimates on the approximating sequence. While initially this is so, once some preliminary properties of the limit objects have been established, additional properties can be deduced purely synthetically i.e. without further reference to the approximating sequence M? This in turn, provides new information on the M? themselves. We now turn to the articles in the present volume. As we have indicated, lower bounds on Ricci curvature and in particular lower bounds on sectional curvature provide a natural setting in which convergence methods playa significant role. Their utility in applications is governed by the extent to which the limit objects and their relation to elements of a limiting sequence is well understood. Such understanding is most complete in the case of bounded sectional curvature; see the article by Rong.

viii

PREFACE

When only a lower sectional curvature bound is imposed, the stabilit~ theorem of Perelman provides good information in the noncollapsing case A (long awaited) complete and detailed exposition of Perelman's stabilit~ theorem is presented in the article by Kapovitch. Although there have been important recent advances in the collapsin~ case, the crucially important class of almost nonnegatively curved manifold: is still poorly understood. Since all limit objects are Alexandrov spaces, the theory of these spaces will be indispensible in future developements; see the article by Petrunin. Convergence theory in the case in which only a lower Ricci curvature bound is imposed, is surveyed in the article by Wei. A class of metric measure spaces, for which a sythetic definitition of lowe: Ricci curvature bounds is possible has long been sought and has recent1~ emerged. This class, includes in particular, weak limits riemannian manifold: with lower Ricci curvature bounds. The general theory will surely underg< considerable further developement; see the article by Lott. Progress in the classical areas of manifolds with nonnegative or positive sectional curvature is discussed in the articles of Wilking and of Ziller. The latter provides a self contained account on all examples known to date. After this volume was completed a milestone was reached in the classica pinching problem: A (pointwise) weakly 1/4 pinched manifold is diffeomor· phic to either a space form or it is isometric to a rank one symmetric space The proof due to Brendle and Schoen is an amazing application of the Rice flow. An account provided by wilking of this exciting development can be found as an "added in proof" section of his article in this volume. In cases where uniform curvature comparisons are available, convergence methods have played a role when only upper curvature bounds are presen" e.g. for simply connected manifolds with nonpositive sectional curvature an< in parts of geometric group theory. A survey on the current state of affair: for general spaces with an upper curvature bound is provided in the article by Schroder and Buyalo. The article by Farrell, Jones and Ontaneda is concerned with geometric and topological rigidity and flexibility issues for negatively curved manifolds An update on the status of the classification problem for manifold: with positive and nonnegative scalar curvature is given in the article b~ Rosenberg. The selection of topics treated in this volume has been influenced b~ several factors, including space, existence of other sources and our succes: (or failure) in attracting contributors. So we will conclude by mentionin!

PREFACE

ix

some topics that might have well been included, but for whatever reason, ended up being omitted, or almost so. In the first place, lower curvature bounds are very much emphasized over upper bounds. While this was not by design, there do exist several excellent surveys which treat nonpositively curved manifolds. Recent progress on Einstein manifolds, in whose proofs metric geometry plays a role, might have been discussed. Nonsmooth calculus on metric measure spaces is another topic which has close relations with material considered here. (For a very informative overview, see Heinonen's recent survey in the Bulletin of the AMS.) Several topics from Perelman's papers on geometrization, such as comparison theorems in generalized and possibly infinite dimensional settings, would have been natural to include had they not been exposed at great length elsewhere. Other natural topics which wound up being left out are geometric group theory, isoperimetric inequalities and "curvature free" metric geometry (including results on systols). The excluded topics could easily fill a second volume. Perhaps, at some future time, they will.

Surveys in Differential Geometry XI

Contents

Preface.........................................................

v

Collapsed Manifolds with Bounded Sectional Curvature and Applications Xiaochun Rong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

N onnegatively and Positively Curved Manifolds Burkhard Wilking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Examples of Manifolds with Non-negative Sectional Curvature Wolfgang Ziller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Perelman's Stability Theorem Vitali Kapovitch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

Semiconcave Functions in Alexandrov's Geometry Anton Petrunin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

Manifolds with a lower Ricci Curvature Bound Guofang Wei....................................................

203

Optimal Transport and Ricci Curvature for MetricMeasure Spaces John Lott. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

Manifolds of Positive Scalar Curvature: A Progress Report Jonathan Rosenberg............................................

259

Spaces of Curvature Bounded Above S. Buyalo and V. Schroeder.....................................

295

Negative Curvature and Exotic Topology F.T. Farrell, L.E. Jones, and P. Ontaneda.......................

329

Surveys in Differential Geometry XI

Collapsed Manifolds with Bounded Sectional Curvature and Applications Xiaochun Rong ABSTRACT. This is a survey article on collapsed Riemannian manifolds with bounded sectional curvature. Instead of attempting to cover many results in related topics, we will concentrate on one path that includes most of the main ideas and techniques developed in the last two decades.

1. Introduction

Consider a complete n-manifold M of sectional curvature normalized to be bounded in absolute value, IsecMI ~ 1. Given € > 0, there is an € thickthin decomposition of M: the thick part consists of points whose injectivity radius is ~€ and the complement is the thin part. According to [12], the local topology of the thick part is under control: after a small perturbation of the boundary, any ball of radius one in the thick part has only finitely many possible topological types. On the other hand, when € < €(n) (a constant depending only on n), there exists a special geometric/topological structure on any unit ball in the thin part [14], consisting of a sort of generalized foliation with orbits consisting of nilmanifolds. Unless otherwise specified, a collapsed manifold means a complete Riemannian manifold M with IsecMI ~ 1, whose injectivity radii are less than €(n) everywhere, i.e., M is thin. Since the 1980s, Riemannian geometry has experienced an explosive development, and one of the most important achievements is the theory of collapsed manifolds. Before discussing collapsing in detail, we recall the Cheeger-Gromov compactness theorem [11, 12, 36], which, in its pointed version, controls the thick part. A sequence of Riemannian n-manifolds, (Mi' gi), is said to converge in the el,a-topology to a el,a manifold (Moo, goo) if there are diffeomorphisms fi : Moo -+ Mi such that the pullback metrics The author was partially supported by NSF Grant DMS 0203164 and by a research found from Capital Normal University. ©2007 International Press I

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2

e

converge to goo in the el,a sense, where goo is a 1 ,a metric. Specifically, there is an atlas on Moo, with 2 ,a transition functions, such that in local coordinates corresponding to each chart, the convergence of the gi,j is in the el,a-topology.

e

THEOREM 1.1. Given positive constants, n, d, v, and a sequence of closed n-manifolds Mi satisfying

diam(Md ~ d,

vol(Md ~ v > 0,

there is a subsequence converging in the el,a topology to a el,a-manifold. An important consequence of Theorem 1.1 is that for any n, d, v, there are only finitely many closed manifolds in the given class up to diffeomorphism. Essentially, this is obtained in [11, 12] by estimating a uniform lower bound on the injectivity radius and by constructing an atlas whose charts are normal coordinate systems defined on balls of a definite size, for which the transition functions are controlled; compare [45]. It is also observed in [11] that assuming additional bounds on higher covariant derivatives of curvature gives correspondingly better control of the transition functions. In an unpublished work of Cheeger (part of which was the subject of a lecture at the Summer Institute on Global Analysis held at Stanford in 1973), under the assumptions of Theorem 1.1, Lipschitz control of the metric was obtained via regularization arguments. One should point out, however, that from the standpoint of regularity normal coordinates systems are far from optimal. In [36], Gromov noted that employing distance function coordinates gives control of one more derivative of the transition functions and of metric, i.e., e 2 and e l control, respectively. He also made the powerful observation that Toponogov's comparison theorem for geodesic triangles has a formulation which passes to limits under such convergence, or even under (the weaker) Gromov-Hausdorff convergence. Given the assumptions of Theorem 1.1, harmonic coordinate systems on balls of a definite size, in which the metric has definite el,a-bounds, were constructed in [40]. Harmonic coordinates were used in [30] to obtain the optimal regularity in Theorem 1.1. A natural question is: What can be said if the assumption of positive lower bound on volume in Theorem 1.1 is removed? In general, one asks the same question when removing bounds on diameter and volume but assuming that the local volume is arbitrarily small (equivalently, the injectivity radii are everywhere uniformly small). The first non-trivial example of collapsing was observed by M. Berger. It is obtained from the standard metric on 8 3 by multiplying the component tangent to the Hopf fibration on 8 3 by E2 while keeping the metric in the orthogonal complement. Then as E ---* 0 the sectional curvature lies in [E 2 ,4 - 3E2] while the injectivity radii converge to zero everywhere. The first

COLLAPSED MANIFOLDS

3

theorem on collapsing is Gromov's characterization of "almost flat" manifolds [32], which became a cornerstone of the subsequent more general collapsing theory (see Section 3). Gromov classified the maximally collapsed situation, i.e., when the diameter of M is very small. He showed that a finite normal covering of M is diffeomorphic to a nilpotent manifold [32J. A simple but powerful idea used in [32] is the notion of Gromov-Hausdorff distance, which measures the closeness of metric spaces, and a compactness theorem for this distance (see Section 2), whose importance, as further developments showed, cannot be overstated. The more general collapsing theory was established in the 1980s in the works of Cheeger-Gromov, [15, 16J, Fukaya [24, 28J and Cheeger-FukayaGromov (see Sections 4 and 5). Since the early 1990s, several interesting applications of collapsing theory have been obtained (see Section 6).

Acknowledgment. The author would like to thank Jeff Cheeger for his invaluable comments and suggestions on this paper. 2. Gromov-Hausdorfl' distances and compactness Let (Z, d) be a metric space. The collection of all compact subsets of Z forms a metric space under the Hausdorff distance, dH(A, B) = max{d(x, A), d(y, B), x E A, y E B}. Comparing this to the distance, d(A, B) = min {d(x,y), x E A, y E B}, notice that d(A, B) = 0 if and only if A n B =1= 0 while dH(A, B) = 0 if and only if A = B. Hence, dH(A, B) measures the "uniform closeness" of A and B. Gromov introduced an abstract version of the Hausdorff distance between any two compact metric spaces X and Y. A metric on the disjoint union, X 11 Y, is called admissible if it extends the metrics on X and on Y. For example, any disjoint isometric embedding of X and Y into the metric product, X x Y x [0, 1], induces an admissible metric on X 11 Y. DEFINITION 2.1 (Gromov-Hausdorff distance). For any two compact metric spaces X and Y, we call

dGH(X, Y) = inf { dH(X, Y), all admissible metrics on X

II Y} ,

the Gromov-Hausdorff distance (simply, the GH-distance). It is easy to check that dGH satisfies the triangle inequality, and dGH(X, Y) = 0 if and only if X is isometric to Y. Let Mete denote the set of isometric classes of all compact metric spaces. Then (Mete, dGH) is a metric space. Observe that for X E Mete and given any € > 0, dGH(X, A) < € for any finite €-dense subset A eX. This shows that dGH may not measure differences in local geometry. The power and usefulness of dGH lie in its

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4

pre-compact criterion below, which applies to many interesting geometric situations in Riemannian geometry (see Lemma 2.1). Let's first observe two obvious properties of a Cauchy sequence, {Xd c Mete: (2.1) There is a uniform upper bound on the diameter of Xi. (2.2) Given E > 0, each Xi has an E-dense subset A(E) of size IAi(E)1 ::; f(E), a constant depending only on EWe now verify that (2.1) and (2.2) are also sufficient conditions for any sequence in Mete to contain a convergent subsequence (so (Mete, dCH) is a complete metric space): To construct a limit for Xi, we may assume, passing to a subsequence if necessary, that for all i, an admissible metric di ,i+l on Xi II Xi+! such that di ,i+l,H(Xi , Xi+l) < 2- i . We then define an admissible metric d on Y = II Xi by assigning an admissible metric di,i+j on each Xi llXi+j:

It is easy to see that Xi is a Cauchy sequence in (Y, d) with respect to dH . Let X = {{xd : equivalent Cauchy sequences in Y, Xi E Xi} with {Xi} equivalent to {yd if d(Xi' Yi) --+ 0 as i --+ 00. Using (2.1) and (2.2), one verifies that X is a compact metric space with the metric dx( {Xi}, {yd) = .lim d(Xi' yd. Finally, we define an admissible metric on Y II X by t-+oo

d(y, {xd)

=

limd(y,xd, and check that dCH(Xi,X)::; dH(Xi, X) --+ O. As

2-+00

a by-product, we see that Xi d GH ) X can be understood as Xi ~ X in the compact metric space Y II X. In particular, it makes sense to say that Xi E Xi and Xi --+ X EX. The Bishop-Gromov volume comparison theorem asserts that if M is a complete n-manifold of Ricci curvature 2: k(n - 1), then for p E M, the ratio of volumes of r-balls, vol(Br(p))/vol(B~), is not increasing in r, where B~ is an r-ball in a simply connected n-space form of curvature k. As an application, Gromov observed that (2.2) is satisfied under the following geometric conditions: LEMMA 2.1 (Precompactness). Any sequence of closed n-manifolds, M i , with Ricci curvature RicMi 2: -k and diameter diam(Mi) ::; d, contains a dCH-convergent subsequence. In the rest of this section, we will discuss the equivariant and pointed Gromov-Hausdorff convergence (a motivation will be given at the end of this subsection). Consider Xi dGH ) X such that each Xi also admits an effective and isometric action by a compact group G i . It is natural to ask if there is a symmetry structure on X related to these Gi-actions. To give a positive answer,

COLLAPSED MANIFOLDS

5

we need the following 'equivalent' definition of dGH: a map f: X --+ Y is called an f-GH approximation if Id(Xl, X2) -d(f(Xl)' f(x2))1 < f and if f(X) is f-dense in Y. Define

dGH(x, Y) = inf {f, :3 f-Hausdorff approximation from X to Y and vice versa}. It turns out that ~dGH ::; dGH ::; 2dGH, and thus we may view "dGH = dGH " as far as the convergence is concerned (but dGH may not satisfy the triangle inequality.) . By the above, Xi dGH ) X is equivalent to the condition that given (decreasing) fi --+ 0, there are fi-GH approximations, Ii : Xi --+ X and hi : X --+ Xi. We now construct a limit group G of Gi as follows: take a sequence of finite fi-dense subsets, A( fi) C X, such that A( fi) c A( fj) for all i < j, and define, for each i, a sequence of maps: 0, the sequence of closed r-balls, Br(Xi) C Xi, dGH-converges to Br(x) C X such that Xi --+ x. For instance, if a closed group Gi acts isometrically on Xi, then (Xi, Xi)

dGH

)

(X, X) implies that (Xi, Xi, Gi )

deqGH )

some closed group of isometries of X and that (Xi/G i , Xi)

(X, X, G) for

dGH

)

(X/G, X).

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6

However, a significant difference for pointed GH-convergence is that for a sequence, different choice of base points may yield different limits. We conclude this section by applying the above results to two convergent sequences associated to a given sequence of Riemannian manifolds Mi d GH ) X:

(2.3) deqGH

(Mi,:h ri) ----'-------+

1

1ft

rri

Mi

dGH

~

deqGH

(X,x,r)

x=x/r

(F(Mi),O(n)) ----'-------+ and

l

(Y,O(n))

1ft

Pi

Mi

dGH

~

X = Y/O(n)

where 7ri : Mi -+ Mi is the Riemannian universal covering, r i = 7rl(Mi) is the group of deck transformations and F(Mi) is the orthogonal frame bundle equipped with a canonical metric: the parallel transport on Mi defines 'horizontal subspaces' on T(F(M)), and thus introduces a canonical metric on F(M) (up to a choice of a bi-invariant metric on O(n)) such that p: F(M) -+ M is a Riemannian submersion. A reason for studying the above associated sequences is that more infor. dGH - r i ) ----'-------+ deqGH ( - matIon on Mi ~ X may b e seen from (M i , Xi, X, X, r ) ( especially when dim(X) < dim(X)) and from (F(Mi),O(n)) (see the next two sections).

d eq GH )

(Y,O(n))

3. Almost flat manifolds

Gromov's theorem on almost flat manifolds is the first result on collapsing, and it has become a cornerstone of the collapsing theory ([32], cf. [4, 14, 28, 53]). A closed manifold M is called almost flat if the scaling invariant, max Isecl . diam 2 (M), is very small. A flat manifold is almost flat, but an almost flat manifold may admit no flat metric (Example 3.1). If we scale the metric so that IsecMI ::; 1, then M is almost flat if and only if M is maximally collapsed, i.e., M is close to a point with respect to de H . THEOREM 3.1 (Almost flat manifolds). There exist positive constants €(n) and w(n) such that if a closed n-manifold satisfies max Isecl·diam2 (M) < €( n), then M is finitely covered by a nilpotent manifold (the quotient of a simply connected nilpotent group N by a cocompact discrete subgroup r) with order ::;w(n).

Ruh [53] improved Theorem 3.1, showing that M itself is diffeomorphic to the quotient, r\N, where r c N [>( Aut(N) (the group of automorphisms on N) and [r : rnN] ::; w(n). Such a manifold is called an infra-nilmanifold. The key ingredient in Theorem 3.1 is the following Margulis lemma [32].

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7

LEMMA 3.1. If G is a connected Lie group, then its identity has a neighborhood Ue such that if r is a discrete subgroup, then r n Ue generates a nilpotent subgroup. Lemma 3.1 follows from the property d(e, g1g2g1 1g;;1) ~ Cd(e, gl)d(e, g2) for any gl, g2 close to the identity e E G (equipped with a left invariant metric), which is seen by (twice) applying the mean value theorem to f(t) = d(e,gl(t)g2(t)g1(t)-l g2 (t)-I), where gi(t) is a geodesic from gi to e and C is a constant. Lemma 3.1 easily implies Theorem 3.1 in the following special situation: let r be a cocompact discrete subgroup of a simply connected Lie group N. Assume that N admits a left invariant metric such that Isecl ~ 1 and diam(r \ N) < E. Then r n Ue generates rand exp;l(r n Ue ) spans the Lie algebra Ii of N. Thus r is nilpotent (Lemma 3.1), which implies that N is nilpotent. On the other hand, given a simply connected nilpotent group, one can construct a family of left invariant metrics via inhomogeneous rescaling so that the diameter of any compact subset goes to zero (Example 3.1). This implies that for any discrete co compact subgroup, the quotient is almost flat. EXAMPLE 3.1. A Lie group G is nilpotent if [G, G k] = 1 for some natural number k, where Gi+1 = [G, Gi ] denotes the commutator of Go = G and Gi . Then GHI is a normal subgroup of Gi such that Gi/Gi+1 is abelian. If ~ denotes the Lie algebra of Gi, then [~, h] C ~+1' and thus one can choose a basis for h, {eik}, such that {ejl' i ~ j} spans ~, and

[eik,ej!l = LLC~kleBP' B~i P

LLIC~kll ~ C (C~kl'C are constants). j$B P

Given any left invariant metric g, one can estimate the curvature tensor, IR(U, V)WI ~ 611 ad ll g • lUI· IVI . IWI, where Iladll g = max{I[U, V]I, lUI = IVI = 1, U, V E Ii}. We now define a one-parameter family of left invariant metrics by assigning {eik}, an orthogonal basis, with norm g€( eik, eik) = E2i (inhomogeneous scaling). It is easy to check that ladl gE ~ C for all E, and thus g€ has the desired property. One can easily extend the above construction in a fibration setting: let M ---+ N be a fibration with fiber a nilpotent manifold with a flat connection, and let M have a metric such that when restricting to a fiber, parallel fields are Killing fields. Thus the structural group is a subgroup of the affine automorphisms of a fiber. By collapsing a fiber to a point as in the above, one obtains a sequence of metrics, g€, on the total space of the fibration such that (M,g€) dGH ) N with IsecgEI ~ 1 [27]. SKETCH OF A PROOF OF THEOREM 3.1. Recall that a Lie group has the unique canonical flat connection, i.e., left invariant fields are parallel, and thus the torsion is parallel. Conversely, if a simply connected manifold

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8

M admits a flat connection with a parallel torsion, then parallel fields form a Lie algebra which then determines a Lie group structure on M. The goal of the proof is to construct a flat Riemannian connection with a parallel torsion on the Riemannian universal covering space £1 such that the deck transformations preserve the flat connection. It then follows that £1 is a Lie group and 7rl(M) C £1 ~ Aut(M). By the discussion following Lemma 3.1, we can conclude the desired result. By an obvious contradiction argument, it suffices to prove Theorem 3.1 for a sequence (see Lemma 2.1), (3.1) Let F(Mi) denote the orthogonal frame bundle. Since we will work on F(Mi) with a canonical metric where a bound on curvature is required, we will need a bound on the covariant derivative of the curvature tensor. Deforming the metric gi on Mi a short (but definite) time along the Ricci flows, one gets another almost flat metric with the required regularity [39, 50]. Hence, without loss of generality, we may assume that gi satisfies this extra regularity. For the sake of exposition, let's assume that (M, gd is obtained by slightly perturbing a 'left invariant' almost flat metric on a nilpotent manifold A \ N, as in Example 3.1. Because the injectivity radius of is infinite, it is expected that

g:

9:

the injectivity radius of (Mi ,9i) is bounded below by a constant p(n) > O. (3.2)

Since IsecMJ ~ 1, (3.2) implies a positive lower bound for the convexity radius of Mi, say pc(n) > O. Given a finite number points {Yj} in a ball Bi C Mi of radius pc(n), the function h(x) = L- j d2 (Yj,x) : Bi -? lR. is strictly convex and thus achieves the minimum at a unique point, call the center of mass for {Yj}. Assuming (3.2) (whose proof will be delayed until the next section), we will first construct a cross section for F(M) -? M via the technique of 'the center of mass': fixing x E £1, a(x) E F(M), by parallel translation of a(x) along radial geodesics in Bp(x), one obtains a cross section, a : Bp(x) -? F(Bp(x)). Of course, a may not be '7rl(M)-invariant,' i.e., a(-y(Y)) 1= ,*(a(y)), where Y E Bp(x)" E 7rl(M) such that ,(Y) E Bp(x). However, for any z E Bp(x), if the following inclusion holds for the finite set A(z),

!

(3.3)

A(z)

= b*(a(y)), Y E C Bpl(a(z)),

Bp(z)"

E

7rdM) such that ,(Y)

= z}

(p' > 0 is the convex radius of F(M)),

COLLAPSED MANIFOLDS

9

then the map z ~ the center of mass of A(z), is well-defined and defines a 1l"l(M)-invariant cross section, and thus a cross section on F(M) ~ M (note that Bp(x) ~ M is onto, because diam(M) « p)). We now verify (3.3). Consider an equivariant sequence, (Mi' Xi, rd d eqGH ) (X, X, r), associated to (3.1) as in (2.3). By (3.1) and (3.2), we may identify (X,x) = (IRn,O), and thus G is a closed subgroup of Isom(IRn) = IRn ~ O(n). Because IRn IG is a point, G = IRn ~ H, where H is a subgroup of O(n). By an argument similar to the proof of (3.2), one concludes that H is finite. This implies, from the equivariant convergence, that a short geodesic loop, "Ii (representing a nontrivial element in ri), has either a nonsmall holonomy or has a very small holonomy compared to its length. This implies a homomorphism, 0 (m = dim(M)) such that if deqGH((M, G), (N, G)) < E :::; E(n, io), then there is a G-invariant fibration map, f : (M, G) -+ (N, G), with a connected fiber F such that:

(4.1.1) d(x, f(x)) < T(E), where d is an admissible metric on M II N, T(E) ~ O. (4.1.2) f : M -+ N is an almost Riemannian submersion: any vector ~ orthogonal to F satisfies that e- T (€) :::; Idf(~)I/I~1 :::; eT (€). (4.1.3) If secM :::; 1, then the second fundamental form of fibers IIIFI :::; c(m). SKETCH OF PROOF. We will present a proof with G = {e}, and the general case can be obtained with suitable 'equivariant' modification. Given an admissible metric don M II N such that dH(M, N) < E, there is a natural projection that maps x E M to yEN which is closest to x, but this projection may not even be continuous if y is not unique. Using the geometric bounds on M and N, one overcomes this ambiguity by constructing a smooth embedding, : N y ~s, and a CI-map, '11 : M -+ ~s, such that W(M) is contained in a tube U of (N) where the projection P : U -+ (N), to the nearest point in (N), is smooth, and then defining f = -1 0 Po'll : M -+ N. Furthermore, f will satisfy (4.1.2) if and '11 are also 'C 1-close' in 'horizontal directions' as follows: let Ui E N, Vi E M such that d(Ul' Ui) = io/l0 (i = 2,3) and d(Ui' Vi) < E, and let ~ and 'f} be tangent vectors of the minimal geodesics from Ul to U2 and from VI to V2 respectively. Then there are constants C, T(E) such that

(4.3) To construct and'll, we first choose a pair of closed 'E-nets' in M and in N (i.e., {Xi} C Nand {Yi} C Mare E-dense in Nand Y respectively such that d(Xi' Xj) 2: E and d(Xi' Yi) < E), and a smooth 'cut-off' function, p(t) 2: 0, with h(( -00,0]) = p(O) = 1, supp(p) = (-00, io/lOO], p'(t) :::; 0 and p'(t) ,. . . , -C 1 near O. Then, define (x) = (p(d(X,Xi))) E ~s and w(y) = (pCI/Ji(y))) E ~s, where s = Inet€(N)1 and 'l/Ji(Y) is the average distance from y to z E B€(Yi) (this guarantees that p(d(y, Yi)) is C 1 smooth). The bounds on curvature and injectivity radius of N guarantee (4.1.1) and that is an embedding (the verification is somewhat tedious). The 'Cl-close' in

COLLAPSED MANIFOLDS

11

(4.3) can be verified from the 'angle-close' from dCH(M, N) < €: Let a be the angle between segments UI Ui, (3 the angle between VI Vi, i = 2,3. Then using the Toponogov comparison theorem, one can show that la - (31
0, there is a constant c(n, 0 such that if a closed symplectic n-manifold of finite fundamental group satisfies Isecl ~ 1,

diam

~

d,

then vol(M) 2: v(n, d). In particular, M has finitely many possible diffeomorphic types depending on nand d.

Note that Theorem 6.2 is false if we remove the restriction on the fundamental group without imposing further restrictions (e.g., a flat torus). SKETCH OF A PROOF OF THEOREM 6.2. Arguing by contradiction, we may assume a sequence satisfying Theorem 6.2, Mi dGH ) X with dim(X) < n. Then Mi admits a pure F-structure with orbits of positive dimensions (Theorem 4.2). Equivalently, the universal covering space !VIi admits a torus action without fixed points, and thus there is a circle subgroup without fixed points, a contradiction to a topological result obtained in [22]: any effective circle action on a closed symplectic manifold has a 0 non-empty fixed point set. Using Theorem 4.2, one can also prove an isomorphism finiteness result for the q-th homotopy groups of closed n-manifolds in terms of n, q and bounds on curvature and diameter [22, 50]. Note that the homotopy group finiteness does not hold if we remove the upper curvature bound (compare to [33]).

b. Manifolds with pinched positive sectional curvature. Let M be a closed n-manifold of positive sectional curvature. Recall that the fundamental group 1I'1(M) is finite, and if n is even and M is orientable, then 1I'1(M) = 1 (cf. [13]). However, in odd dimensions no general constraint on 1I'1(M) is known that could distinguish positive curvature from non-negative curvature. A conjectured obstruction is that there is 'Y E 11'1 (M) such that the ratio 11I'1(M)1/1'Y1 ~ wen), a constant depending only on n, where I'YI is the order of'Y [50]. Based on Theorem 4.2, one can partially verify the above conjecture. THEOREM 6.3. [50, 51] Let M be a closed n-manifold of 8-pinched curvature. If 11I'1(M)1 2: wen, 8), then 1I'1(M) has a non-trivial normal cyclic subgroup of index at most wen). SKETCH OF A PROOF OF THEOREM 6.3. We may assume dim(M) is odd. The condition 'secM 2: 8' implies that the diameter ~ 11'1 J8 (Bonnet theorem) and the volume ~ ':(~fl (volume comparison) which is small when w(n,8) is large, where 8'; is the n-sphere of constant curvature 8. Hence,

COLLAPSED MANIFOLDS

19

M admits a pure F -structure of rank k ~ 1 and a nearby invariant metric (Theorem 4.1). As it turns out, it is crucial to have an invariant metric of positive curvature. Based on the regularity of the Ricci flows, one can get a nearby invariant metric with 8/2-pinched curvature [50]. The above symmetry structure of a positively curved metric is all one needs to prove the desired property of 7rl (M). For instance, if k = 1 and circle orbits on M form a fibration, then the homotopy class of a circle orbit generates a normal cyclic subgroup < a >~ 7rl(M) such that 7rl(M)/ < a >~ 7rl(M*), where M* denotes the orbit space. Note that 7rl(M*) = 1 or Z2 because dim(M*) is even and M* has a unique metric so that M -+ M* is a Riemannian submersion (thUS secM" ~ 8/2, Gray-O'Neill submersion equations) . The proof in general is quite involved, and the constant w(n) is related to Gromov's Betti number estimate [33]. 0

Recall that the injectivity radius of a closed even-dimensional manifold of 0 < secM ::; 1 is at least 7r /2 while there is no positive lower bound in odd dimensions (e.g., Berger sphere). The Klingenberg-Sakai conjecture says once 8 is fixed, there is a positive lower bound on the injectivity radius of a 8-pinched metric depending on 8. However, there are infinitely many simply connected 7-manifolds of uniformly pinched positive sectional curvature [1]. Hence, to have a possible universal lower bound, i.e., one depending only on nand 8, additional restrictions are required. THEOREM 6.4. [46] Let M be a closed n-manifold of 8-pinched curvature. If M is 2-connected, then the injectivity radius of M is at least €(n, 8) > 0, a constant depending only on nand 8. Note that in the above-mentioned 7-manifolds, each second homotopy group has rank one. SKETCH OF A PROOF OF THEOREM 6.4. We argue by contradiction, assuming a sequence Mi satisfying Theorem 6.4 such that Mi dGH ) X, where dim(X) < n. From the proof of Theorem 6.1, we may assume a manifold M c::: Mi admitting a Tk-action without fixed points and a sequence of invariant metrics 9i such that 9i collapses along F, i.e., the diameters of all F converge uniformly to zero and the induced metrics di on M /Tk converge to d pointwise. As seen in the proof of Theorem 6.3, we may assume the invariant metrics are 8/2-pinched. By now, we are in a situation similar to the collapsing of Berger's sphere (where, however, the minimal curvature converges to zero). Indeed, given any sequence of metrics 9i on a manifold, A ::; secgi ::; 1, collapsing along a (fixed) F-structure, one can construct a complete noncompact length space with curvature ~A, and the non-compactness forces

20

X. RONG

>. ::; 0 (a generalized Bonnet theorem, [45]). By now, one sees a contradiction to the above. For simplicity, we will explain the idea with the special case k = 1. To get a contradiction, we take a finite open cover {UaJ for M such that each Ua. is a tube ofradius p > 0 with respect to d. Clearly, (Ua., gi) converges to a p-ball in X. For Ua. n U(3 =1= 0, let cPa.(3 : Ua. ~ Ua. n U(3 -t Ua. n U(3 C U(3 denote a gluing map. On ua.nu(3nu-y =1= 0, these maps satisfy cPa.(30cP(3-Y 0cP-ya. = id. Let U -t Ua. denote the Riemannian universal covering space. Then there is a lifting map, 4>0'.(3 : Ua. ~ Ua. n U(3 -t Ua. n U(3 C U(3. However, these lifting maps do not satisfy the compatibility condition: 4>0'.(3 ° 4>(3-y ° 4>-ya. = ~a.(3-y =1= id. We may view {~a.(3-y} as an obstruction to gluing {Ua.} together. The key observation is that when taking limits, (Ua., Xa., gi) d CH ) (Ua.,oo, xa.) and 4>0'.(3 -t 4>0'.(3,00 simultaneously, the collapsing condition implies that ~a.(3-y -t id as i -t 00. Consequently, using {4>a.(3,oo} , one can glue {Ua.,oo} together to form a complete non-compact manifold (because Ua.,oo ~ D2 X JR) with curvature 2:>' (e.g., for Berger's sphere, N = S'I x JR). 0 "2

c. Collapsed manifolds with non-positive sectional curvature. A fiat manifold (of small volume) is a trivial example of a collapsed manifold with (bounded) non-positive curvature, and a nontrivial example is that any graph 3-manifold whose fundamental group contains no cyclic subgroup of finite index admits a collapsed metric with (bounded) non-positive curvature [34J. S. Buyalo studied a collapsed 3-manifold M with -1 ::; secM ::; 0 and found that there are a finite number of totally geodesic fiat tori, eM, such that each component Uj of M - Ui T; is a metric product, Uj = ~; X Sl [5-7]. By definition, M is a graph manifold with a graph system {Tn. Note that {(Ui' T1)} actually defines an F-structure, called a Crstructure, with the additional properties that Ui is a product (in general, up to a finite covering space) and the fundamental group of an orbit injects into 11"1 (M). One may describe the local metric product structure in terms of the subgroups of 1I"1(M) generated by loops in orbits (up to a finite covering). Recall that for each non-trivial abelian subgroup A C 11"1 (M), there is an isometric immersion of a fiat torus, i : Trank(A) -t M, such that the induced maps on the fundamental groups satisfy i*1I"1 (pank(A») = A [31, 41J. More generally, the minimal set splits, Min(A) = D x JRrank(A), where Min(A) denotes the set of points in the Riemannian universal covering of M at which the displacement of any element in A achieves the minimum, and whose projection contains the immersed fiat torus. Let A = {Aa.} denote a collection of abelian subgroups Aa. C 11"1 (M) which are preserved by conjugation. We say that A determines an abelian structure (resp. a local splitting structure) on M if the following conditions

T;

COLLAPSED MANIFOLDS

21

{resp. (6.5.1)) hold: (6.1)

The Riemannian universal covering space

(6.2)

Min{Aa)

Nt =

U Min{A

Q}

AaEA

n Min{Aj3) i= 0 if and only if Aa and Aj3 commute.

If 1T1{M) has a nontrivial normal abelian subgroup A, then A = {A} determines an abelian structure for any non-positively curved metric on M [31, 41]. THEOREM 6.5. [55] Let M be a closed n-manifold with -1 :S seCM :S o. If vol{Bdx)) < t:{n) for all x E M, then there is A = {Aa} C 1T1{M) that determines an abelian structure. Moreover, any abelian structure determines a canonical Cr-structure (i.e., one whose orbits are totally geodesic fiat submanifolds).

Theorem 6.5 was essentially conjectured by Buyalo, who also verified it for n = 3,4. Indeed, one easily concludes that if a graph manifold admits a Cr-structure compatible with one metric of non-positive curvature, then it is compatible with every non-positively curved metric. Recall that geometrical rigidity results often assert that a class of certain metrics are unique up to a scaling (e.g., the higher rank rigidity, [2]). In this spirit, one may view the above as a weak rigidity: the underlying Cr-structure captures the local splitting structure of every metric of nonpositive curvature. (In this sense, all these metrics are alike.) It is conjectured that such a weak rigidity should hold in all dimensions. The following result partially supports the conjecture. THEOREM 6.6. [10] Let M be a closed manifold which admits a metric of non-positive sectional curvature. If M admits an F-structure, then every metric of non-positive sectional curvature has a splitting structure.

Here we omit the outline of proofs. A remaining problem is to show that the local splitting structure in Theorem 6.6 satisfies (6.2). References [1] S. Aloff and N.R. Wallach, An infinite family of 7-manifolds admitting positive curved Riemannian structures, Bull. Amer. Math. Soc., 81 (1975), 93-97. [2] W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Basel, Birkhiiuser, Boston-Basel-Stuttgart, 1985. [3] M. Berger, Riemannian geometry during the second half of the twentieth century, Jahrbericht. Deutsch. Math.-Verein, 100 (1998), 45-208. [4] P. Buser and H. Karcher, Gromov's almost flat manifolds, Ast'erisque, 81, Societe Mathmatique de France, Paris, 198!. [5] S. Buyalo, Collapsing manifolds of non-positive curvature, I, Leningrad Math. J., 1(5) (1990), 1135-1155. [6] S. Buyalo, Collapsing manifolds of non-positive curvature, II, Leningrad Math. J., 1(6) (1990), 1371-1399.

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[7] S. Buyalo, Homotopy invariance of some geometric properties of nonpositively curved three-manifolds, St. Petersburg Math. J., 3 (1992), 791-808. [8] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357. [9] J. Cao, J. Cheeger, and X. Rong, Splittings and Cr-structure for manifolds with nonpositive sectional curvature, Invent. Math., 144 (2001), 139-167. [10] J. Cao, J. Cheeger, and X. Rong, Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3, Comm. Anal. Geom., 12 (2004),389-415. [11] J. Cheeger, Thesis, Princeton Univ., 1969. [12] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math., 92 (1970), 61-75. [13] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, Elservie, North-Holland, 1975. [14] J. Cheeger, K. Fukaya, and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. AMS, 5 (1992), 327-372. [15] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bound, I, J. Differential Geom., 23 (1986), 309-364. [16] J. Cheeger and M. Gromov, Collapsing Riemannian manifold while keeping their curvature bounded, II, J. Differential Geom., 32 (1990), 269-298. [17] J. Cheeger and M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, H.E. Rauch Mem., Vol. I (Chavel and Farkas, Eds.), Springer, Berlin, 1985, 115-154. [18] J. Cheeger and X. Rong, Collapsed manifolds with bounded diameter and bounded covering geometry, GaFa (Geometrical and functional analysis), 5(2) (1995), 141-163. [19] J. Cheeger and X. Rong, Existence of polarized F-structure on collapsed manifolds with bounded curvature and diameter, GaFa (Geometrical and functional analysis), 6(3) (1996), 411-429. [20] F. Fang and X. Rong, Positive pinching, volume and homotopy groups, Geom. Funct. Anal., 9 (1999), 641-674. [21] F. Fang and X. Rong, Fixed point free circle actions and finiteness theorems, Comm. Contemp. Math., 2(1) (2000), 75-86. [22] F. Fang and X. Rong, Curvature, diameter, homotopy groups and cohomology rings, Duke Math. J., 107(1) (2001), 135-158. [23] F. Fang and X. Rong, The twisted second Betti number and convergence of collapsing Riemannian manifolds, Invent. Math., 150 (2002), 61-109. [24] K. Fukaya, Collapsing of Riemannian manifolds to ones of lower dimensions, J. Differential Geom., 25 (1987), 139-156. [25] K. Fukaya, A boundary of the set of Riemannian manifolds with bounded curvature and diameter, J. Differential Geom., 28 (1988), 1-2l. [26] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, II, J. Math. Japan, 41 (1989), 333-356. [27] K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Recent Topics in Differential and Analytic Geometry (T. Ochiai, ed.), Kinokuniya, Tokyo, 1990. [28] K. Fukaya, Metric Riemannian Geometry, preprint, 2006. [29] P. Ghanaat, Geometric construction of holonomy coverings for almost fiat manifolds, J. Differential Geome., 34 (1991), 571-579. [30] R.E. Green and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math., 131 (1988), 119-14l. [31] D. Gromoll and J. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Am. Math. Soc., 77(4) (1971), 545-552.

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[32] M. Gromov, Almost flat manifolds, J. Differential Geom., 13 (1978), 231-241. [33] M. Gromov, Curvature diameter and Betti numbers, Comment. Math. Helv., 56 (1981),179-195. [34] M. Gromov, Volume and bounded cohomology, IRES Pul. Math., 56 (1983), 213-307. [35] M. Gromov, Manifolds of negative curvature, J. Differential Geom., 3 (1978), 223-230. [36] M. Gromov, J. Lafontaine, and P.Pansu, Structures metriques pour les varietes riemannienes, CedicFernand, Paris, 1981. [37] K. Grove and H. Karcher, How to conjugate CI-close group actions, Math. Z., 132 (1973),11-20. [38] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math., 152 (2000), 331-367. [39] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306. [40] J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-prioriSchranken fur harmonische Abbildungen, Manu. Math., 19 (1982), 27-77. [41] B. Lawson and S.-T. Yau, On compact manifolds of nonpositive curvature, J. Differential Geom., 7 (1972). [42] S. Peters, Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math., 349 (1984), 77-82. [43] A. Petrunin and W. Tuschmann, Diffeomorphism finiteness, positive p'inching, and second homotopy, Geom. Punet. Anal., 9 (1999), 736-774. [44] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below. Geom. Funet. Anal. 8 (1998), no. 1, 123-148. [45] A. Petrunin, X. Rong, and W. Tuschmann, Collapsing vs. positive pinching, Geom. Punet. Anal., 9 (1999), 699-735. [46] X. Rong, The limiting eta invariant of collapsed 3-manifolds, J. Differential Geom., 37 (1993), 535-568. [47] X. Rong, The existence of polarized F-structures on volume collapsed 4-manifolds, Geom. Punet. Anal., 3(5) (1993), 475-502. [48] X. Rong, Bounding homotopy and homology groups by curvature and diameter, Duke. Math. J., 2 (1995), 427-435. [49] X. Rong, On the fundamental group of manifolds of positive sectional curvature, Ann. of Math., 143 (1996), 397-411. [50] X. Rong, The almost cyclicity of the fundamental groups of positively curved manifolds, Invent. Math., 126 (1996), 47-64. [51] X. Rong, Lecture notes on the convergence and collapsing theory in Riemannian geometry, to appear. [52] E. Ruh, Almost flat manifolds, J. Differential Geom., 17 (1982), 1-14. [53] T. Shioya and T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann., 333(1) (2005), 131-155. [54] W. Tuschmann, Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness, Math. Ann., 322(2) (2002), 413-420. [55] A. Weinstein, On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel), 18 (1967), 523-524. [56] Q. Cai; X. Rong, Collapsing construction with nilpotent structures Preprint. MATHEMATICS DEPARTMENT, RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903, USA E-mail address: ronghath. rutgers. edu MATHEMATICS DEPARTMENT, CAPITAL NORMAL UNIVERSITY, BEIJING, P.R.C.

Surveys in Differential Geometry XI

N onnegatively and Positively curved manifolds Burkhard Wilking

The aim of this paper is to survey some results on nonnegatively and positively curved Riemannian manifolds. One of the important features of lower curvature bounds in general is the invariance under taking Gromov Hausdorff limits. Celebrated structure and finiteness results provide a partial understanding of the phenomena that occur while taking limits. These results however are not the subject of this survey since they are treated in other surveys of this volume. In this survey we take the more classical approach and focus on "effective" results. There are relatively few general "effective" structure results in the subject. By Gromov's Betti number theorem the total Betti number of a nonnegatively curved manifold is bounded above by an explicit constant which only depends on the dimension. The Gromoll Meyer theorem says that a positively curved open manifold is diffeomorphic to the Euclidean space. In the case of nonnegatively curved open manifolds, the soul theorem of Cheeger and Gromoll and Perelman's solution of the soul conjecture clearly belong to the greatest structure results in the subject, as well. Also relatively good is the understanding of fundamental groups of nonnegatively curved manifolds. A theorem of Synge asserts that an even dimensional orient able compact manifold of positive sectional curvature is simply connected. An odd dimensional positively curved manifold is known to be orient able (Synge), and its fundamental group is finite by the classical theorem of Bonnet and Myers. The fundamental groups of nonnegatively curved manifolds are virtually abelian, as a consequence of Toponogov's splitting theorem. However, one of the "effective" conjectures in this context, the so called Chern conjecture, was refuted: Shankar [1998] constructed a positively curved manifold with a non cyclic abelian fundamental group. As we will discuss in the last section the known methods for constructing nonnegatively curved manifolds are somewhat limited. The most important tools are the O'Neill formulas which imply that the base of a Riemannian submersion has nonnegative (positive) sectional curvature if the total space has. We recall that a smooth surjective map u: M -+ B between two Riemannian manifolds is called a Riemannian submersion if the dual ©2007 International Press

25

26

B. WILKING

O"~d: Ta(p)B -t TpM of the differential of 0" is length preserving for all p EM. Apart from taking products, the only other method is a special glueing technique, which was used by Cheeger, and more recently by Grove and Ziller to construct quite a few interesting examples of nonnegatively curved manifolds. By comparing with the class of known positively curved manifolds, the nonnegatively curved manifolds form a huge class. In fact in dimensions above 24 all known simply connected compact positively curved manifolds are diffeomorphic to rank 1 symmetric spaces. Due to work of the author the situation is somewhat better in the class of known examples of manifolds with positive curvature on open dense sets, see section 4. Given the drastic difference in the number of known examples, it is somewhat painful that the only known obstructions on positively curved compact manifolds, which do not remain valid for the nonnegatively curved manifolds, are the above quoted results of Synge and Bonnet Myers on the fundamental groups. Since the list of general structure results is not far from being complete by now, the reader might ask why a survey on such a subject is necessary. The reason is that there are a lot of other beautiful theorems in the subject including structure results, but they usually need additional assumptions. We have subdivided the paper in five sections. Section 1 is on sphere theorems and related rigidity results some notes on very recent significant developments were added in proof and can be found in section 6. In section 2, we survey results on compact nonnegatively curved manifolds, and in section 3, results on open nonnegatively curved manifolds. Then follows a section on compact positively curved manifolds with symmetry, since this was a particularly active area in recent years. Although we pose problems and conjectures throughout the paper we close the paper with a section on open problems. We do not have the ambition to be complete or to sketch all the significant historical developments that eventually led to the stated results. Instead we will usually only quote a few things according to personal taste.

1. Sphere theorems and related rigidity results

A lot of techniques in the subject were developed or used in connection with proving sphere theorems. In this section we survey some of these results. We recall Toponogov's triangle comparison theorem. Let M be a complete manifold with sectional curvature K 2: K, and consider a geodesic triangle Ll in M consisting of minimal geodesics with length a, b, C E R Then there exists a triangle in the 2-dimensional complete surface M~ of constant curvature K, with side length a, b, c and the angles in the comparison triangle bound the corresponding angles in Ll from below. 1.1. Topological sphere theorems. We start with the classical sphere theorem of Berger and Klingenberg.

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

27

THEOREM 1.1 (Quarter pinched sphere theorem). Let M be a complete simply connected manifold with sectional curvature 1/4 < K ~ 1. Then M is homeomorphic to the sphere. The proof has two parts. The first part is to show that the injectivity radius of M is at least 7r /2. This is elementary in even dimensions. In fact by Synge's Theorem any even dimensional oriented manifold with curvature o < K ~ 1 has injectivity radius ~7r. In odd dimensions the result is due to Klingenberg and relies on a more delicate Morse theory argument on the loop space. The second part of the proof is due to Berger. He showed that any manifold with injectivity radius ~7r /2 and curvature > 1 is homeomorphic to a sphere. In fact by applying Toponogov's theorem to two points of maximal distance, he showed that the manifold can be covered by two balls, which are via the exponential map diffeomorphic to balls in the Euclidean space. Grove and Shiohama [1977] gave a significant improvement of Berger's theorem, by replacing the lower injectivity radius bound by a lower diameter bound. THEOREM 1.2 (Diameter sphere theorem). Any manifold with sectional curvature ~ 1 and diameter >7r /2 is homeomorphic to a sphere. More important than the theorem was the fact the proof introduced a new concept: critical points of distance functions. A point q is critical with respect to the distance function d(p, .) if the set of initial vectors of minimal geodesics from q to p intersect each closed half space of TqM. If the point q is not critical it is not hard to see that there is a gradient like vectorfield X in a neighborhood of q. A vectorfield is said to be gradient like iffor each integral curve c of X the map t M d(p, c( t)) is a monotonously increasing bilipschitz map onto its image. An elementary yet important observation is that local gradient like vectorfields can be glued together using a partition of unity. PROOF OF THE DIAMETER SPHERE THEOREM. We may scale the manifold such that its diameter is 7r /2 and the curvature is strictly > 1. Choose two points p, q of maximal distance 7r /2, and let z be an arbitrary third point. Consider the spherical comparison triangle (p, ij, z). We do know that the side length of (p,z) and (ij,z) are less or equal to 7r/2 whereas d§2(p,ij) = 7r/2. This implies that the angle of the triangle at is ~7r /2. By Toponogov's theorem any minimal geodesic triangle with corners p, q, z in M has an angle strictly larger than 7r /2 based at z. This in turn implies that the distance function d(p, .) has no critical points in M \ {p, q}. Thus there is a gradient like vectorfield X on M \ {p, q}. Furthermore without loss of generality X is given on Br(P) \ {p} by the actual gradient of the distance function d(p, .), where r is smaller than the injectivity radius. We may also assume IIX(z)1I ~ d(q, z)2 for all z E M \ {p, q}. Then the flow q, of X exists for all

z

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B. WILKING

future times and we can define a diffeomorphism

as follows: for a unit vector v E TpM and a nonnegative number t put 1/J(t . v) = exp(tv) if t E [0, r] and 1/J(t . v) = cI>t-r(exp(rv)) if t ~ r. Clearly this implies that M is homeomorphic to a sphere. 0 There is another generalization of the sphere theorem of Berger and Klingenberg. A manifold is said to have positive isotropic curvature if for all orthonormal vectors e1, e2, e3, e4 E TpM the curvature operator satisfies

By estimating the indices of minimal 2 spheres in a manifold of positive isotropic curvature, Micallef and Moore [1988] were able to show that THEOREM 1.3. A simply connected compact Riemannian manifold of positive isotropic curvature is a homotopy sphere.

A simple computation shows that pointwise strictly quarter pinched manifolds have positive isotropic curvature. Thus the theorem of Micallef and Moore is a generalization of the quarter pinched sphere theorem. A more direct improvement of the quarter pinched sphere theorem is due to Abresch and Meyer [1996]. THEOREM 1.4. Let M be a compact simply connected manifold with sectional curvature 4(1+110 6)2 ~ K ~ 1. Then one of the following holds

• M is homeomorphic to a sphere . • n is even and the cohomology ring H*(M, '1.2) is generated by one element. It is a well known result in topology that the '1.2 cohomology rings of spaces which are generated by one element are precisely given by the '1.2-cohomology rings of rank 1 symmetric spaces lI~.lpm, Clpm , lHIpn, Cap2 and §n, cf. [Zhizhou, 2002]. The proof of Theorem 1.4 has again two parts. Abresch and Meyer first establish that the injectivity radius of l\tf is bounded below by the conjugate radius which in turn is bounded below by 7r. From the diameter sphere theorem it is clear that without loss of generality diam(M, g) ~ 7l'(1 + 10- 6 ). They then establish the horse shoe inequality, which was conjectured by Berger: for p E M and any unit vector v E TpM one has

d(exp(7l'v), exp( -7l'v))

< 7l'.

In particular exp(7l'v) and exp( -7l'v) can be joined by a unique minimal geodesic. Once the horse shoe inequality is established it is easy to see that there is a smooth map f: lRpn -+ M n such that in odd dimensions the

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29

integral degree is 1 and in even dimensions the Z2-degree is 1. The theorem then follows by a straightforward cohomology computation. The horse shoe inequality relies on a mixed Jacobi field estimate. We only state the problem here in a very rough form. Let c be a normal geodesic in M and J a Jacobifield with J(O) = O. Suppose that at time to = 2; the value IIJ(to)11 is quite a bit smaller than one would expect by Rauch's comparison from the lower curvature bound. Can one say that IIJ(t)1I is also quite a bit smaller than in Rauch's comparison for t ~ [to, 7l"]? Abresch and Meyer gave an affirmative answer. If one wants to improve the pinching constant one certainly needs to improve their Jacobifield estimate. 1.2. Differentiable sphere theorems. It is not known whether there are exotic spheres with positive sectional curvature. A closely related question is whether one can improve in any (or all) of the above mentioned topological sphere theorems the conclusion from homeomorphic to being diffeomorphic to a sphere. In other words, can one turn the topological sphere theorems into differentiable sphere theorems. In each case this is an open question. However, there are quite a few differentiable sphere theorems, which hold under stronger assumptions. The first differentiable sphere theorem was established in his thesis by Gromoll. He had a pinching condition 7r /2+c: for some c: > 0. We claim that we can find a continuous family of metrics with go = 9 and K t 2: 1 such that (M, gt) converges for t -+ 1 to the suspension of 8B7r / 2 (p). We consider the suspension X of M, i.e., X = [-7r/2,7r/2] x M/ rv where the equivalence classes of,...., are given by p+ := {7r/2} X M, p_ := {-7r /2} x M and the one point sets {(t,p)} for It I i= 7r /2. Recall that X endowed with the usual warped product metric is an Alexandrov space with curvature 2:1. We consider the curve c(t) = ((1 - t)7r/2,p) as a curve in X, r(t) := 7r/2 + c:(1 - t) and the ball Br(t)(c(t)) c X. Put Nt := 8Br(t)(c(t)). Since X\Br(t) (c(t)) is strictly convex and Nt is contained in the Riemannian manifold X \ {P±} for all t i= 1, it follows that Nt is an Alexandrov space with curvature 2:1 for all t E [0,1]. Clearly No is up to a small scaling factor isometric to M. Moreover N1 is isometric to the suspension of 8B7r/2(P) eM. Using that Nt is strictly convex in the Riemannian manifold X \ {P±} for t E [0, 1), it follows that the family Nt can be approximated by a family of strictly convex smooth submanifolds Nt C X\ {P±}, t E [0,1). Furthermore, one may assume that limt~l Nt = N1 = limt~l Nt. We found a family of metrics gt of curvature >1 such that (M,gt) converges to the suspension of 8B7r / 2(p). We may assume that 8B7r / 2(P) has weak k-th packing radius >7r /2 and k 2: 2. We now choose a curve of points qt E M converging for t -+ 1 to a point on the equator q1 E 8B7r/2(P) of the limit space such that there is a point q2 in 8B7r/2(P) whose intrinsic distance to q1 is >7r/2. We now repeat the above construction for all t E (0,1) with (M,g,p) replaced by (M, gt, qt). This way we get for each t an one parameter family of smooth metrics g(t, s) with K 2: 1 which converges for s -+ 1 to the suspension of the boundary of B 7r / 2(qt) C (M,gt). It is then easy to see that one can choose the metrics such that they depend smoothly on sand t. Moreover, after a possible reparameterization of g(s, t) the one parameter family t t-+ g(t, t) converges to the double suspension of the boundary of B7r / 2(q1) C 8B7r / 2 (p). Clearly the theorem follows by iterating this process. 0 SKETCH OF THE PROOF OF THEOREM

We recall that to each Riemannian manifold (M, g) and each point p E M one can assign a curvature operator R: A2TpM -+ A2TpM. We call the operator 2-positive if the sum of the smallest two eigenvalues is positive. It is known that manifolds with 2-positive curvature operator have positive isotropic curvature. THEOREM 1.10. Let (M, g) be a compact manifold with 2-positive curvature operator. Then the normalized Ricci flow evolves 9 to a limit metric of constant sectional curvature.

In dimension 3 the theorem is due to Hamilton [1982]. Hamilton [1986] also showed that the theorem holds for 4-manifolds with positive curvature

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operator. This was extended by Chen to 4-manifolds with 2-positive curvature operator. In dimension 2 it was shown by Hamilton and Chow that for any surface the normalized Ricci flow converges to limit metric of constant curvature. In dimensions above 4 the theorem is due to B6hm and Wilking [2006]. For n 2: 3 the proof solely relies on the maximum principle and works more generally in the category of orbifolds. We recall that a family of metrics gt on M is said to be a solution of the Ricci flow if ttgt = -2 Ric t Hamilton showed that if one represents the curvature operator R with respect to suitable moving orthonormal frames, then

ttR = flR + 2(R2 + R#) where R# = ad oR /\ R 0 ad*, ad: A2.50(TpM) -+ .5o(TpM) is the adjoint representation and where we have identified A2TpM with the Lie algebra .5o(TpM). Hamilton's maximum principle allows to deduce certain dynamical properties of the PDE from dynamical properties of the ODE

ftR=R2+R#. SKETCH OF THE PROOF OF THEOREM 1.10. We let S~(.5o(n)) denote the vectorspace of algebraic curvature operators satisfying the Bianchi identity. We call a continuous family C(S)SE[O,l) C S~(.5o(n)) of closed convex O(n)-invariant cones of full dimension a pinching family, if (1) each R E C(s) \ {O} has positive scalar curvature, (2) R 2 + R # is contained in the interior of the tangent cone of C (s) at R for all R E C(s) \ {O} and all s E (0,1), (3) C(s) converges in the pointed Hausdorff topology to the onedimensional cone lR +I as s -+ 1. The argument in [B6hm and Wilking, 2006] has two parts. One part is a general argument showing for any pinching family C(s) (s E [0,1)) that on any compact manifold (M,g) for which the curvature operator is contained in the interior of C(O) at every point the normalized Ricci flow evolves 9 to a constant curvature limit metric. In the proof of this result one first constructs to such a pinching family a pinching set in the sense Hamilton which in turn gives the convergence result. The harder problem is actually to construct a pinching family with C(O) being the cone of 2-nonnegative curvature operators. Here a new tool is established. It is a formula that describes how this ordinary differential equation R' = R2 + R# changes under O(n)-equivariant linear transformations. By chance the transformation law is a lot simpler than for a generic O(n)-invariant quadratic expression. The transformation law often allows to construct new ODE-invariant curvature cones as the image of a given

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33

invariant curvature cone under suitable equivariant linear transformation l: S~(so(n)) ~ S~(so(n)). This in turn is used to establish the existence of a pinching family. 0 1.3. Related rigidity results. We first mention the diameter rigidity theorem of Gromoll and Grove [1987J THEOREM 1.11 (Diameter rigidity). Let (M, g) be a compact manifold with sectional curvature K ~ 1 and diameter ~7r /2. Then one of the following holds: a) M is homeomorphic to a sphere. b) M is locally isometric to a rank one symmetric space.

The original theorem allowed a potential exceptional case • M has the cohomology ring of the Cayley plane, but is not isometric to the Cayley plane. This case was ruled out much later by the author, see [Wilking, 2001J. The proof of the diameter rigidity theorem is closely linked to the rigidity of Hopffibrations which was established by Gromoll and Grove [1988J as well THEOREM 1.12 (Rigidity of Hopf fibrations). Let u: §n ~ B be a Riemannian submersion with connected fibers. Then u is metrically congruent to a Hopf fibration. In particular the fibers are totally geodesic and B is rank one symmetric space. Similarly to the previous theorem, the original theorem allowed for a possible exception, Grove and Gromoll assumed in addition (n, dim B) =1= (15,8). Using very different methods, the rigidity of this special case was proved by the author in [Wilking, 2001 J. This in turn ruled out the exceptional case in the diameter rigidity theorem as well. SKETCH OF THE PROOF OF THE DIAMETER RIGIDITY THEOREM. The proof of the diameter rigidity theorem is the most beautiful rigidity argument in positive curvature. One assumes that the manifold is not homeomorphic to a sphere. Let p be a point with N2 := 8B7r/2(P) =1= 0. One defines Nl = 8B7r / 2(N2) as the boundary of the distance tube B 7r / 2(N2) around N2. It then requires some work to see that Nl and N2 are totally geodesic submanifolds without boundary satisfying N2 = 8B7r / 2(N1 ). Not both manifolds can be points, since otherwise one can show that M is homeomorphic to a sphere. If one endows the unit normal bundle 1/ 1 (Ni) with its natural connection metric, then Grove and Gromoll show in a next step that the map Ui: 1/ 1 (Nd ~ N j , V H exp( 7r /2v) is a Riemannian submersion, {i,j} = {1,2}. Furthermore Ui restricts to a Riemannian submersion 1/~(Ni) ~ N j for all q E N i .

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In the simply connected case one shows that Ni is simply connected as well, i = 1,2. By the rigidity of submersions defined on Euclidean spheres (Theorem 1.12) we deduce that Ni is either a point or a rank one symmetric space with diameter 7r/2. Going back to the definition of Nl, it is then easy to see that Nl = {pl. Using that 0"1: §n-l -t N2 is submersion with totally geodesic fibers, one can show that the pull back metric exp; 9 on B 7r / 2 (0) C TpM is determined by 0"1. Thus M is isometric to a rank one symmetric space. In the non simply connected case one can show that either the universal cover is not a sphere and thereby symmetric or dim(Nl) + dim(N2) = n - 1. In the latter case it is not hard to verify that M has constant curvature ~. 0 Since the proof of the differentiable sphere theorem for manifolds with 2-positive curvature follows from a Ricci flow argument it is of course not surprising that it has a rigidity version as well. THEOREM 1.13. A simply connected compact manifold with 2-nonnegative curvature operator satisfies one of the following statements. • The normalized Ricci flow evolves the metric to a limit metric which is up to scaling is isometric to §n or ClPn / 2 . • M is isometric to an irreducible symmetric space. • M is isometric to nontrivial Riemannian product.

Of course in the last case the factors of M have nonnegative curvature operators. By Theorem 2.2 (M,9) admits a possibly different metric 91 such that (M,9I) is locally isometric to (M,9) and (M,91) is finitely covered by a Riemannian product Td x M' where M' is simply connected and compact. This effectively gives a reduction to the simply connected case. The theorem has many names attached to it. Of course Theorem 1.10 (Hamilton [1982,1986]' B6hm and Wilking [2006]) enters as the 'generic' case. This in turn was used by Ni and Wu [2006] to reduce the problem from 2-nonnegative curvature operators to nonnegative curvature operators. One has to mention Gallot and Meyer's [1975] investigation of manifolds with nonnegative curvature operator using the Bochner technique. Berger's classification of holonomy groups, as well as Mori's [1979], Siu and Yau's [1980] solution of the Frankel conjecture are key tools. Based on this Chen and Tian [2006] proved convergence of the Ricci flow for compact Kahler manifolds with positive bisectional curvature. SKETCH OF A PROOF OF THEOREM 1.13. Consider first the case that the curvature operator of M is not nonnegative. We claim that then the Ricci flow immediately evolves 9 to a metric with 2-positive curvature operator. We consider a short time solution 9(t) of the Ricci flow and let f: [0, IS) X M -t JR, denote the function which assigns to (t, p) the sum of the lowest two eigenvalues of the curvature operator of (A1, 9( t)) at p. We first want to

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

35

show that f(t,·) is positive somewhere for small t > O. We may assume that f(O,p) = 0 for all p. It is straightforward to check that f satisfies

From the invariance of 2-nonnegative curvature operators it is known that q(R) ~ O. In fact a detailed analysis of the proof shows that q(R) ~ 2(Al(R))2. In the present situation we deduce by a first order argument that f(t,p) becomes positive somewhere for small t > O. Now it is not hard to establish a strong maximum principle that shows that f(t,·) is everywhere positive for small t > 0, see Ni and Wu [2006]. In other words (M, gt) has 2-positive curvature operator for t > 0 and the result follows from Theorem 1.10. We are left with the case that the curvature operator of (M, g) is nonnegative. Essentially this case was already treated by Gallot and Meyer using the Bochner technique, see [Petersen, 2006]. We present a slightly different argument following Chow and Yang (1989). Using Hamilton's [1986] strong maximum principle one deduces that for t > 0 the curvature operator of (M, gt) has constant rank and that the kernel is parallel. Thus either Rt is positive or the holonomy is non generic. We may assume that M does not split as a product. Hence without loss of generality M is irreducible with non generic holonomy. Since (M, gt) clearly has positive scalar curvature Berger's classification of holonomy groups implies that Hol(M) ~ U(n/2), Sp(1)Sp(n/4) unless (M, g) is a symmetric space. In the case of Hol(M) ~ Sp(1)Sp(n) we can employ another theorem of Berger [1966] to see that M is up to scaling isometric to lHIlpm/4, since in our case the sectional curvature of (M, gt) is positive. In the remaining case Hol(M) = U(n/2) it follows that M is Kahler and (M, gt) has positive (bi-)sectional curvature. By Mori [1979] and Siu and Yau's [1980] solution of the Frankel conjecture M is biholomorphic to cpn/2. In particular, M admits a Kahler Einstein metric. Due to work of Chen and Tian [2006] it follows, that the normalized Ricci flow on M 0 converges to the Fubini study metric which completes the proof.

2. Compact nonnegatively curved manifolds The most fundamental obstruction to this date is Gromov's Betti number theorem. THEOREM 2.1 (Gromov, 1981). Let M n be an n-dimensional complete manifold with nonnegative sectional curvature, and let IF be a field. Then the total Betti number satisfies n

b(M,IF):= Lbi(M,IF) < 1010n4 . i=O

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Gromov's original bound on the total Betti number was depending double exponentially on the dimension. The improvement is due to Abresch [1987]. However, this bound is not optimal either. In fact Gromov posed the problem whether the best possible bound is 2n , the total Betti number of the n-dimensional torus. The statement is particularly striking since the nonnegatively curved manifolds in a fixed dimension ?:.7 have infinitely many homology types with respect to integer coefficients. More generally Gromov gave explicit estimates for the total Betti numbers of compact n-manifolds with curvature ?:.-1 and diameter "5:D. The proof is an ingenious combination of Toponogov's theorem and critical point theory. SKETCH OF THE PROOF OF THEOREM 2.1. The most surprising part in the proof is a definition: Gromov assigns to every ball Br(P) C M a finite number called the corank of the ball. It is defined as the maximum over all k such that for all q E B2r(P) there are points ql, . .. , qk with

d(q, ql) ?:. 2n+3 r, d(q, qi+l) ?:. 2nd(p, qi) and qi is a critical point of the distance function of q in the sense of Grove and Shiohama. One can show as follows that the corank of a ball is at most 2n: Choose a minimal geodesic Cij from qi to qj, i < j and minimal geodesic Ci from q to qi, i = 1, ... , k. Since qi is a critical point we can find a possibly different minimal geodesic Ci from q to qi such that the angle of the triangle (Ci' Cj, Cij) based at qi is "5:7r /2. Therefore L(Cj)2 "5: L(Cij)2 + L(cd 2. Applying Toponogov's theorem to the triangle (Ci, Cj, Cij) gives that the angle 'Pij between Ci and Cj satisfies tan( 'Pij) ?:. 2n. Thus 'Pij ?:. 7r /2 - 2- n . The upper bound on k now follows from an Euclidean sphere packing argument in TqM. By reverse induction on the corank, one establishes an estimate for the content of a ball cont(Br(p)) which is defined as the dimension of the image of H*(Br(p)) in H*(B5r(P)). A ball Br(P) with maximal corank is necessarily contractible in B5r(P) since for some q E B2r(P) the distance function of q has no critical points in Bsr(q) \ {q}. This establishes the induction base. It is immediate from the definition that corank(Bp(q)) ?:. corank(Br(p)) for all q E B 3r / 2 (P) and all p "5: r/4. In the induction step one distinguishes between two cases. In the first case, one assumes that corank(Bp(q)) > corank(Br(p)) for all q E Br(P) and p := In. Using the Bishop Gromov inequality it is easy to find a covering of Br(q) with at most 4n(n+2) balls of radius p. By the induction hypothesis the balls Bp(q) have a bounded content. Using a rather involved nested covering argument one can give an explicit estimate of the content of Br (p). In the remaining case there is one point q E Br(P) such that corank(Bp(q)) = corank(Br(p)) with p = Thus for some point x E B2p(q) there is no critical point of the distance function of x in Bsr (x) \ B 2-n+3 r (x). This implies that one can homotop Br(P) to a subset of B r/ 4 (x) in B5r(P)'

In.

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37

From this it is not hard to deduce that cont(Br/ 4 {x)) 2:: cont{Br{p)). We have seen above corank(Br/ 4 {x)) 2:: corank{Br{p)). One can now apply the same argument again with Br{p) replaced by B r/ 4 {x). Since small balls are contractible, the process has to stop after finitely many steps unless possibly cont{Br{p)) = 1. 0

Fundamental groups. Fundamental groups of nonnegatively curved manifolds are rather well understood. On the other hand, the known results are essentially the same as for compact manifolds with nonnegative Ricci curvature. In fact there is a general belief that the general structure results for fundamental groups should coincide for the two classes. One of the main tools in this context is the splitting theorem of Toponogov, resp. the splitting theorem of Cheeger and Gromoll [1971]. Recall that a line is a normal geodesic c: IR -+ (M, g) satisfying d(c{t), c(s)) = It - sl for all t, s E R By Cheeger and Gromoll's splitting theorem complete manifolds of nonnegative Ricci curvature split as products IR x M' provided they contain lines. In the special case of nonnegative sectional curvature, the result is due to Toponogov. By the work of Cheeger and Gromoll [1971], the splitting theorem implies that a nonnegatively curved manifold M is isometric to IRk x B where B has a compact isometry group. The same results holds for the universal cover of a compact manifold M of nonnegative Ricci curvature. As a consequence they deduced that the fundamental group of M is virtually abelian, Le., it contains an abelian subgroup of finite index. Moreover one can show THEOREM 2.2. Let (M, g) be a compact manifold of nonnegative Ricci curvature or an open manifold of nonnegative sectional curvature. Then there is a family of complete metrics gt on M with go = g, {M, gt} is locally isometric to (M,g) for all t and (M,gl) is finitely covered by a Riemannian product Td x M', where M' is simply connected and Td is a fiat torus. The theorem is due to author [2000] but is based on a slightly weaker version of Cheeger and Gromoll [1971]. Moreover, it was shown in [Wilking, 2000] that any finitely generated virtually abelian fundamental group occurs in some dimension as the fundamental group of a nonnegatively curved manifold. However, the more interesting and challenging problem is what one can say about fundamental groups in a fixed dimension. To the best of the authors knowledge the only other "effective" result known for fundamental groups of nonnegative sectional curvature is THEOREM 2.3 (Gromov, 1978). The fundamental group of a nonnegatively curved n-manifold is generated by at most n . 2n elements. The proof of the theorem is a simple application of Toponogov's theorem applied to the short generating system of 7r1 (M, p).

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Although we mentioned in the introduction that we will report on results which are based on collapsing techniques, we quote, for the sake of completeness, the following recent theorem of Kapovitch, Petrunin and Thschmann [2005]. THEOREM 2.4. For each n there is a constant C(n) such that the fundamental group of any compact nonnegatively curved n-manifold (M, g) contains a nilpotent subgroup of index at most C(n).

The theorem remains valid for almost nonnegatively curved manifolds and it improves a similar theorem of Fukaya and Yamaguchi from "solvable" to "nilpotent". The proof relies on a compactness result and it remains an open problem whether one can make the bound effective, in other words whether one can give explicit estimates on C(n). It is also remains open whether in case of nonnegative curvature one can improve it from "nilpotent" to "abelian". Other structure results. By the Gauss-Bonnet formula a compact nonnegatively curved compact surface is given by lR1P'2, §2, T2 or the Klein bottle. Due to Hamilton [1982] a compact 3-manifold of nonnegative Ricci curvature and finite fundamental group is diffeomorphic to spherical space form, see Theorem 1.13. In dimension 4 a classification remains open. The best result is a theorem in Kleiner's thesis. THEOREM 2.5 (Kleiner). Let (M,g) be a nonnegatively curved simply connected 4-manifold. If the isometry group is not finite then M is homeomorphic to §4, C1P'2, §2 X §2 or to a connected sum C1P'2# ± C1P'2.

The Bott conjecture (see last section) would imply that the theorem remains valid if one removes the assumption on the isometry group. It would be interesting to know whether one can improve the conclusion in Theorem 2.5 from homeomorphic to diffeomorphic. Kleiner never published his thesis, but Searle and Yang [1994] reproved his result. We present a slightly shorter proof which has also the advantage that it does not make use of a signature formula of Bott for four manifolds with Killing fields. This in turn implies that part of the proof carries over to simply connected nonnegatively curved 5-manifolds with an isometric 2-torus action. In fact using minor modifications it is not hard to check that the second rational Betti number of such a manifold is bounded above by 1. LEMMA 2.6. Let p: 51 -+ 0(4) be a representation such that there is no trivial subrepresentation. Consider the induced action of 51 on the standard sphere §3.

a) Any four pairwise different points PI, ... ,P4 E B

L l~i<j9

d(Pi,pj) ~ 27r.

:= §3/5 1

satisfy

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and equality occurs if and only if B is isometric to the 2-sphere §2(1/2) of constant curvature 4 and {Pl,P2,P3,pd = {±p, ±q}. b) The diameter of B is equal to 7r /2. In fact for P E B there is most one point q E B with d(p, q) ~ 7r/2.

We may assume that p is faithful. If the action of 51 is given by the Hopf action, then B := §3/5 1 is the 2-sphere §2(1/2) of constant curvature 4. Recall that a triangle in §2(1/2) has perimeter ::;7r and that equality can only occur if two of the points on the boundary triangle have distance 7r/2. Using this for all triangles {ql,q2,q3} c {Pl,P2,P3,P4}, we get the claimed inequality. Equality can only occur if the four points are on a great circle. A more detailed analysis shows that equality implies {Pl,P2,P3,P4} = {±p, ±q}. In general it is easy to construct a distance non-increasing homeomorphism PROOF.

For the proof notice that B admits an isometric action of a circle Tl, since the centralizer of p( 5 1) in SO (4) acts isometrically on B, the quotient space B /Tl is isometric to the interval [0, 7r /2]. The same holds for the quotient space §2(1/2)/Tl. It is now easy to see that the orbits of the Tl action on §2(1/2) are at least as long as the corresponding orbits in B. Finally if the action is not given by the Hopf action, then we can not find four different points Pl,P2,P3,P4 E B with d(Pl,P2) = d(P3,P4) = 7r/2. 0 Since f is distance non-increasing part b) follows as well. PROOF OF THEOREM 2.5. By Freedman's classification of simply connected topological 4-manifolds, it suffices to show that X(M) ::; 4. Since the Eulercharacteristic of M equals the Eulercharacteristic of the fixed point set of 51 C Iso(M,g), it suffices to estimate the latter. We now consider the orbit space A3 := M4/5 1 as an Alexandrov space. We first want to rule out that 51 has more than four isolated fixed points. Suppose PI, ... , P5 are pairwise different isolated fixed points in AI. We can view these points also as points in the orbit space A. Choose a fixed minimal normal geodesic 'Yij: [0, 1] -+ A between Pi and Pj for i i= j. We may assume 'Yij and 'Yji are equal up to a change of direction. We also consider all angles O'.ijk between 'Yij and 'Yik for all pairwise different i, j and k. A simple counting argument shows that there are precisely 30 angles. We next prove two different estimates for the sum of these angles. For any three points in {Pl,P2,P3,P4,P5} we get a triangle. The sum of the angles in the triangle is ~7r, as X is nonnegatively curved in the Alexandrov sense. Therefore the sum of all 30 angles is ~ 107r. On the other hand we can consider for a fixed point Pi all 6 angles based at Pi. The angles are given as the pairwise distances of four distinct points

B. WILKING

40

in the space of directions ~Pi X. Since ~Pi X is isometric to the quotient of §3 by a 5 1-action, we infer from Lemma 2.6 that the sum of these 6 angles is ~27r. This proves that the sum of all 30 angles is at most 107r. Hence equality must hold everywhere. It follows that the space of directions at Pi is given by a sphere of constant curvature 4. There are precisely 10 angles of size 7r /2 and for each triangle corresponding to three points in {PI, ... , P5} the sum of the angles is 7r and hence precisely one angle in such a triangle equals 7r/2. We may assume d(Pl,P2) = mini#j d(Pi,Pj). For one point q E {P3,P4,P5} the triangle (Pl,P2,q) has neither an angle 7r/2 at PI nor an angle 7r /2 at P2. Thus there is an angle 7r /2 at q. Since equality holds in Toponogov's comparison theorem we see

a contradiction since d(Pl, P2) was minimal. Suppose next that the fixed point set Fix(5 1) of 51 contains at least two 2-dimensional components. These components form totally geodesic submanifolds of the Alexandrov space A. Since they do not intersect it is easy to see that A is isometric to F x [0, lJ where F is a fixed point component. In particular 51 has no fixed points outside the two components. Since each component has Eulercharacteristic ~2 the result follows. It remains to consider the case that 51 has precisely one 2-dimensional fixed point component F. We have to show that the 5 1-action has at most two isolated fixed points. Notice that F is the boundary of the Alexandrov space A and the distance function h := d(F,·): A -+ ~ is concave. Let pEA denote one isolated fixed point with minimal distance r to the boundis convex. Let v E ~pA be the initial direction of ary. The set h- l ([r, a minimal geodesic from P to F. The tangent cone Cp h- l ([r, consist of 'vectors' which have an angle ~7r /2 with v. From Lemma 2.6 we deduce that Cp h- l ([r, is at most one dimensional. Thus the convex set h- l ([r, is either a point or an interval. By construction h- l ([r, contains all isolated fixed points of 51. Since for each fixed point the space of direction has diameter 7r /2, we deduce that there are at most two isolated fixed points. D

ooD

ooD

ooD

ooD

ooD

Gursky and LeBrun [1999J obtained strong restrictions on 4-dimensional nonnegatively curved Einstein manifolds. One might ask whether any nonnegatively curved compact manifold with finite fundamental group also admits nonnegatively curved metrics with positive Ricci curvature. A partial result in direction was proved recently. THEOREM 2.7 (Bohm and Wilking, 2005). Let (M, g) be a compact nonnegatively curved manifold with finite fundamental group, and let gt be a solution of the Ricci flow. Then for all small t > 0, gt has positive Ricci curvature.

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

41

The proof applies a dynamical version of Hamilton's maximum principle to a family of curvature conditions lying in between nonnegative sectional curvature and nonnegative Ricci curvature. It then follows that 9t has nonnegative Ricci curvature for t E [0, eJ with e depending on an upper curvature bound. Then the theorem follows easily from a strong maximum principle. In the same paper it was also shown that there is no Ricci flow invariant curvature condition in between nonnegative sectional curvature and nonnegative Ricci curvature in dimensions above 11. This in turn generalized previous results saying that neither nonnegative Ricci curvature nor nonnegative sectional curvature are invariant under the Ricci flow in dimensions above 3, see [Ni, 2004J. In particular, any compact nonnegatively curved manifold with finite fundamental group satisfies all obstructions coming from positive Ricci curvature. In the simply connected case the only general known obstruction for positive Ricci curvature is that the manifold admits a metric with positive scalar curvature. By the work of Gromov and Lawson and Stolz the latter statement is equivalent to saying: Either M is not spin or M is a spin manifold with a vanishing a-invariant. For more details and references we refer the reader to the surveys of Jonathan Rosenberg and Guofang Wei published in this volume. Grove-Ziller examples. Recently Grove and Ziller generalized a gluing technique which by the work of Cheeger [1973J was previously only known to work in the special case of connected sums of two rank one symmetric spaces. Since they are discussed in more detail in the survey of Wolfgang Ziller we will be brief. THEOREM 2.8 (Grove and Ziller, 2000). Let G be a compact Lie group, and let G//H be a compact biquotient. Suppose there are two subgroups K± c G x G such that K±/H ~ §1 and the action of K± on G is free. Then the manifold obtained by gluing the two disc bundles associated to the two sphere bundles G//H -+ G//K± along their common houndary G//H has a metric of nonnegative sectional curvature. The stated theorem is slightly more general than the original version of Grove and Ziller, who considered cohomogeneity one manifolds or equivalently the case where all groups H, K± act from the right on G and hence the corresponding quotients are homogeneous. Of course it would be interesting to know whether the generalization gives rise to any interesting new examples. One can actually reduce the more general statement to the one of Grove and Ziller as follows PROOF. We consider the manifold M which admits a cohomogeneity one action of G x G with principal isotropy group H and singular isotropy groups K± c G x G. By Grove and Ziller this manifold has an invariant metric of nonnegative sectional curvature, see the survey of Wolfgang Ziller for details. By assumption the diagonal ~G c G x G acts freely on M. Clearly

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the manifold in the theorem is the quotient M / ~G. Thus the result follows from the O'Neill formulas. 0 THEOREM 2.9 (Grove and Ziller). Any principal SO(n)-bundle over §4 admits a cohomogeneity one action of S3 x SO(n) with singular orbits of codimension 2. The proof uses the classification of bundles over §4 in terms of characteristic classes. Grove and Ziller endow §4 with the unique cohomogeneity one action of S3 with singular orbits of co dimension 2. Then they compute for all S3 x SO(n)-cohomogeneity one manifolds which are SO(n)-principal bundles over the given cohomogeneity one manifold §4 all characteristic classes. By comparing the set of invariants, it follows that one gets all bundles this way. The details are involved. By taking quotients of such principal bundles it follows that any sphere bundle over §4 admits a metric of nonnegative sectional curvature. This is particular striking since 10 of the 14 exotic spheres in dimension 7 can be realized as such bundles. Grove and Ziller conjectured in their paper that any cohomogeneity one manifold admits an invariant nonnegatively curved metric. A partial answer was given by Schwachhofer and Tuschmann [2004] who showed that these manifolds admit metrics of almost nonnegative sectional curvature. However, counterexamples to the Grove-Ziller conjecture were recently found by Grove, Verdiani, Wilking and Ziller [2006]. The counterexamples contain all higher dimensional Kervaire spheres and therefore all exotic spheres of cohomogeneity one. Additional counterexamples are given but to this day it remains an open question how big the class of nonnegatively curved cohomogeneity one manifolds is. 3. Open nonnegatively curved manifolds Noncompact non negatively curved spaces often occur as blow up limits of sequences of manifolds converging with lower curvature bound -1 to a limit. Also recall a result of Hamilton and Ivey saying that for any singularity of the Ricci flow in dimension 3 the corresponding blow up limit has nonnegative sectional curvature. This in turn was one key feature which allowed Hamilton and Perelman to classify the possible singularities of the Ricci flow in dimension 3. By a result of Gromov [1986] any noncompact manifold admits a positively curved metric. However Gromov's metrics are not complete and we assume throughout the paper that all metrics are complete. The structure of open manifolds of nonnegative (positive) sectional curvature is better understood than the compact case. By a theorem of Gromoll and Meyer [1969] a positively curved open manifold is diffeomorphic to the Euclidean space. For a nonnegatively curved manifold there is the soul theorem.

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43

3.1 (Cheeger and Gromoll, 1971). For an open nonnegatively curved manifold M there is a totally geodesic submanifold ~ called the soul such that M is diffeomorphic to the normal bundle of~. THEOREM

The basic observation in the proof is that for each point p E !v! the function fo(q) := limr-too d(8Br(P), q) - r is concave, proper and bounded above. Hence the maximal level of fo is a convex closed subset C 1 of M. Cheeger and Gromoll showed that C1 is a totally geodesic compact submanifold with a possibly non-empty and non-smooth intrinsic boundary 8C1 . One can then show that if 8C1 =1= 0, then the function !I (q) = d(8Cl' q) is concave on C 1 • As before the maximal level set C 2 of !I is a convex subset of M. Since dim(C2) < dim(C1) one can iterate the process until one arrives at a convex level set Ck without intrinsic boundary. Then ~ := Ck is a soul of M. One can show that the distance function rE := d(~,·) has no critical points on M \ ~ in the sense of Grove and Shiohama, for a definition see section 1. Thus there is a gradient like vectorfield X on M \~, with IIXII ::; 1. Similarly to the proof of the diameter sphere theorem one can now construct a diffeomorphism 'IjJ: 1I(~) -t M. 0 SKETCH OF THE PROOF.

We emphasize that the diffeomorphism 1I(~) -t M is in general not given by the exponential map. On the other hand it was shown by Guijarro [1998], that there is always at least one complete nonnegatively curved metric on M such that this is the case. From the soul construction it is clear that there is a Hausdorff continuous family (C(S))SE[O,oo) of convex compact subsets of M such that C(O) = ~, C(sI) c C(S2) for Sl < S2 and Us>o C(s) = M. In fact from the above sketch this family can be obtained-by collecting all nonempty sublevels f i- 1 ([c,oo[) of the functions fo, ... , A-I in one family. Given such a family, Sharafutdinov [1979] showed, independent of curvature assumptions, that there is a distance non-increasing retraction P: M -t ~. THEOREM 3.2 (Perelman, 1994). Let ~ be a soul of M, bundle and P: M -t ~ a Sharafutdinov retraction. Then

1I(~)

its normal

a) Po eXPv(E) = Jr, where Jr: 1I(~) -t ~ denotes the projection. b) Each two vectors u E IIp(~) and v E Tp~ are tangent to a totally geodesic immersed fiat 1R2 . c) P is a Riemannian submersion of class C 1 . The theorem also confirmed the soul conjecture of Cheeger and Gromoll: A nonnegatively curved open manifold with positive sectional curvature at one point is diffeomorphic to IRn. Although this conjecture was open for more than two decades, the proof of the above theorem is very short and just uses Rauch's comparison theorem. Guijarro [2000] showed that P is of class C 2 and it was shown in [Wilking, 2005] that P is of class Coo. The latter result is a consequence of another

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structure theorem on open nonnegatively curved manifolds whose explanation requires a bit of preparation: One defines a dual foliation ;:# to the foliation F given by the fiber decomposition P: M -1- ~ as follows. For a point p E M we define the dual leaf c,# (p) as the set of all points which can be connected with p by a piecewise horizontal curve. We recall that a curve is called horizontal with respect to P, if it is everywhere perpendicular to the fibers of P. Because of Theorem 3.2 each dual leaf can also be obtained as follows. Consider a vector v in the normal bundle 1I(~) of the soul. Let S(v) denote set of all vectors in 1I(~) which are parallel to v along some curve in ~. Then exp(S(v)) = c'#(exp(v)). The structure of the dual foliation is thus closely linked to the normal holonomy group of the soul. If the normal holonomy group is transitive on the sphere, then the dual leaves are just given by distance spheres to the soul. If the holonomy group is trivial, then by a result of Strake [1988] and Vim [1990] M n splits isometrically as ~k X (l~n-k, g) and the dual leaves are just given by isometric copies of ~. In general the holonomy group is neither transitive nor trivial. In fact, by an unpublished result of the author, any connected subgroup of SO(n - k) can occur as the normal holonomy group of a simply connected soul. THEOREM 3.3 (Wilking, 2005). Let M,~, P,;:# be as above. a) Then F# is a singular Riemannian foliation, i. e., geodesics emanating perpendicularly to dual leaves stay perpendicularly to dual leaves. b) If u E TpM is horizontal with respect to P and v E TpM is perpendicular to the dual leaf c,# (p), then u and v are tangent to a totally geodesic immersed fiat ]R2. An analogous theorem holds for Riemannian submersions on compact nonnegatively curved manifolds. A consequence of the above theorem is that any non-contractible open nonnegatively curved manifold has an honest product as a metric quotient. COROLLARY 3.4. Let M be an open non negatively curved manifold and a soul of M. Then there is a noncompact Alexandrov space A and a submetry

~

a:M-1-~xA

onto the metric product ~ x A. The fibers of a are smooth compact submanifolds without boundary.

We recall that a: M -1- B is called a submetry if a(Br(P)) = Br(a(p)) for all p and r. The space A is given by the space of closures of dual leaves, which by Theorem 3.3 can be endowed with a natural quotient metric. The main new tool used to prove these results is a simple and general observation which may very well be useful in different context as well.

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45

It allows to give what we call transversal Jacobi field estimates. Let c: 1-+ (M, g) be a geodesic in a Riemannian manifold (M, g), and let A be an (n - I)-dimensional family of normal Jacobi fields for which the corresponding Riccati operator is self adjoint. Recall that the Riccati operator L(t) is the endomorphism of (c(t))1. defined by L(t)J(t) = J'(t) for J E A. Suppose we have a vector subspace YeA. Put T~(t)M := {J(t)

IJ

E Y} EB

{J'(t) I J E Y, J(t) = O}.

Observe that the second summand vanishes for almost every t and that T~t)M depends smoothly on t. We let Tctt)M denote the orthogonal complement of T~(t)M, and for v E Tc(t)M we define v1. as the orthogonal projection of v to Tctt)M. If L is non-singular at t we put

At: T~(t)M -+ Tctt)M, J(t)

H-

J'(t)1. for J E Y.

It is easy to see that A can be extended continuously on I. For a vector field X(t) E Tctt)M we define V'~/ = (X'(t))1.. THEOREM 3.5. Let J E A - Y and put Y(t) := J1.(t). Then Y satisfies the following Jacobi equation

(~~t Y(t) + (R(Y(t), c(t))c(t))1. + 3AtA~Y(t) = o. One should consider (R(., c( t) )c( t)) 1. + 3A t A; as the modified curvature operator. The crucial point in the equation is that the additional O'Neill type term 3At A; is positive semidefinite. COROLLARY 3.6. Consider an n - I-dimensional family A of normal Jacobi fields with a self adjoint Riccati operator along a geodesic c: IR -+ M in a nonnegatively curved manifold. Then

A = spanffi.{ J E A I J(t) = 0 for some t} EB {J E A I J is parallel}. 3.1. Which bundles occur? The major open problem in the subject is PROBLEM 1. Let (~, g) be a nonnegatively curved compact manifold. Which vectorbundles E over ~ admit nonnegatively curved metrics such that the zero section of the bundle is a soul? If L is a nonnegatively curved compact manifold with a free isometric

O(k) action, then the corresponding bundle L x O(k) IRk has a nonnegatively curved metric with the zero section being the soul. It is remarkable that all examples of open nonnegatively curved manifolds constructed so far are diffeomorphic to examples arising in this way. On the other hand the above method is rather flexible already. From Theorem 2.8 it follows.

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THEOREM 3.7 (Grove and Ziller). All vectorbundles over §4 admit complete nonnegatively curved metrics. It is not known whether one can find nonnegatively metrics such that the souls are isometric to the round sphere. The souls of the Grove-Ziller metrics have lots zero curvature planes. All of the relatively few vectorbundles over §5 also admit nonnegatively curved metrics [Rigas, 1985]. However, in general Cheeger and Gromoll's question which bundles over a sphere admit nonnegatively curved metrics remains open. We mention in some cases one can say a bit more about which bundles occur: if either the soul has infinite fundamental group or if one fixes the isometry type of the soul. Ozaydin and Walschap [1994] observed that a flat soul necessarily has a flat normal bundle. If one has an open manifold with infinite fundamental group then, by Theorem 2.2 one can deform the metric within the space of nonnegatively curved metrics such that a finite cover is isometric to Td x M, where M is simply connected. This in turn shows that the normal bundle of the soul Td x ~/ is canonically isomorphic to the pull back of a bundle over the simply connected factor ~/. The question whether such a bundle can also be written as a twisted bundle over Td x ~/ was studied in great detail by Belegradek and Kapovitch [2003] using rational homotopy theory. Moreover one can analyze the situation if the soul is isometric to a simply connected product ~ = ~l X ~2. Although this is just an observation due to the author we carry out some details here since they can not be found in the literature. If Ui E Tp~ is tangent to the i-th factor (i = 1,2), then R(Ul' U2)V = 0 for v E vp(o} By "integrating" this equation we deduce that for a closed curve I' (t) = hI (t), 1'2 (t)) the normal parallel transport Par'Y decomposes Par'Y = Par'Yl 0 Par'Y2 = Par'Y2 0 Par'Yl' Thus the normal holonomy group is given as the product of two commuting subgroups. Each subgroup gives rise to a principle bundle over ~ which is isomorphic to the pull back bundle of a principle bundle over ~i under the natural projection ~ -t ~i. If we decompose the normal bundle into parallel subbundIes v(~) = VI (~) EEl ... EEl VI (~) such that on each summand the holonomy group is irreducible, then each summand is isomorphic to a tensor product Vi(~) = ViI (~) Q90c Vi2(~) where Vij(~) is isomorphic to the pull back of a IK vectorbundle bundle over ~j under the natural projection ~ -t ~j, j = 1,2 and IK E {IR, C, lHI} depends on i. Since any vector bundle over §3 is trivial, we deduce.

COROLLARY 3.8. Suppose the soul is isometric to a product §3 x §3 then the normal bundle of the soul is trivial. 3.2. The space of nonnegatively curved metrics. Perelman's theorem indicates that the moduli space of metrics should be rather small. On the other hand one can not expect too much. Belegradek used the method of Grove and Ziller to exhibit the following phenomena.

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47

THEOREM 3.9 (Belegradek). There is a non-compact manifold M that admits a sequence of complete nonnegatively curved metrics (gk)kEN such that the souls of (M, gk) are pairwise non-diffeomorphic. The theorem shows that the moduli space of nonnegatively curved metrics on A1 has infinitely many components. This is in sharp contrast to the space of nonnegatively curved metrics on §2 x ]R2. THEOREM 3.10 (Gromoll and Tapp). Up to a diffeomorphism a nonnegatively curved metric on §2 x ]R2 is either a product metric or the metric is invariant under the effective action of a two torus and it can be obtained as a quotient of a product metric on §2 x ]R2 X ]R by a free ]R-action. For a nontrivial 2-dimensional vector bundles over §2 the space of nonnegatively curved metrics is somewhat more flexible. In fact Walschap [1988] showed that given an open four manifold with a soul §2 for which any zero curvature plane is tangent to one of the Perelman flats from Theorem 3.2 the following holds: Let :cp denote one of the two unit vectorfields in M \ 2: tangent to the fibers of the Sharafutdinov retraction and whose integral curves have constant distance to the soul. If f is an arbitrary function on M with compact support contained in M \ 2:, then the following metric has nonnegative sectional curvature as well, gt(u, v) := g(u, v)

+ tf(p)g(u, :cp)g(v, :cp)

for all 1l, v E TpM and all small t. A partial rigidity result was established by Guijarro and Petersen [1997]' THEOREM 3.11. Let (M, g) be an open nonnegatively curved manifold and p EM. Suppose that for any sequence Pn E !vI converging to (Xl the corresponding sequence scal(Pn) of scalar curvatures tends to O. Then the soul of M is fiat. 4. Positively curved manifolds with symmetry Grove (1991) suggested to classify manifolds of positive sectional curvature with a large isometry group. The charm of this proposal is that everyone who starts to work on this problem is himself in charge of what 'large' means. One can relax the assumption if one gets new ideas. One potential hope could be that if one understands the obstructions for positively manifolds with a 'large' amount of symmetry, one may get an idea for a general obstruction. However the main hope of Grove's program is that the process of relaxing the assumptions should lead toward constructing new examples. That this can be successful was demonstrated by the classification of simply connected homogeneous spaces of positive sectional curvature carried out by Berger [1961]' Wallach [1972], Aloff Wallach [1975] and Berard

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Bergery [1976]. The classification led to new examples in dimension 6, 7 and 12, 13 and 24. For the sake of completeness it should be said that the only other source of known positively curved examples are biquotients, i.e., quotients G//H, where G is a compact Lie group and H is a subgroup of G x G acting freely on G from the left and the right. Eschenburg [1982] and Bazaikin [1996] found infinite series of such examples in dimensions 7 and 13. We refer the reader to the survey of Wolfgang Ziller for more details. Another motivation for Grove's proposal was the following theorem. THEOREM 4.1 (Hsiang and Kleiner, 1989). Let M4 be an orientable compact 4-manifold of positive sectional curvature. Suppose that there is an isometric nontrivial action of 51 on M4. Then M4 is homeomorphic to §4 or CJP>2. The theorem is a special case of Theorem 2.5. Grove and Searle [1994] realized that the proof of the above theorem can be phrased naturally in terms of Alexandrov geometry of the orbit space M4/5 1 . A careful analysis of the orbit space also allowed them to establish the following result. THEOREM 4.2 (Grove and Searle). Let M n be an orientable compact Riemannian manifold of positive sectional curvature. Then symrank(M,g) := rank(Iso(M, g)) ~

[7111]

and if equality holds, then M is diffeomorphic to §7I, CJP>7I/2 or to a lens space.

The inequality is a simple consequence of a theorem of Berger saying that a Killing field on an even dimensional positively curved manifold has a zero. For the equality discussion Grove and Searle first show, that there in an isometric 51 action on M such that the fixed point set has a component N of co dimension 2. They then prove that the distance function d(N,·) has no critical points in !vI \ N except for precisely one 5 1-orbit where it attains its maximum. This is used to recover the structure of the manifold. Another result which essentially relies on the study of the orbit space is due to Rong [2002]. He showed that a simply connected positively curved 5-manifold with symmetry rank 2 is diffeomorphic to §5. Recently, the author made the following basic observation, see [Wilking, 2003]. THEOREM 4.3 (Connectedness Lemma). Let M7I be a compact Riemannian manifold with positive sectional curvature. a) Suppose N n - k c M n is a compact totally geodesic embedded submanifold. Then the inclusion map N n - k -+ M n is n - 2k + 1 connected. If there is a Lie group G that acts isometrically on M n and fixes N n - k pointwise, then the inclusion map is n - 2k + 1 + 8(G) connected where 8(G) is the dimension of the principal orbit.

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49

b) Suppose Nf- k1 , N:;-k 2 c M n are two compact totally geodesic embedded submanifolds, kl ~ k2, kl + k2 ~ n. Then the intersection Nf- k1 n N:;-k 2 is a totally geodesic embedded submanifold as well and the inclusion

Nn-kl 1

n N,n-k2 2

-7

N,n- k2 2

is n - kl - k2 connected. Theorem 4.3 turns out to be a very powerful tool in the analysis of group actions on positively curved manifolds. In fact by combining the theorem with the following lemma, one sees that a totally geodesic submanifold of low co dimension in a positively curved manifold has immediate consequences for the cohomology ring of the manifold.

4.4. Let M n be a closed differentiable oriented manifold, and let N n - k be an embedded compact oriented submanifold without boundary. Suppose the inclusion N n- k -7 Mn is n - k -l connected and n - k - 21 > O. Let [N] E Hn-k(M, Z) be the image of the fundamental class of N in H*(M, Z) and let e E Hk(M, Z) be its Poincare dual. Then the homomorphism LEMMA

Ue: Hi(M,Z) -7 HHk(M,Z)

is surjective for l

~

i < n - k -I and injective for I < i

~

n - k -I.

Notice that in the case of a simply connected manifold M the submanifold N is simply connected as well and hence it is orient able. Recall that the pull back of e to Hk(N, Z) is the Euler class of the normal bundle of N in M. Part b) of the Theorem 4.3 says in particular that Nf- k1 n N:;-k 2 is not empty which is exactly the content of Frankel's Theorem. In fact similarly to Frankel's Theorem a Synge type argument is crucial in the proof of Theorem 4.3. The proof of Theorem 4.3 is a very simple Morse theory argument in the space of all curves from N to N, respectively from Nl to N2. The critical points of the energy functional are geodesics starting and emanating perpendicularly to the submanifolds. Using the second variation formulas it is then easy to give lower bounds on the indices of the nontrivial critical points. The above result is the main new tool that is used in [Wilking, 2003] to show.

4.5. Let M n be a simply connected n-dimensional manifold of positive sectional curvature, n 2: 8, and let d 2: ~ + 1. Suppose that there is an effective isometric action of a torus Td on Mn. Then M is homotopically equivalent to Clpm/2 or homeomorphic to IHIpn/4 or §n. THEOREM

In dimensions 8 and 9 the theorem is due to Fang and Rong [2005]. Thus dimensions 6, 7 remain the only dimensions where one needs maximal symmetry rank assumptions for a classification.

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If M n is an odd-dimensional manifold, that is not simply connected but satisfies all other assumptions of the theorem, then its fundamental group is cyclic, see Rong [2000]. A conjecture of Mann [1965] asserts that an exotic sphere ~n can not support an effective smooth action of ad-dimensional torus with d ~ ~ + 1. Notice that F4 , the isometry group ofCa]p>2 has rank 4. Thus in dimension 16 the result is optimal. Similarly the isometry group of the 12-dimensional Wallach flag has rank 3. In dimension 13 the Berger space SU(5)jSl ·Sp(2) is an optimal counterexample. There are three major constants to measure the amount of symmetry of a Riemannian manifold (M,g):

symrank(M,g) = rank(Iso(M,g)), symdeg(M,g) = dim(Iso(M,g)) cohom(M, g) = dim( (M, g)j Iso(A1, g)). So far we have mostly considered the first of these constants. THEOREM 4.6 (Wilking, 2006). Let (Mn,g) be a simply connected Riemannian manifold of positive sectional curvature. If symdeg( M n , g) ~ 2n - 6, then (M, g) is tangentially homotopically equivalent to a rank 1 symmetric space or isometric to a homogeneous space of positive sectional curvature. Notice that all homogeneous spaces of positive sectional curvature satisfy the assumptions of the theorem. In dimension 7 the theorem gives the optimal bound as there are positively curved Eschenburg space SU(3)jjSl with a seven dimensional isometry group. Finally we consider the cohomogeneity of a Riemannian manifold. THEOREM 4.7 (Wilking, 2006). Let k be a positive integer. In dimensions above 18(k + 1)2 each simply connected Riemannian manifold M n of cohomogeneity k ~ 1 with positive sectional curvature is tangentially homotopically equivalent to a rank one symmetric space. The proof of Theorem 4.7 actually establishes the existence of an infinite sequence of (connected) Riemannian manifolds M=MoCM1 c··· such that dim(Mi) = n + ih, where h ~ 4k + 4 is a positive integer that depends on M. All inclusions are totally geodesic, all manifolds are of cohomogeneity k and all have positive sectional curvature. One then considers Moo := U Mi. On the one hand one can use the connectedness lemma to show that Moo has h-periodic integral cohomology ring. On the other hand, using Alexandrov geometry of the orbit space, one can show that A100 , has

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

51

the homotopy type of the classifying space of a compact Lie group. The results combined show that Moo is either contractible or has the homotopy type of ClPoo or lHIlPoo . The connectedness lemma then implies that M has the corresponding homotopy type. The details are quite involved and we refer the reader to [Wilking, 2006]. Of course one might hope that for small k one can use similar techniques to get a classification in all dimensions, or at least a classification up to some potential candidates for positively curved manifolds. The following theorem carries out such a program in the case of k = 1. THEOREM 4.8 (Verdiani, Grove, Wilking, Ziller). Let M n be a simply connected compact Riemannian manifold of positive sectional curvature. Suppose that a connected Lie group G acts by isometries with cohomogeneity one, i.e., the orbit space Mn /G is one dimensional. Then one of the following holds: • M n is equivariantly diffeomorphic to one of the known positively curved biquotients endowed with a natural cohomogeneity action. • n = 7 and M is the two fold cover of a 3-Sasakian manifold that corresponds to one of the self dual Einstein 4-orbifolds of cohomogeneity one that were found by Hitchin. • n = 7 and M is equivariantly diffeomorphic to one particular cohomogeneity one manifold.

In even dimensions the theorem is due to Verdiani [2001]' in this case only rank 1 symmetric spaces occur. The odd dimensional case is more involved and is due to Grove, Wilking and Ziller [2006]. This is partly due to the fact that in dimensions 7 and 13, there are infinitely many positively curved biquotients of cohomogeneity one. It remains open whether the last two cases can indeed occur. The proof of the theorem uses a lot of the techniques that we have mentioned above. We refer to the survey of Ziller for a more detailed discussion. Very different results on positively curved manifolds with symmetry were found by Dessai [2005]. THEOREM 4.9. Suppose (M, g) is a positively curved spin manifold of dimension ~12. Let G be a connected Lie group acting smoothly and suppose a subgroup Z~ c G acts by isometries. Then the characteristic number A(M, TM) vanishes. The proof is a clever combination of Frankel's theorem on the intersection of totally geodesic submanifolds and a vanishing theorem of Hirzebruch and Slodowy. The non-vanishing of A(M, TM) would by that result ensure that each of the three involutions in Z~ has a fixed point set of codimension 4. By Frankel these three components have a common intersection and the contradiction arises by inspecting the isotropy representation of Z~ at a fixed point.

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In the presence of stronger symmetry assumptions he can show the vanishing of more characteristic numbers. These numbers occur naturally as coefficients of a power series expanding the elliptic genus. 4.1. Manifolds with positive sectional curvature almost everywhere. As mentioned before there are relatively few known examples of positively curved manifolds. The lists of examples is quite bit longer if one includes manifolds which have positive sectional curvature on an open dense set. The most interesting example in the class is the Gromoll Meyer sphere. Gromoll and Meyer [1974] considered the subgroup H c Sp(2) x Sp(2) given by H:= {(diag(l,q),diag(q,q)) I q E S3} and the induced free two sided action of H on Sp(2). They showed that the corresponding biquotient ~7 := Sp(2)//H is diffeomorphic to an exotic sphere. Furthermore, by the O'Neill formulas the metric on Sp(2)//H induced by the biinvariant metric g on Sp(2) has nonnegative sectional curvature. In fact it is not hard to see that there is a point p E ~7 such that all planes based at p have positive curvature. It was shown later by Wilhelm [1996] that there is a left invariant metric on Sp(2) such that the induced metric on ~7 has positive sectional curvature on an open dense set of points. Gromoll and Meyer mention in their paper the so called deformation conjecture: PROBLEM 2. (Deformation conjecture) Let M be a complete nonnegatively curved manifold for which there is point p E M such that all planes based at p have positive sectional curvature. Does (M, g) admit a positively curved metric, as well? In the case of an open manifold M the conjecture is by Perelman's solution of the soul conjecture valid. However in general counter examples were found in [Wilking, 2002]. THEOREM 4.10. The projective tangent bundles ~T~lP,n, pcTCJP>n and FHTlHIJP>n of the projective spaces admit metrics with positive sectional curvature on open dense sets. It is easy to see that the projective tangent bundle of ~T~JP>2n+l is odd dimensional and not orientable. By a theorem of Synge it can not admit a metric with positive sectional curvature. In dimensions 4n-1, (n;::: 3) there are infinitely many homotopy types of simply connected compact manifolds with positive sectional curvature on open dense sets. In fact one 'half' of the circle bundles over pcTCJP>n give rise to such examples. It is also interesting to note that the natural inclusions among these examples remain totally geodesic embeddings and that the isometry groups

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

53

of the manifolds act with cohomogeneity 2. By the results on positively manifolds with symmetry, these properties could not persist for positively curved metrics. Another consequence is that §2 x §3, the universal cover of f>JRTlRIP'3, admits a metric with positive sectional curvature on an open dense set. Finally we should mention that prior to [Wilking, 2002], Petersen and Wilhelm [1999] constructed a slightly different metric on the unit tangent bundle of §4, the universal cover of f>JRTlRr, with positive curvature on an open dense set. 5.

Open Problems

In this final section we mention some of the major open problems in the subject. The authors favorite conjecture in this context is the so called Bott-conjecture which was promoted by Grove and Halperin. CONJECTURE

5.1. Any nonnegatively curved manifold is rationally

elliptic.

We recall that a manifold is called rationally elliptic if 11"*(M) I8lQ is finite dimensional. The conjecture would for example show that the total rational Betti number of a nonnegatively curved manifold M is bounded above by 2n with equality if and only if M is a flat torus. There is a conceptual reason why the Bott-conjecture holds for all known nonnegatively curved manifolds. Up to deformation of metrics all known nonnegatively manifolds are constructed from Lie groups endowed with biinvariant metrics using the following three techniques • One can take products of nonnegatively curved manifolds. • One can pass from a nonnegatively curved manifold endowed with a free isometric group action to the orbit space endowed with its submersion metric. • Due to work of Cheeger[1973] and Grove and Ziller [2000] one can sometimes glue disc bundles, i.e., if M is a nonnegatively curved manifold which is in two ways the total space of a sphere bundle (with the structure group being a Lie group), then sometimes the manifold obtained by glueing the two corresponding disc bundles has nonnegative curvature as well. It is well known that Lie groups are rationally elliptic. Furthermore, by the exact homotopy sequence the class of rationally elliptic manifolds is invariant under taking quotients of free actions. By the work of Grove and Halperin [1987], a manifold obtained by gluing two disc bundles along their common boundary is rationally elliptic if and only if the boundary is. Grove suggested that the conjecture should hold more generally for the class of simply connected almost nonnegatively curved manifolds. Here we call a manifold almost nonnegatively curved if it admits a sequence 9k of

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metrics with diameter 1 and sectional curvature "2-c(k) -t O. The latter class contains more known examples. On the other hand the only additional technique needed to construct all of the known simply connected almost nonnegatively curved manifolds is:

• If M is an almost nonnegatively curved manifold and P is a principal G-bundle over M with G being a compact Lie group, then P has almost nonnegative sectional curvature as well. Clearly with this method one can not leave the class of rationally elliptic manifolds either. Grove suggested that it might be possible to prove the Bott conjecture by induction on the dimension. In this context he posed the problem whether any compact nonnegatively curved manifold has a nontrivial collapse: Is there a sequence of metrics 9n on M with diameter "5:D and curvature "2 - 1 such that (lvI,9n) converges to a k-dimensional Alexandrov space with 0 < k < n. Of course it would be also interesting if there is a property that is shared by all nonnegatively curved Alexandrov spaces, and which in the case of manifolds is equivalent to saying that the space is rationally elliptic. Alexandrov spaces are more flexible since one can take quotients of non free group actions and in the case positive curvature joins of spaces. Totaro [2003] posed the problem whether any nonnegatively curved manifold has a good complexification, i.e., is M diffeomorphic to the real points of complex smooth affine variety defined over lR such that the inclusion into the complex variety is a homotopy equivalence. Totoro's work shows that these manifolds share many properties of rationally elliptic manifolds. The Bott conjecture would also imply that the Eulercharacteristic of a nonnegatively curved manifold is nonnegative and positive only if the odd rational Betti numbers vanish. The former statement is part of the Hopf conjecture. CONJECTURE 5.2 (Hopf). A compact nonnegatively (positively) curved manifold has nonnegative respectively positive Eulercharacteristic. Slightly more modest (and vague) one might ask QUESTION 5.3. Is there any obstruction that distinguishes the class of simply connected compact manifolds admitting nonnegatively curved metrics from the corresponding class admitting positively curved metrics? Of course the huge difference in the number of known examples suggests that plenty of such obstructions should exist, but to this day there is not a single dimension where such an obstruction has been found. Closely related is another Hopf conjecture. CONJECTURE 5.4 (Hopf). sectional curvature.

§2

x

§2

does not admit a metric of positive

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

55

Unlike on §2 x §3 it is not known whether there is metric on §2 x §2 with positive curvature almost everywhere. For that reason one could hope that the nonnegatively curved metrics on §2 x §2 are rather rigid. In fact a partial confirmation of this view was given by Bourguignon, Deschamps, and Sentenac [1972]. They showed that for a product metric on §2 x §2 without Killing fields any analytical deformation which preserves nonnegative curvature is up to diffeomorphisms given by a deformation through product metrics. However, one should be careful to expect too much rigidity in this context. The author learned the following observation from Bruce Kleiner. We consider §2 x §2 endowed with the Muter metric (§2

X

§2,g)

= 51

X

51 x {1}\SO(3) x SO(3) x 80(3)/~80(3)

where 80(3)3 is endowed with a biinvariant metric. Clearly the metric is of cohomogeneity one, since there is an 80(3)-action from the left on 80(3)3 commuting with the left action of 51 x 51. The two singular orbits are given by two 2-dimensional spheres and we let Mreg C §2 X §2 denote the union of all principal orbits. Muter [1987] showed for each point p E Mreg that there is precisely one zero curvature plane based at p. Moreover the plane is tangent to a totally geodesic torus in M. In particular the generic part of the manifold Mreg C §2 X §2 is foliated by totally geodesic flat submanifolds. We now consider a symmetric (2,0) tensor b, whose compact support is contained in Mreg and for which b(v,·) = o for all v contained in a zero curvature plane. It is then straightforward to check that the foliation of Mreg by totally geodesic fiats remains a totally geodesic foliation by flats for all metrics in the family g( t) = 9 + tb. Therefore, the set of zero curvature planes of (M, g( t)) contains the set of zero curvature planes in (M,g(O)). What is more: the zero curvature planes remain critical points of the sectional curvature. Since the zero curvature planes in (Mreg , g(O)) form a sub manifold of the Grassmannian Gr2(Mreg) and the Hessian of the sectional curvature function is nondegenerate transversal to this submanifold, it is clear that (M,g(t)) has nonnegative sectional curvature for all small t. This shows that the space of nonnegatively curved metrics of §2 x §2 is somewhat larger than one would expect at first glance. One way to give new impulses to the subject is to construct newexamples. In this context we pose the following question. QUESTION 5.5. Are there any positively curved compact Alexandrov spaces satisfying Poincare duality which are not homeomorphic to one of the known positively curved manifolds?

Of course an easy way to check that an Alexandrov space satisfies Poincare duality is to show that the space of directions at each point is

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homeomorphic to a sphere. One could try to look at non free isometric group actions on nonnegatively curved manifolds and ask whether the orbit space is homeomorphic to a manifold without boundary. It would be also interesting to know, whether in special circumstances one can resolve the metric singularities of a positively curved Alexandrov space while keeping positive curvature.

6. Added in Proof One of the most significant developments in the subject took place after this survey was completed. We will briefly explain it here. We recall that a manifold is strictly pointwise quarter pinched if at each point p E M there is a constant K,(p) 2: 0 such that for all planes based at p have curvature strictly between K,(p) and 4K,(P). THEOREM 6.1 (Brendle and Schoen, 2007). For any strictly pointwise quarter pinched manifold (M, g), the normalized Ricci flow evolves g to a limit metric of constant sectional curvature.

We use the notation that we introduced in section 1 in connection with Theorem 1.10. The theorem relies on the following result. THEOREM 6.2 (B6hm and Wilking, 2006). Let C be an O(n)-invariant cone C in the vector space of curvature operators S~(.so(n)) with the following properties

• C is invariant under the ODE ftR = R2 + R#. • C contains the cone of nonnegative curvature operators or slightly weaker all nonnegative curvature operators of rank 1. • C is contained in the cone of curvature operators with nonnegative sectional curvature. Then for any compact manifold (M, g) whose curvature operator is contained in the interior of C at every point p EM, the normalized Ricci flow evolves g to a limit metric of constant sectional curvature. It actually suffices to assume that the curvature operator of (M, g) is contained in C at all points, and in the interior of C at some point, cf. [Ni and Wu, 2006]. We should remark that the theorem was not stated like this in [B6hm and Wilking 2006]. However by Theorem 5.1 in that paper it suffices to construct a pinching family with C(O) = C. Furthermore, the construction of a pinching family for the cone of nonnegative curvature operators only relied on the above three properties. In other words, one can define a pinching family C(8) with C(O) = C by

C(8) := la(s),b(s) ({R

E

C I Ric 2: s~alp(8)})

NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS

57

where the parameters a(s), b(s) defining the linear map la(s),b(s): S~(50(n)) -+ S~(50(n)) and p(s) are chosen exactly as in [Bohm and Wilking, 2006J. SKETCH OF THE PROOF OF THEOREM 6.1.. The most important step was proved independently by Nguyen [2007J and Brendle and Schoen [2007J: Nonnegative isotropic curvature defines a Ricci flow invariant curvature condition. Both proofs are similar. By Hamilton's maximum principle it suffices to show that the cone C of curvature operators with nonnegative isotropic curvature is invariant under the ODE

ftR = R2 +R#. The idea is to make use of the second variation formula at a four frame where the isotropic curvature attains a zero - that is one uses the fact that the Hessian of the isotropic curvature function is positive semidefinite. Although the computation is elementary it is quite long and that it succeeds comes close to being a miracle. Brendle and Schoen then proceed as follows. They consider the condition that a Riemannian manifold crossed with R2 has nonnegative isotropic curvature. It is easy to see that the cone C of curvature operators corresponding to this curvature condition satisfies the hypothesis of Theorem 6.2. Finally Brendle and Schoen establish that any pointwise quarter pinched manifold (M, g) has the property that (M, g) x R2 has nonnegative isotropic curvature. This is again a lengthly computation. 0 REMARK 6.3. Ni and Wolfson [2007J observed that the condition that the manifold crossed with R2 has nonnegative isotropic curvature is equivalent to saying that (M, g) has nonnegative complex sectional curvature. They also give an alternative shorter argument that nonnegative complex curvature is preserved by the Ricci flow. Finally they remark that the statement that quarter pinched manifolds have nonnegative complex curvature was essentially known. In fact Yau and Zheng showed in a different context that a curvature operator with sectional curvature between -4 and -1 has nonpositive complex sectional curvature.

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Surveys in Differential Geometry XI

Examples of Riemannian Manifolds with Non-negative Sectional Curvature Wolfgang Ziller

Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on §2 x §2, also fit into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking a quotient of a compact Lie group with a biinvariant metric and another a gluing procedure due to Cheeger and recently significantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and find new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones. The techniques used to construct them are fairly simple. In addition to the above, the main tool is a deformation described by Cheeger that, when applied to non-negatively curved manifolds, tends to increase curvature. Such Cheeger deformations can be considered as the unifying theme of this survey. We can thus be fairly explicit in the proof of the existence of all known examples which should make the basic material understandable at an advanced graduate student The author was supported by a grant from the National Science Foundation. ©2007 International Press

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64

level. It is the hope of this author that it will thus encourage others to study this beautiful subject. This survey originated in the Rudolph Lipschitz lecture series the author gave at the University of Bonn in 2001 and various courses taught at the University of Pennsylvania.

1. General structure theorems

To put the examples discussed in this survey into context, we mention the main structure theorems and conjectures in this subject. See the survey by Burkhard Wilking in this volume for further information. • (Gromov) If M n is a compact manifold with sec 2: 0, then there exists a universal constant c( n) such that bi (Mn ,F) S c( n) for all i and any field of coefficients F. Furthermore, the fundamental group has a generating set with at most c( n) elements. • (Cheeger-Gromoll) If M n is a compact manifold that admits a metric with non-negative sectional curvature, then there exists an abelian subgroup of 71'1 (Mn) with finite index. • (Lichnerowicz-Hitchin) The obstructions to positive scalar curvature imply that a compact spin manifold with A(M) i= 0 or a(M) i= o does not admit a metric with non-negative sectional curvature. This holds in particular for the unique exotic sphere in dimension 9. • (Cheeger-Gromoll) If M n is a non-compact manifold with a complete metric with sec 2: 0, then there exists a totally geodesic compact submanifold Sk, called the soul, such that Mn is diffeomorphic to the normal bundle of Sk. Surprisingly, for positive curvature one has in addition only the classical obstructions: • (Bonnet-Myers) A manifold which admits a metric with positive curvature has finite fundamental group. • (Synge) An even dimensional manifold with positive curvature has fundamental group 0, if orientable, and Z2, if non-orientable. In odd dimensions a positively curved manifold is orientable. If we allow ourselves to add an upper as well as a lower bound on the sectional curvature it is convenient to introduce what is called the pinching constant which is defined as 6 = min sec/max sec. One then has the following recognition and finiteness theorems:

• (Berger-Klingenberg) If Mn is a compact simply connected manifold with 6 2: then M is either homeomorphic to §in or isometric to cpn, JH[pn or Cap2 with their standard Fubini metric. • (Cheeger) Given a positive constant €, there are only finitely many diffeomorphism types of compact simply connected manifolds M2n with 6 2: €.

i,

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 65

• (Fang-Rong, Petrunin-Thschmann) Given a positive constant E, there are only finitely many diffeomorphism types of compact manifolds M2n+1 with 7l"l(M) = 7l"2(M) = 0 and 8 ~ E. We finally mention some conjectures. • (Hopf) There exists no metric with positive sectional curvature on §2 x §2. More generally, there are no positively curved metrics on the product of two compact manifolds, or on a symmetric space of rank at least two. • (Hopf) A compact manifold with sec ~ 0 has non-negative Euler characteristic. An even dimensional manifold with positive curvature has positive Euler characteristic. • (Bott) A compact simply connected manifold M with sec ~ 0 is elliptic, i.e., the sequence of Betti numbers of the loop space of M grows at most polynomially for every field of coefficients. The latter conjecture, and its many consequences, were discussed in the literature for the first time in [37]. 2. Compact examples with non-negative curvature There are two natural constructions to produce new metrics with nonnegative curvature from given ones. If Ml and M2 are endowed with metrics of non-negative curvature, the product metric on Ml x M2 clearly has nonnegative curvature. The second construction is by taking quotients, or more generally by considering Riemannian submersions. Recall that if M and B are two Riemannian manifolds, then a smooth map 7l": M -+ B is called a Riemannian submersion if 7l"* is an isometry on horizontal vectors, i.e., on vectors orthogonal to the fibers. For such submersions one has the 0 'N eill formula:

where x, yare orthonormal horizontal vectors, i.e., orthogonal to the fibers, X, Y are horizontal vector fields extending x, y, and [X, y]V denotes the vertical part of [X, Y], i.e., the component tangent to the fiber. If M has non-negative curvature, so does B, and one can hope that in some cases B is even positively curved. The most basic examples of Riemannian submersions are given by taking quotients 7l": M -+ MIG where G is a compact Lie group acting freely and isometrically on M. We often call the induced metric on MIG the 'quotient' metric. Before we describe a third method, let us first recall some standard ways of putting metrics on homogeneous spaces. If a compact Lie group G acts transitively on M and p EM, we can write M = GIH where H is the isotropy group at p. We will often fix a biinvariant metric Q on {I, the Lie algebra of G. Note that adX: Y -+ [X, Y] is then a skew symmetric endomorphism with respect to Q. Thus the Q-orthogonal decomposition {I = ~ + m

W. ZILLER

66

satisfies [~,~] C ~ and [~, m] em. The more restrictive condition [m, m] C ~ corresponds to the case where the metric Q induces a locally symmetric metric on G / H. We identify m with TpM via action fields: X E m -+ X* (p) where X*(q) = ft1t=o exp(tX)q. The action of H on TpM is then identified with the action of Ad H on m. A metric on G / H, invariant under the action of G, corresponds to an inner product on m ~ TpM which is Ad(H)invariant. This inner product can be expressed as Q(P X, Y) where X, Y E m and P: m -+ m is a Q-symmetric endomorphism. A third method that produces new non-negatively curved metrics from a given one is obtained via a Cheeger deformation. This process was first used by M. Berger who considered metrics on spheres, shrunk in the direction of the Hopf fibration, to produce odd dimensional manifolds with small injectivity radius and positively pinched curvature. A systematic general description was given in [18]. Let (M,g) be a Riemannian manifold and G a Lie group acting by isometries on M. We then consider the Riemannian submersion 7r:MxG-+M This can also be viewed as a quotient construction via the action

g* (p,g) = (gp,gg) or simply M = (M x G)/t::,.G. We can thus start with a non-negatively curved metric 9 on M, take a product with a biinvariant metric, and then the quotient metric defines a new metric on M. To describe this process as a deformation, fix a biinvariant metric Q on 9 and let gt be the metric obtained as a quotient of the product metric 9 + tQ on M x G. Tangent to the orbit Gp = G/Gp, we write the original metric as above in the form Q(PX, Y), where X, Y E mp , with ffip the orthogonal complement of the Lie algebra of Gp • The symmetric endomorphism P: mp -+ mp is changed into a new symmetric endomorphism Pt describing gt in terms of Q and we claim:

(2.1) To see this, observe that 7r* (X* , Y) = X* - Y*. Thus a horizontal lift of X E mp ~ Tp(G/Gp) C TpM, under the Riemannian submersion 7r, is equal to

and the length squared of this vector is Q((P- 1 + tId)-l X, p-1(p- 1 + tId)-l X)

+ ~Q(t(P-l + tId)-lX, t(p-l + tId)-l X) = Q((P- 1 +tId)-lX,X).

Orthogonal to the orbit Gp, the metric is unchanged since a horizontal lift of X E C TpM is equal to (X*(p),O).

mi

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 67

This process can be considered as a deformation, since we obtain the original metric 9 when t = O. If Ai are the eigenvalues of P, the eigenvalues of Pt are 1;£>"i' i.e. the metric is shrunk in the direction of the orbits. This process will in general destroy symmetries, although the group G clearly still acts by isometries induced by right multiplication in M x G. We can thus also consider iterated Cheeger deformations corresponding to a chain of subgroups HI C ... C Hk C G. The deformation 9 ---+ 9t tends to improve curvature. If the original metric 9 has non-negative curvature, the deformed metric does also by O'Neill's formula. If X, Y span a O-curvature 2-plane of 9, then, by considering the G-components of vectors in M x G, its curvature becomes positive in the metric 9t if [PXm, PYmJ ~ 0, where we have associated to X E TpM a vector Xm E mp such that X~(p) is the component of X in the orbit direction. Although this will not be needed in this survey, one finds a detailed study of the basic properties of this deformation in Muter [61J. We mention here only that, if we let Ct = p- 1Pt on mp and Ct1mil- = Id be the symmetric endomorphism that expresses 9t in terms of 9 = 90, then secgt ( C t- 1 X, Ct- 1 Y) > 0 for t > 0 unless secg(X, Y) = 0, [PXm, PYmJ = 0 and dwz(X, Y) = 0 for all Z E g, where Wz is the one form dual to the Killing vector field Z. Thus the O-curvature planes tend to "move" with Ct 1 . Furthermore, 2-planes which are tangent to a totally geodesic fiat 2-torus, and which contain a vector orthogonal to the G orbit, remain fiat. As a starting point for finding examples, one considers compact Lie groups G endowed with a biinvariant metric since their curvature satisfies:

secc(x,y) =

1

411 [x,yJ 1122: 0

for x,y E g orthonormal.

Combining this fact with O'Neill's formula, we obtain non-negative curvature on every isometric quotient of a compact Lie group. In particular, all homogeneous spaces G/ H, where H is a closed subgroup of G, have metrics with non-negative curvature. Since the identity component of the isometry group of a compact Lie group G, endowed with a biinvariant metric, consists of left and right translations, it is natural to generalize the class of homogeneous manifolds to what are called biquotients. Consider H C G x G and define an action of H on G by

The action is free if and only if hI conjugate to h2, for (hI, h2) E H, implies that hI = h2 = e. If this is the case, the quotient is a manifold, which we denote by G / / H and is called a biquotient. If H = L x KeG x G, we will also write L\G/ K. Thus we obtain: 2.2. A biinvariant metric on G induces a metric with nonnegative sectional curvature on every homogeneous space G / H and every biquotient G / / H . THEOREM

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The first time where biquotients were considered in geometry, was in [36J, where it was shown that an exotic 7-sphere admits non-negative curvature. To describe this example, consider the action of the unit quaternions Sp(l) on the symplectic group Sp(2) via: q * A = diag(q, q)A diag(q, 1)-1

q E Sp(l), A E Sp(2).

This action is clearly free and we have: THEOREM 2.3 (Gromoll-Meyer). The non-negatively curved manifold Sp(2)11 Sp(l) is homeomorphic, but not diffeomorphic, to §7. In order to prove this, one observes that Sp(2)11 Sp(l) Sp(l) = diag(q, q) Adiag(r,1)-1, q,r E Sp(l), is diffeomorphic to §4 since the action of diag(q, q) on Sp(2)1 diag(r, 1) = §7 is the Hopf action by S3. Thus Sp(2)11 Sp(l) can be considered as an §3 bundle over §4. One then identifies which sphere bundle it represents by using Milnor's description of certain exotic 7-spheres as §3 bundles over §4. We also point out that in [77], and independently in [55J, it was shown that the only exotic sphere which can be written as a biquotient is the Gromoll-Meyer sphere. Another special class of non-negatively curved examples were constructed in [18J: THEOREM 2.4 (Cheeger). The connected sum of any two rank one symmetric spaces carries a metric with non-negative sectional curvature. In [77J it was shown that some of these Cheeger manifolds, but not all, can be viewed as biquotients as well. The gluing construction used in order to prove Theorem 2.4, was recently significantly generalized to what are called cohomogeneity one manifolds. Recall that if G is a Lie group that acts on a manifold M, the cohomogeneity of the action is defined as cohom( M, G) = dim MIG. Thus an action with cohom(M, G) = 0 is an action that is transitive, i.e., the manifold is a homogeneous spaces. Cohomogeneity one manifolds can thus be considered as the next simplest kind of group actions to study. They are also special among all group actions since, as we will see, the manifold can be reconstructed from its isotropy groups. The geometry and topology of homogeneous spaces is fairly well understood by now, whereas this is not yet the case for cohomogeneity one manifolds. One should point out though, that this class of manifolds does not contain the homogeneous spaces as a subset. In fact only very few homogeneous spaces carry a cohomogeneity one action. Let G be a compact group acting by cohomogeneity one on a compact manifold M. Since MIG is one dimensional, it is either a circle sl, or an interval I. In the first case all G orbits are principal and 71': M --+ MIG = S1 is a fiber bundle with fiber a principal orbit G I H, and thus the fundamental group is infinite. One also easily sees that such fiber bundles carry a

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE

69

G invariant metric with non-negative curvature. In the second more interesting case there are precisely two non principal G-orbits corresponding to the endpoints of I, and M is decomposed as the union of two tubular neighborhoods of these nonprincipal orbits, with common boundary a principal orbit. Let f_ and f+ be the co dimension of the nonprincipal orbits. We have the following existence theorem [42]: THEOREM 2.5 (Grove-Ziller). A compact cohomogeneity one G-manifold with f± ~ 2 has a G-invariant metric with non-negative sectional curvature. One easily sees that f± > 1 if M is simply connected. Although the assumption on the co dimensions seems rather special, it turns out that the class of cohomogeneity one manifolds with f± = 2 is surprisingly rich. An immediate application is: COROLLARY 2.6. Each of the 4 oriented diffeomorphism types of ~1P'5 's carries a metric with non-negative sectional curvature. This follows since §5 carries (non-linear) cohomogeneity one actions by SO(2) SO(3), discovered by G.Calabi, with codimension 2 singular orbits. They are a special case of the action on Kervaire spheres described below. The involution in SO(2) acts freely and, using surgery theory, one shows that anyone of the exotic ~1P'5,s can be obtained in this fashion, see [60]. In [42] it was also conjectured that Theorem 2.5 holds without any assumption on the codimensions. This turns out to be false. One has [40]: THEOREM 2.7 (Grove-Verdiani-Wilking-Ziller). For each pair (f_, f+) with (f_, f+) =1= (2,2) and f± 2': 2 there exist infinitely many cohomogeneity one G-manifolds that do not carry a G invariant metric with non-negative sectional curvature. The most interesting example in this Theorem are the Kervaire spheres, which are the only exotic spheres that can carry a cohomogeneity one action [74]. They are described as a 2n - 1 dimensional Brieskorn varietiy: d Zo

+ Zl2 + ... Zn2 = 0 ,

It carries a cohomogeneity one action by SO(2)SO(n) defined by (e iB , A) (zo, ... , zn) = (e2iBzo, eidB A(Zl,"" zn)t) whose co dimensions are (L, f+) = (2, n -1). For n odd and d odd, they are homeomorphic to spheres, and are exotic spheres if 2n - 1 == 1 mod 8. If n 2': 4, d 2': 3 one shows that there exists no G-invariant metric with non-negative curvature. To prove Theorem 2.7, one needs to describe the set of all G-invariant metrics explicitly. They depend on a finite collection of functions, 6 in the case of the Kervaire spheres, which need to satisfy certain smoothness conditions at the endpoint. For each choice of 2-planes tangent to the principal orbit, one obtains differential inequalities on these functions and their first

70

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derivatives from the Gauss equations. By a suitable choice of 2-planes one obtains upper and lower bounds on the first derivatives which contradict the smoothness conditions at one of the singular orbits. We mention that in the case of Ricci curvature one has the positive result that every cohomogeneity one manifold carries an invariant metric with non-negative Ricci curvature, and with positive Ricci curvature if and only if the fundamental group is finite [43J. To discuss the proof of Theorem 2.5 and some of its applications, we first recall the basic structure of cohomogeneity one actions. We will only consider the most interesting case, where MIG = I and let 7r: M ~ MIG be the projection. In order to make the description more explicit, we choose an arbitrary but fixed G-invariant Riemannian metric on M, normalized so that with the induced metric, MIG = [-1, IJ. Fix a point Xo E 7r- 1 (O) and let c : [-1, IJ ~ M be a geodesic orthogonal to the orbit through Xo, and hence to all orbits, and parameterized such that 7r 0 C = id[-I,lj. Let B± = 7r- 1 (±1) = G· x± be the two nonprincipal orbits, where x± = c(±I). It then follows that c : [2n - 1, 2n + IJ ~ M, n E Z are minimal geodesics between the two nonprincipal orbits B± since G acts transitively on the set of all geodesics orthogonal to the orbits. Let K± = G x ± be the isotropy groups at x± and H = G xo = Gc(t), -1 < t < 1, the principal isotropy group. By the slice theorem, we have the following description of the tubular neighborhoods D(B_) = 7r- 1 ([-I,O]) and D(B+) = 7r-l([O, 1]) of the nonprincipal orbits B± = G I K ±:

where Df.± is the normal (unit) disk to B± at x±. Here the action of K± on G X Df.± is given by k* (g,p) = (gk- 1, kp) where k acts on Df.± via the slice representation. Hence we have the decomposition

where E = 7r- 1 (O) = G . Xo = GI H is a principal orbit which is canonically identified with the boundaries 8D(B±) = G XK± §f.±-I, via the maps G ~ G X §f.±-I, 9 ~ (g,=j=c(±I)). Note also that 8Df.± = §f.±-1 = K±IH since the boundary of the tubular neighborhoods must be a G orbit and hence 8Df.± is a K± orbit. All in all we see that we can recover M from G and the subgroups Hand K±. We caution though that the isotropy types, Le., the conjugacy classes of the isotropy groups K± and H do not determine M. An important fact about cohomogeneity one actions is that there is a converse to the above construction. Suppose G is a compact Lie group and H c K± c G are closed subgroups, which we sometimes denote by H c {K _, K+} c G. Assume furthermore that K±I H = §f.± -1 are spheres. It is well known that a transitive action of a compact Lie group K on a sphere Sf. is conjugate to a linear action and is determined by its isotropy group H c K. We can thus assume that K± acts linearly on §f.± with

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE

71

isotropy group H at P± E §e± -1 and define a manifold M

= G XK_

DC

UG/H

G

XK+

D e+,

where we glue the two boundaries by sending [g,p_] to [g,p+]. G acts on M via g*[g,p] = [g*g,p] on each half and one easily checks that it has isotropy groups K± at [e,O] and H at [e,po] and is thus cohomogeneity one. Theorem 2.5 clearly follows from the following geometric result by gluing two such metrics on the tubular neighborhoods D(B±) along their common boundary G / H. PROPOSITION 2.8. Let H eKe G be Lie groups with K / H = §1 = 8D 2 and fix a biinvariant metric Q on G. On the disc bundle G x K D2 there exists a G-invariant metric with non-negative sectional curvature, which is a product near the boundary G x K§1 = G / H with metric on G / H induced by Q. The crucial ingredient in the proof of Proposition 2.8 is the following result about left invariant metrics. LEMMA 2.9. Let G be a compact Lie group and t egan abelian subalgebra. Consider the left invariant metric on G whose value at TeG = g is given by Qt = tQlt + Qlt.L, where Q is a biinvariant metric on G. Then Qt has non-negative sectional curvature as long as t ~ 4/3. PROOF. The curvature formula for a left invariant metric (X, Y) = Q(P X, Y) is given by (see e.g. [61]): (R(X, Y)Y, X))

1

3

= "2 Q([P X, Y] + [X, PY], [X, Y]) - :1 Q(P[X, Y], [X, Y])

+ Q(B(X, Y), p-l B(X, Y)) -

Q(B(X, X), p- 1B(Y, Y)),

where B(X,Y) = !([X,PY]- [PX,Y]). In our case, let X = A + R, Y = B + S with A, B E m = t..l and R, SEt and hence P(A + R) = A + tR. We can now split up the expressions into components in direction of m and of t A computation shows that the m component is given by

:11 II [A, B]m + t[X, B] + t[A, Y]II Q2

~ 0,

where we have used the bi-invariance of Q and the Jacobi identity to show that ([X, B], [A, Y]) = ([X, A], [Y, B]). On the other hand, the t component is given by

which is non-negative as long as t ~ ~.

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W. ZILLER

72

Proof of Proposition 2.8 We have inclusions He KeG with K/ H = §1 and define Q-orthogonal decompositions 9 = e+ m and e= ~ + p. As usual, we identify the tangent spaces T(H)K/ H ~ P and T(H)G/ H ~ P + m. Since p is one dimensional, Lemma 2.9 implies that the left invariant metric on G defined by Qa = aQlp +QIHm has non-negative curvature as long as a ~ 4/3. Since [p, p] = 0 and [~, p] C p, the sub algebra p is an ideal of e and hence Ad K invariant, and thus Qa is right K-invariant as well. In addition we choose a metric = dt 2 + f(t)2d(J2 on D2 which is clearly invariant under the action of K on D2 and has non-negative curvature if f is concave. The product metric Qa + on G x D2 thus induces a non-negatively curved metric ga,! on the homogeneous disk bundle G x K D2. We now claim that given 1 < a ~ 4/3, we can choose f such that ga,! is a product near the boundary with metric on G / H induced by Q. To see this, consider the Riemannian submersion G x (K / H) -+ G x K K / H ~ G / H where we endow K/ H = §1 with the metric of a circle of radius f(t). The induced metric on G / H is the metric ga,! restricted to the boundary of a tube of radius t. We compute this metric as in the case of a Cheeger deformation (2.1). If 211"so is the length of the circle K/ H in the metric Qlp, the metric on K/ H is given by (f / So )2Q and it follows that the metric on G / H is given by Q

g,

g,

on m and by l+a(f/so)

2

Q = P~:S5 Q on p. Hence we obtain the desired

metric by choosing a concave function

t 2: to.

f and a to such that

P (t) =

2

:~ol' for 0

Remark. We can view this construction as a "scaling up, scaling down" procedure. The natural metric on GXKD induced by a biinvariant metric Q on G shrinks the metric on the boundary G / H in the direction of K / H, as in the case of a Cheeger deformation. This needs to be compensated by scaling the metric Q up in the direction of p in order to recover the metric Q on G / H. This explains the difficulty of proving Proposition 2.8 for .e > 2 since left invariant metrics as in Lemma 2.9 in general have some sectional curvature positive when t > 1. In fact we believe: PROBLEM 1. Let G be a compact simple Lie group and K a non-abelian subgroup. Show that a metric Qt as in Lemma 2.9 has some negative sectional curvatures for any t > 1.

Nevertheless, it is possible that there are other special homogeneous disk bundles for which Proposition 2.8 holds with codimension .e > 2. As was observed by B. Wilking, Theorem 2.5 can be generalized to the situation where the homogeneous orbits are replaced by biquotients. In other words, if K± c G x G acts freely on G and H c K± with K±/ H = §1, then the resulting manifold carries a metric with non-negative curvature. This follows by applying Theorem 2.5 to the cohomogeneity one manifold H C {K_,K+} c G x G and then dividing by !:J.G c G x G on the left, which acts freely by assumption.

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 73

We now apply this result to some concrete cohomogeneity one manifolds in order to prove: THEOREM 2.10 (Grove-Ziller). Every principal SO(k) bundle P over §4 carries a cohomogeneity one action by SO(3) x SO(k) with codimension two principal orbits and hence an invariant metric with non-negative curvature. Thus, by O'Neill's formula, every associated bundle P XSO(k)X with X a non-negatively curved manifold on which SO(k) acts by isometries, also carries a non-negatively curved metric. COROLLARY 2.11. Every sphere bundle over §4 carries a metric with non-negative sectional curvature. Of particular interest are §3 bundles over §4 since Milnor discovered the first exotic spheres among these manifolds. It implies: COROLLARY 2.12. Of the 14 (unoriented) exotic 7-spheres, 10 carry a metric with non-negative curvature. The group of exotic spheres, under the group operation of connected sums, is isomorphic to Z28, but a change of orientation corresponds to taking an inverse. It is not known whether the remaining 4 exotic spheres carry non-negative curvature metrics as well. Proof of Theorem 2.10: Let the cohomogeneity one manifold Pr,s be given by the isotropy groups: H

= t::,.Q C {(eire, eie ). H, (d Se , eie . H}

C S3

X

S3,

where t::,.Q = {±(1, 1), ±(i, i), ±(j,j), ±(k, k),} is the quaternion group and eire = cos(rO) + i sin(rO) is an embedding of a circle into the unit quaternions S3. In order for H to be a subgroup of K±, we need to assume that r, s == 1 mod 4. We then have K±/H = S1 and thus Theorem 2.5 implies that Pr,s carries an S3 x S3 invariant metric with non-negative curvature. The subgroup S3 = S3 X { e} C S3 X S3 acts freely on Pr,s since its isotropy groups are the intersection of S3 x{e} with K± and H, which by construction are trivial. We now claim that Pr,s/ S3 is §4. To see this, observe that the second S3 factor induces a cohomogeneity one action with group diagram Q C {e ie . Q, eie . Q} C S3 on the quotient. The element -1 E S3 acts trivially and the effective version of the action has isotropy groups Z2 x Z2 c {S(0(2) 0(1)), S(O(l) 0(2))} c SO(3). But there is a well known linear action by SO(3) on §4 given by conjugation on the set of 3 x 3 symmetric real matrices with trace O. Since every matrix is conjugate to a diagonal one, it follows that the two singular orbits are given by symmetric matrices with two equal eigenvalues, positive for one and negative for the other, and the principal orbits by matrices with 3 distinct eigenvalues. One now easily

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checks that the isotropy groups are the same as for the above action and hence Pr,s/8 3 is equivariantly diffeomorphic to §4. Thus Pr,s can be viewed as an 8 3 principal bundle over §4. These are classified by an integer k, namely the Euler class of the bundle evaluated on a fixed orientation class of §4. To recognize which bundle it is, one observes that the Gysin sequence implies H4(E, Z) = Zlkl for such a bundle. For a co homogeneity one manifold one can compute the cohomology groups by using Meyer-Vietoris on the decomposition into the disk bundles D(B±). The disk bundles are homotopy equivalent to G / K ± and their intersection to G / H. Using well known methods for computing the cohomology groups of homogeneous spaces one shows that H4(Pr,s, Z) is a cyclic group of order (r2 - s2)/8. But for r, s == 1 mod 4 the values of (r2 - s2)/8 can take on any integer. Thus every 8 3 principal bundle over §4 is of the form Pr,s for some r, s. 8ince every 80(3) principal bundle over §4 is spin, i.e., has a lift to an 8 3 principal bundle, this implies Theorem 2.10 for k = 3. The case of k = 4 one obtains by repeating the above argument for G = 8 3 X 8 3 X 8 3 with K± again one dimensional and identity component of say K_ equal to (e ir18 , eir28 , e i8 ) with ri E Z. For principal bundles P over §4 with k > 4 it is well known that their structure group reduces to 80(4). Thus there exists an 80(4)-principal bundle pI with P = plxso(4)80(k) on which 80(k) acts on the right. Hence the lift of 80(3) to pI also lifts to P and commutes with 80(k). 0 We finally indicate how the proof of Cheeger's Theorem 2.4 fits into the above framework. Of the connected sums considered in his theorem, only cpn# - cpn admits a cohomogeneity one action. But a similar idea as in the proof of Proposition 2.8 applies to all cases. A rank one projective space M n with a small ball removed, is diffeomorphic to the disk bundle of the canonical line bundle over the projective space of one dimension lower. This bundle is a homogeneous disk bundle with boundary diffeomorphic to a sphere. One now uses the same "scaling up, scaling down" method as in the proof of Proposition 2.8 to show that these disk bundles have a metric with non-negative curvature which is a product near the boundary and has constant curvature one on the boundary. One can then glue together any two rank one projective spaces along this boundary. The methods described in the proof of Theorem 2.10 can also be applied to other 4-manifolds as base [44]: THEOREM 2.13 (Grove-Ziller). Every principal80(k) bundle P overCp2 which is not spin, i.e., W2(P) =1= 0, carries a cohomogeneity one action with codimension two principal orbits and hence an invariant metric with nonnegative curvature. Thus, so does every associated sphere bundle.

To prove this, one uses the linear cohomogeneity one action on CP2 given by 80(3) c 8U(3), which one easily verifies has group diagram Z2 C {8(0(2) 0(1)), 80(2)} c 80(3) and constructs a group diagram with G = 8 3 X 8 3 as above, but with H = {(±1, ±1), (±i, ±i)}. The topological

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 75

considerations needed to identify what bundle the cohomogeneity one manifold represents, are significantly more difficult. It also raises the following general question, which the above examples show is important in the context of cohomogeneity one manifolds. Given a principal L bundle P -+ M over a G-manifold M. When does the action of G on M have a commuting lift, i.e., a lift to an action of G, or possibly a cover of G, on the total space P, such that the lift commutes with L.

This problem has been studied extensively. However, apart from the general result that every action of a semi simple group admits a commuting lift to the total space of every principal circle or more generally torus bundle [63], the results seem to be difficult to apply in concrete cases. For a cohomogeneity one manifold Me with isotropy groups H C {K_, K+} C G, one has a natural description of the lifts to an L-principal bundle over Me in terms of the isotropy groups. Simply choose embeddings of K± into L x G such that they agree on H and are given in the second component by the original embeddings into G. The action by L x {e} is then clearly free, and the quotient is Me since the induced G action has the same isotropy groups. As long as one allows the action of G on Me to be ineffective, all lifts are described in this fashion. The difficulty is then to decide what the isomorphism type of this L-principal bundle is. Theorem 2.10 and Theorem 2.13 can be restated as saying that the linear actions of SO(3) on §4 and CJlD2 have a commuting lift to every principal SO(k) bundle, respectively principal SO(k) bundle which is not spin. In [44] one finds a classification of which cohomogeneity one actions on simply connected 4-manifolds M4 have a commuting lift to a given principal SO(k) bundle over M4. In particular, it is shown that in the spin case the action of SO(3) on CJlD2 only lifts to half of all SO(3) principal bundles. This shows the limitations of our principal bundle method which finds metrics on their total space with sec 2: o. A particularly interesting case of the above Problem are SO(k) principal bundles over §k since cohomogeneity one actions on spheres are numerous and have been classified in [52]. PROBLEM 2. Which cohomogeneity one actions on §n admit a commuting lift to a given SO(k) principal bundle over §n?

An answer to this question could potentially produce further sphere bundles over spheres, and hence possibly higher dimensional exotic spheres, which carry metrics with non-negative curvature. In light of the existence Theorem 2.5 and the non-existence Theorem 2.7, it is natural to pose the following somewhat vague but important: PROBLEM 3. How large is the class of cohomogeneity one manifolds that admit an invariant metric with non-negative curvature?

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Are there other obstructions, and how strong are the obstructions developed in the proof of Theorem 2.7? As far as existence is concerned, one would need to understand how to put non-negative curvature on cohomogeneity one manifolds without making the middle totally geodesic. In [69J Schwachhofer showed that for the adjoint action of SU(3) on §7 C 5u(3) there exist no invariant metric with non-negative curvature such that the middle is totally geodesic (for any homogeneous metric on the principal orbit!). But there of course exists an invariant metric with positive curvature. We end this section with the following natural problem. Many examples are obtained by taking a quotient of a compact Lie group, equipped with a left invariant metric with sec ~ 0, by a group acting by isometries. It thus seems to be important to know what all such metrics look like. PROBLEM 4. Classify all left invariant metrics with non-negative sectional curvature on compact Lie groups.

Surprisingly, the only examples known so far are obtained by combining the following: Cheeger deformations of a biinvariant metric along a subgroup K, i.e. the metric on G = G x K K induced by Q + If the subgroup is 3-dimensional, we can more generally consider the metric on G induced by Q+ where g is a left invariant metric on K with positive curvature. Finally, we can scale a biinvariant metric up in the direction of an abelian subalgebra as in Lemma 2.9. The only Lie groups where a complete answer is known, are SU(2) and U(2), see [15]' and with partial results for SO(4), [51J. In the latter paper it was also shown, as another application of Cheeger deformations, that every left invariant non-negatively curved metric g on a compact Lie group G can be connected by an "inverse linear" path of non-negatively curved left invariant metrics to a fixed biinvariant metric Q. Indeed, in the Cheeger deformation (2.1) applied to the right action of G on itself, we can let t --+ 00 and then the rescaled metric tgt --+ Q since the eigenvalues of tgt in terms of Q are l!~~i' Thus the main interest lies in deciding what derivatives are allowed at Q for an inverse linear path of left invariant metrics with sec ~ O. This approach is discussed in detail in [51J.

tQ.

tg

Topology of non-negatively curved manifolds For the following we assume that our manifolds are compact and simply connected. Recall the Bott conjecture which states that a non-negatively curved manifold is elliptic. Even rationally elliptic, i.e., where the condition on the Betti numbers of the loop space is only assumed for rational coefficients, already has strong consequences. By Sullivan's theory of minimal models in rational homotopy theory, rationally elliptic is equivalent to saying that there are only finitely many homotopy groups which are not finite. Rationally elliptic implies that the sum of the Betti number of M n is at most 2n , which is the optimal upper bound in Gromov's Betti number theorem. Furthermore, the Euler characteristic is non-negative (one half of

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 77

the Hopf conjecture), and positive if and only if the odd Betti numbers are O. Thus it is natural to conjecture that an even dimensional manifold with positive curvature has vanishing odd Betti numbers. See [37] where geometric consequences for rationally elliptic as well as for the remaining class of simply connected so-called rationally hyperbolic manifolds were first discussed in detail. In dimension four, rationally elliptic manifolds are homeomorphic to one of the known examples with non-negative curvature, i.e., one of §4, ClP2, §2 X §2 or ClP2# ± ClP2. It is natural to conjecture that a non-negatively curved manifold is indeed diffeomorphic to one of these, and that only the first two can admit positive curvature. In [65] it was shown that an elliptic 5-manifold is diffeomorphic to one of the known examples with non-negative curvature, i.e., one of §5, SU(3)j SO(3), §3 x §2 or the non-trivial §3 bundle over §2. Thus the Bott conjecture in dimension 5 states that a non-negatively curved manifold is diffeomorphic to one of these models, and it is natural to conjecture that only the first admits positive curvature. We remark that in dimension two a non-negatively curved manifold is diffeomorphic to §2 by Gauss-Bonnet and in dimension three to §3 by Hamilton's theorem [46]. We now describe some topological properties of the known examples with non-negative curvature. A homogeneous manifold M is 2-connected iff M = G j H with G and H semisimple and hence there are only finitely many such manifolds in each dimension. If it is not 2-connected, M is a torus bundle over a 2-connected one. In [77] it was shown that both statements also hold for biquotients. On the other hand, the class of biquotient manifolds is significantly larger than the class of homogeneous spaces. For example, in [78] it was shown that there exist infinitely many 6-dimensional biquotients of the form (S3)3 jj( Sl)3 with non-isomorphic rational cohomology rings. On the other hand, compact simply connected homogeneous spaces in dimension 6 are either diffeomorphic to a product of rank one symmetric spaces or to the Wallach manifold SU(3)jT2. The class of cohomogeneity one manifolds, including associated bundles and quotients, is again much larger than both. Indeed, there are infinitely many homotopy types of 2-connected cohomogeneity one manifolds, starting in dimension 7, since all §3 bundles over §4 admit non-negative curvature. In Section 6 one also finds an infinite family of 7 dimensional cohomogeneity one manifolds, depending on 4 arbitrary integers, which are 2-connected and have singular orbits of co dimension two. In [24] it was shown that there are infinitely many non-negatively curved manifolds lying in distinct cobordism classes. One starts with one of the principal SO(3) bundles P over §4 in Theorem 2.10 and considers the associated bundle P xSO(3) ClP2, where SO(3) C SU(3) acts linearly on ClP2. It clearly has non-negative curvature and a computation of the Pontryagin classes shows that they have distinct Pontryagin numbers and hence lie in

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different cobordism groups. On the other hand, this is not possible for homogeneous spaces and biquotients since circle bundles are the boundary of the associated disk bundle and hence have vanishing Pontryagin numbers. In [49J C. Hoelscher classified compact simply connected cohomogeneity one manifolds of dimension at most seven. In dimension 4 this was done in [64J (dimension 2 and 3 being trivial). In dimension 5, 6 and 7 there are many cohomogeneity actions with singular orbits of co dimension 2. In dimension 7 there are also some new families whose codimensions are not both two, where it is not known if they carry invariant metrics with nonnegative curvature. On the other hand, one also has the exotic Kervaire spheres in dimension 7, which by Theorem 2.7 does not admit an invariant metric with non-negative curvature.

3. Non-compact examples with non-negative curvature For non-compact manifolds one has the well known Soul Theorem [19J: THEOREM 3.1 (Cheeger-Gromoll). If M n is a non-compact manifold with a complete metric with sec 2: 0, then there exists a totally geodesic compact submanifold Sk such that M n is diffeomorphic to the normal bundle of Sk. The submanifold Sk is called the soul of Mn. A major open problem in this part of the subject is hence: PROBLEM 5. What vector bundles over compact manifolds with nonnegative curvature admit a complete metric with non-negative sectional curvature? This is particularly interesting for vector bundles over spheres. Any homogeneous vector bundle G XK V, where K acts orthogonally on a vector space V, clearly admits such a metric by O'Neill's formula. Thus T§n = SO(n + 1) x SO(n)]Rn also does. Every vector bundle over §n, n = 1,2,3 is a homogeneous vector bundle and hence carries non-negative curvature. In [68J Rigas showed that every vector bundle of §1! is stably, i.e., after taking the connected sum with a trivial bundle of sufficiently large dimension, a homogeneous vector bundle and hence carries non-negative curvature. As a consequence of Theorem 2.10 and Theorem 2.13 one obtains nonnegative curvature on the vector bundles P xSO(k) ]Rk associated to the principal bundles P: COROLLARY 3.2 (Grove-Ziller). Every vector bundle over §4, and every vector bundle over C]p>2 which is not spin, carnes a complete metric with non-negative sectional curvature. This class of vector bundles is quite large since they are classified by one arbitrary integer when the fiber dimension is three and by 2 if the fiber dimension is four.

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 79

As far as vector bundles are concerned over the remaining known 4manifolds with non-negative curvature, i.e. §2 x §2 and C;P2# ± CJP>2, most of them also admit non-negative curvature since their structure group reduces to a torus and circle bundles over these manifolds are known to admit nonnegative curvature, see [88, 77]. For vector bundles over §n, n > 4, one knows that all vector bundles over §5, and most of the vector bundles over §7, admit non-negative curvature [42]. But in both cases there are only finitely many such bundles. If the base does not have finite fundamental group, there are obstructions to the existence of complete metrics with non-negative curvature due to Ozaydin-Walschap [62], in the case where the soul is flat, and BelegradekKapovitch [6, 7] in general. The simplest examples are: THEOREM 3.3. Every orientable vector bundle over T2 or §3 x §l with non-negative curvature is trivial. In [6, 7] the authors give many more examples of vector bundles over ~ 1 which do not admit non-negative curvature. For example, if k ~ 4, there exist infinitely many vector bundles over C x Tk of every fixed rank at least two, whose total space do not admit any complete metric with non-negative curvature. No obstructions are known when the base is simply connected. Although it is known that for a given metric on M n any two souls are isometric, M can have two distinct non-negatively curved metrics with souls that are not even homeomorphic. In fact Belegradek [5] proved:

ex Tk with C compact and simply connected and k

THEOREM 3.4 (Belegradek). For each n ~ 5, there exist infinitely many complete Riemannian metrics on §3 x §4 X lRn with sec ~ 0 and pairwise non-homeomorphic souls. To prove this, consider the principal 80(3) bundle Pk -t §4 corresponding to k E 71'3(80(3)) ~ Z and let E'k = Pk XSO(3)lRn and S'k = Pk XSO(3)§n-l be the associated vector bundle and sphere bundle coming from the standard inclusion 80(3) C 80(n). Then the bundle b.*(Sf. x E"!:..k) , where b.: §4 -t §4 X §4 is the diagonal embedding, can be regarded as a bundle over §4 associated to the principal 80(3) x 80(3) bundle b.*(Pt x P::' k ) , which by Theorem 2.10 carries an invariant metric with non-negative curvature. On the other hand, it can also be regarded as an n-dimensional vector bundle over Sf. and its soul, since it is an associated vector bundle, is equal to the O-section Sf.. Now one uses surgery theory to show that for k == k' mod 12 and n ~ 5, the manifold b.*(Sf. x E"!:..k) is diffeomorphic to §3 x §4 X lR n and a computation of the Pontryagin classes shows that Sf. is homeomorphic to Sf., if and only if k = ±k'. 8ee [5] and [53] for further examples of this type.

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4. Known examples with positive curvature Known examples with positive curvature are surprisingly rare. What is even more surprising is that they are all obtained as quotients of a compact Lie group equipped with a biinvariant or a Cheeger deformation of a biinvariant metric divided by a group of isometries. One may view the following theorem as an explanation of why it is so difficult to find new examples [85]. THEOREM 4.1 (Wilking). If Mn admits a positively curved metric with an isometric action of cohomogeneity k 2: 1 with n > IS(k + 1)2, then M is homotopy equivalent to a rank one symmetric space. Thus, for any new examples, the larger the dimension, the bigger the cohomogeneity. This may increase the difficulty of computing its curvature tensor and estimating the sectional curvature. In fact, known examples exist only in low dimensions. They consist of certain homogeneous spaces in dimensions 6,7,12,13 and 24 due to Berger [10], Wallach [86], and AloffWallach [2], and of biquotients in dimensions 6, 7 and 13 due to Eschenburg [27, 28] and Bazaikin [3]. The purpose of this section is to discuss these examples. The main ingredient for all known examples is the following Cheeger deformation of a fixed biinvariant metric Q on G, of a type we already considered in Lemma 2.9. Let KeG be a closed Lie subgroup with Lie algebras t c 9 and 9 = t + m a Q-orthogonal decomposition. Recall that (G, K) is a symmetric pair if K is, up to components, the fixed point set of an involutive automorphism. For our purposes, the property that [m, m] C t is all that is needed, and is equivalent to being a symmetric pair if G / K is simply connected. For the biinvariant metric Q a O-curvature 2-plane is characterized by [X, Y] = O. The following deformation thus decreases the set of O-curvature 2-planes [28]. LEMMA 4.2 (Eschenburg). Let Qt be a left invariant metric on G defined by Qt = tQlt + Qlm' Then secQt 2: 0 as long as t ~ 1. If we assume in addition that (G, K) is a symmetric pair, X, Y span a O-curvature 2-plane of gt, for t < 1, if and only if [X, Y] = [Xt, Ye] = [Xm, Ym ] = O. PROOF. The metric Qt can be viewed as a Cheeger deformation as in (2.1) with respect to the right action of K on G and hence has non-negative curvature for t ~ 1. As we saw, the metric Q+ ~Q on G x K induces a metric of the form Qt with t = < 1 and the horizontal lift of X = X t + Xm E t+m = 9 is equal to X = (Xm + l!sXt, -l!sXt) E g+t Since the metric on G x K is biinvariant, a horizontal 2-plane spanned by X, Y has 0 curvature if and only if [X, Y] = O. Since the O'Neill tensor is also given in terms of Lie brackets, the same is true for the 2-plane spanned by X, Y E g. If G / K

S!l

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 81

is a symmetric pair, we have [m, m] C t, which, together with [t, t] [t, m] c m, easily implies the claim.

ct

and 0

Given Lie subgroups H eKe G, we define a homogeneous fibration KIH ----+ GIH ----+ GIK

gH -+ gK.

Using the Q-orthogonal decompositions 9 = t + m and t = ~ + p, we can identify the tangent spaces p ~ T(H)KIH , m ~ T(K)GIK and p + m ~ T(H)G I H. In terms of these identifications, we define a one parameter family of homogeneous metrics on GI H by

which scales the fibers of the homogeneous fibrations by t. Notice that they can also be viewed as a Cheeger deformation of the metric Q on GI H in direction of the left action of K on G I H. It is natural to ask, if one has such a metric with positive curvature on the base and on the fiber, when does gt have positive curvature. A partial answer to this question is given by [86]: PROPOSITION

4.3 (Wallach). Given a homogeneous fibration as above,

assume that: (a) The base (G, K) is a compact symmetric pair of rank one. (b) The metric on the fiber K I H induced by Q has positive curvature. (c) For any non-zero vectors X E P and Y E m we have [X, Y] =1= a. Then the metric gt with t < 1 has positive sectional curvature. PROOF. The interpretation as a Cheeger deformation implies that secgt 2 a for t :S 1. If we define the left invariant metric Qt on G by Qt = tQlt + Qlm, the projection G -+ G I H is a Riemannian submersion with respect to the metrics Qt and gt. Thus, if X, YEp + m ~ THGIH span a a-curvature 2-plane of gt, they span a a-curvature 2-plane of Qt as well and hence [X, Y] = [Xt, Yt] = [Xm , Ym ] = a by Lemma 4.2. The vectors Xt, Yt E P can be viewed as spanning a 2-plane of the fiber KI H and since it is assumed to have positive curvature, Xt, Yt must be linearly dependent. Similarly, since the base has positive curvature, X m , Ym are linearly dependent. Hence we can find a new basis of this plane with X E P and Y E m. But now condition (c) implies that [X, Y] = a is impossible and thus secgt > a. 0

The condition that [X, Y] =1= a is equivalent to the positivity of the curvature of the 2-plane spanned by X, Y, i.e. the vertizontal sectional curvatures. This condition is the fatness condition we will discuss in Section 6. Homogeneous examples with positive curvature

Homogeneous spaces which admit a homogeneous metric with positive curvature have been classified by Wallach in even dimensions [86] and by

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Berard-Bergery in odd dimensions [9]. We now describe these examples, leaving out the compact rank one symmetric spaces as well known. In all cases except for one, we will show that they indeed carry a metric with positive curvature as a consequence of Proposition 4.3. 1) The first class of examples are the homogeneous flag manifolds due to Wallach: W 6 = SU(3)jT2, W12 = Sp(3)jSp(1)3 and W24 = F4jSpin(8). They are the total space of the following homogeneous fibrations: §2 -+ SU(3)jT2 -+ C1P'2, §4 -+ Sp(3)jSp(1)3 -+ lHI1P'2, §8 -+ F 4 j Spin(8) -+ CaJID2. We now show that W 6 = SU(3)j T2 has positive curvature, the other cases being similar. Consider the inclusions T2 C U(2) C SU(3) giving rise to the above homogeneous fibration. Here we embed U(2) as the upper 2 x 2 block, i.e., U(2) = {diag(A, det.4) I A E U(2)}. A vector in m is of the form Y

=

(~v ~)

with v E C 2 and one easily shows that [A, Y] = Av+tr(A)v

for A E u(2). Hence if X E P C 5u(2) and Y E m, [X, Y] = 0 iff X = 0 or Y = O. This shows that part (c) of Proposition 4.3 holds. As for (a) and (b) the fiber and base are symmetric spaces of rank 1 and thus SU(3)j T2 has positive curvature. On the other hand, one easily sees that there are vectors X, Y E m + p with [X, Y] = O. Thus the biinvariant metric has non-negative curvature but with some O-curvature 2-planes. The Cheeger deformation deforms this metric into one with positive curvature. 2) The Berger space B13 = SU(5)j Sp(2) . SI admits a fibration ]R1P'5 -+ SU(5)jSp(2)· SI -+ Cr, coming from the inclusions Sp(2) . Sl C U(4) C SU(5). Here Sp(2) C SU(4) is the usual embedding and SI is the center of U(4). Furthermore, the fiber is U(4)j Sp(2) . Sl = SU(4)j Sp(2) . Z2 = SO(6)j 0(5) = ]R1P'5. A biinvariant metric on SU(5) restricts to a biinvariant metric on SO(6) which induces a metric with constant curvature on the fiber ]R1P'5. The base is clearly a symmetric space of rank 1 and condition (c) is verified as in the previous case. 3) The Aloff-Wallach spaces W;'q = SU(3)j diag(zP, zq, zp+q) , (p, q) = 1, form an infinite family. We claim that they have positive curvature iff pq(p + q) =I- O. They admit a fibration

§3jZp+q -+ Wp,q -+ SU(3)jT2, coming from the inclusions diag(zP, zq, zp+q) C U(2) C SU(3). Hence, as long as p+q =I- 0, the fiber is the lens space U(2)j diag(zP, zq) = SU(2)j diag(zP, zq) with zp+q = 1.

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 83

A vector in m again has the form Y =

(~v ~). 8ince the Lie algebra

of H is spanned by diag(ip, iq, -(ip + iq)), we can write an element in p as q + p )a z 2) ) wIth . a E lR X = dIag(A, - tr A) where A = .( + -z_ -1, q Pa and z E C. Hence [X, Y] = Av + i(q - p)av, i.e., A has an eigenvalue i(p - q)a if v =P O. But one easily shows that this is only possible when A = 0 or pq = O. Hence condition (c) is satisfied and since (a) and (b) clearly hold, Wp,q has positive curvature as long as pq(p + q) =P O. If on the other hand one of p, q or p + q is 0, say p = 0, one easily shows that the fixed point set of diag(l, -1, -1) E H = diag(zP, zq, zp+q) = diag(l, z, z) is equal to U(2)/ diag(z, z) = §2 X §1/7l..2 . 8ince fixed point sets of isometries are totally geodesic, and since §2 x §1/7l..2 does not carry a metric with positive curvature, these Aloff-Wallach spaces cannot carry a homogeneous metric with positive curvature. 4. Finally we have the Berger space: B7 = 80(5)/80(3). To describe the embedding 80(3) C 80(5), we recall that 80(3) acts orthogonally via conjugation on the set of 3 x 3 symmetric traceless matrices. This space is special since 80(3) is maximal in 80(5) and hence does not admit a homogeneous fibration. It is also what is called isotropy irreducible, i.e., the isotropy action of H on the tangent space is irreducible. This implies that there is only one 80(5) invariant metric up to scaling. Now a direct calculation is necessary in order to show that a biinvariant metric on 80(5) induces positive curvature on B7.

.

(i(2

Remark. a) The examples B7 and Bl3 were found by Berger [10] when classifying normal homogeneous metrics with positive curvature i.e., metrics on G/H induced by a biinvariant metric on G. But in [83] B. Wilking observed that the Aloff-Wallach space WI,1 is missing since it can be written as 8U(3) 80(3)/ U(2) where a biinvariant metric induces positive curvature. b) In [86] Wallach also proved that if one adds to the assumptions in Proposition 4.3 that the fiber is a symmetric pair as well, then the metrics 9t with 1 < t < 4/3 have positive curvature also. This applies to the flag manifolds and the Berger space B13. We do not know of a simple geometric proof of this fact, similar to the one we gave in Proposition 4.3. It is also mysterious that the limiting value 4/3 is the same as in Lemma 2.9. The number 4/3 shows up again if one considers homogeneous metrics on spheres, scaled in the direction of one of the Hopf fibrations with fibers §I, §3 or §7. As was shown in [81], they have positive curvature as long as the scale is less than 4/3. In the cases where the fiber is 3 or 7 dimensional, the proof again requires detailed curvature estimates. It would be interesting to obtain a uniform and less computational understanding why the number 4/3 appears in all 3 cases.

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After a classification of all homogeneous spaces which admit a metric with positive curvature, one can ask for the best homogeneous metric, i.e., the one with largest pinching O. It admits a large group acting by isometries. Indeed, G = SU(2) x SU(2) acting on SU(3) on the left and on the right, acts by isometries in the Eschenburg metric and commutes with the Sl action. Thus it acts by isometries on Ep and one easily sees that Ep/G is one dimensional, i.e., Ep is cohomogeneity one. A second family consists of the cohomogeneity two Eschenburg spaces Ea,b,c = Ek,l with k = (a, b, c) and l = (1,1, a + b + c). Here c = -(a+b) is the subfamily of Aloff-Wallach spaces. The action is free iff a, b, c are pairwise relatively prime and the Eschenburg metric has positive curvature iff, up to permutations, a ~ b ~ c > 0 or a ~ b > 0, c < -a. For these spaces G = U(2) acts by isometries on the right and Ea,b,c/G is two dimensional. For a general Eschenburg space G = T3 acts by isometries and Ek,!/G is four dimensional. In [39J it was shown that these groups G are indeed the id component of the full isometry group of a positively curved Eschenburg space (unless it is an Aloff-Wallach space). To see that the biquotient SU(3)// T2 has positive curvature, we can view it as an Sl quotient of the Eschenburg spaces diag(zP, zq, zp+q)\ SU(3)/ diag(l, 1, z2p +2q )-1 which has positive curvature when pq > O. 3) We finally have the 13-dimensional Bazaikin spaces B q , which can be considered as a generalization of the Berger space B13. Let q = (q1,"" qs) be a 5-tuple of integers with q = L: qi and define

where A E Sp(2) c SU(4) c SU(5). Here we follow the treatment in [89J of Bazaikin's work [3J. First, one easily shows that the action of Sp(2) . Sl is

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 87

free if and only if

for all permutations a E 8 5 . On SU(5) we choose an Eschenburg metric by scaling the biinvariant metric on SU(5) in the direction of U(4) C SU(5). The right action of Sp(2) . Sl is then by isometries. Repeating the same arguments as in the previous case, one shows that the induced metric on SU(5)11 Sp(2) . Sl satisfies sec >

°

if and only if qi

+ qj >

°

(or < 0) for all i < j.

The special case of q = (1,1,1,1,1) is the homogeneous Berger space. One again has a one parameter subfamily that is cohomogeneity one, given by Bp = B(1,1,1,1,2p-l) since U(4) acting on the left induces an isometric action on the quotient. It has positive curvature when p ~ 1. Unlike in the homogeneous case, there is no general classification of positively curved biquotients, except in the following cases. We call a metric on Gil H torus invariant if it is induced by a left invariant metric on G which is also right invariant under the action of a maximal torus. The main theorem in [28] states that an even dimensional biquotient Gil H with G simple and which admits a positively curved torus invariant metric is diffeomorphic to a rank one symmetric space or SU(3)IIT2. In the odd dimensional case he shows that Gil H with a positively curved torus invariant metric and G of rank 2 is either diffeomorphic to a homogeneous space or a positively curved Eschenburg space. In particular, the sufficient conditions in Proposition 4.4 are also necessary not only for Eschenburg metrics, but more generally torus invariant metrics. The classification of the remaining odd dimensional positively curved biquotients with rk G > 2 was taken up again in [12], where it was shown that if one assumes in addition that H = HI· H2 with HI of rank one and such that H2 has no rank one factors and operates only on one side of G, the manifold is diffeomorphic to a homogeneous space, an Eschenburg space, or a Bazaikin space with positive curvature. The case where G is not simple, on the other hand, is wide open. As we will see in Section 5, one obtains a large number of examples with almost positive curvature in this more general class of biquotients. Not much is known about the pinching constants of the positively curved metrics on biquotients. One easily sees that for a sequences of Eschenburg spaces Ek,l where (k/lkl, lllll) converges to ((1,1, -2)/v'6, (0,0,0)), the pinching of the Eschenburg metric converges to 1/37. In [25] W. Dickinson proved that for a general positively curved Eschenburg space Ek,l with its Eschenburg metric one has 8 :s; 1/37 with equality only for Wl,l. Furthermore, the pinching constant for the cohomogeneity one Eschenburg spaces Ep goes to when p -+ 00.

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Fundamental groups of positively curved manifolds

A cla.ssical conjecture of S.S. Chern states that, analogously to the Preismann theorem for negative curvature, an abelian subgroup of the fundamental group of a positively curved manifold is cyclic. This is in fact not true. The first counter examples were given in [71], and further ones in [38] (see also [4]): THEOREM 4.6. The following groups act freely on a positively curved manifold: (a) (Shankar) Z2 E9 Z2 on the Aloff Wallach space WI,1 and the cohomogeneity one Eschenburg space E2. (b) (Grove-Shankar) The group Z3 E9 Z3 on the Aloff Wallach space Wp,q if 3 does not divide pq(p + q).

In the ca.se of WI,1 = G/H = SU(3)/diag(z,z,z2) this follows since N(H)/ H = U(2)/Z(U(2)) = SO(3) acts isometrically in the Eschenburg metric and the right action of N (H) / H is free on any G / H. Thus a finite subgroup of SO(3) acts freely a.s well. Further example are given in [39] for the cohomogeneity two Eschenburg spaces. But there are no examples known where 1I"1(M) = Zp E9 Zp, p > 3 a prime, acts freely on a positively curved manifold. Topology of positively curved examples

In dimension 7 and 13 we have infinitely many homotopy types of positively curved manifolds since an Eschenburg spaces satisfies H 4 (Ek,L, Z) = Zr with r = Cf2(k) - Cf2(l) and for a Bazaikin spaces one ha.s H 6(B q , Z) = Zr with 8r = Cf3(q) - CfI(q)Cf2(q) where Cfi is the elementary symmetric polynomial of degree i. On the other hand, for a fixed cohomology ring, there are only finitely many known examples [20, 32]. A cla.ssification of 7-dimensional manifolds whose cohomology type is like that of an Eschenburg space wa.s obtained by Kreck-Stolz [58] in terms of certain generalized Eells-Kuiper invariants. They also computed these invariants for the Aloff Wallach spaces and obtained examples that are homeomorphic but not diffeomorphic. Kruggel [57] computed the Kreck-Stolz invariants for a general Eschenburg space in terms of number theoretic sums and Chinburg-Escher-Ziller [20] found further examples of this type. 4.7. One has the following examples with positive curvature: (a) The pair of Aloff Wallach spaces Wk,l with (k, I) = (56.788, 51.561) and (k, I) = (61.213, 18.561) and the pair of Eschenburg spaces Ek,l with (k; l) = (79, 49, -50; 0, 46, 32) and (k; 1) = (75, 54, -51; 0, 46, 32) are homeomorphic to each other but not diffeomorphic. (b) The pair of Aloff Wallach spaces Wk,l with (k, I) = (4.638.661, 4.056.005) and (k, l) = (5.052.965, 2.458.816) and the pair of

THEOREM

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 89

Eschenburg spaces Ek,l with (k; l) = (2.279, 1.603, 384; 0,0, 4.266) and (k; l) = (2.528, 939, 799; 0,0, 4.266) are diffeomorphic to each other but not isometric. The diffeomorphic pairs of Aloff-Wallach spaces give rise to different components of the moduli space of positively curved metrics [59], in fact these two metrics cannot be connected even by a path of metrics with positive scalar curvature. The diffeomorphic pair of Eschenburg spaces are interesting since such cohomogeneity two manifolds also carry a 3-Sasakian metric (see Section 6) and they give rise to the first known manifold which carries two non-isometric 3-Sasakian metrics. The situation for the Bazaikin spaces seems much more rigid. A computation of the Pontryagin classes and the linking form indicates, verified for the first 2 Billion examples, that Bazaikin spaces are all pairwise diffeomorphically distinct, see [32]. There is also only one Bazaikin space, the Berger space B 13 , which is homotopy equivalent to a homogeneous space. The Berger space B7 = SO(5)/ SO(3) plays a special role. It is, apart from spheres, the only known odd dimensional positively curved manifold which is 2-connected, which should be compared with the finiteness theorem by Fang-Rong and Petrunin-Tuschmann mentioned in Section 1. It is also, apart from the Hopf bundle, the only §3 bundle over §4 which is known to have positive curvature since it was shown in [35] that it is diffeomorphic to such a bundle. The topology of §3 bundles over Cp2 is studied in [31] where it is shown that they are frequently diffeomorphic to positively curved Eschenburg spaces when the bundle is not spin. It is thus natural to ask: PROBLEM 6. Does every §3 bundles over §4, and every §3 bundle over CP2 which is not spin, admit a metric with positive curvature. Do S3_ principal bundles over §4, and SO(3)-principal bundles over CP2 which are not spin, admit a metric with positive curvature invariant under the principal bundle action.

Notice that the existence in the latter case would imply infinitely many homotopy types of positively curved manifolds in dimension 6. Also recall that there are two SO(3)-principal bundles over Cp2 with such positively curved metrics and that all bundles in Problem 6 have a metric with nonnegative curvature.

5. Examples with almost positive or almost non-negative curvature As was suggested by Fred Wilhelm, there are two natural classes of metrics that lie in between non-negative curvature and positive curvature. In an initial step of deforming a non-negatively curved metric into one with positive curvature one can first make the curvature of all two planes at a point positive. We say that a metric has quasi positive curvature if there

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exists an open set such that all sectional curvatures in this open set are positive. In a second step one can try to deform the metric so that all sectional curvatures in an open and dense set are positive. We say that a metric with this property has almost positive curvature. It is natural to suggest that there should be obstructions to go from non-negative to quasi positive curvature and from almost positive to positive curvature, but that one should always be able to deform a metric from quasi positive to almost positive curvature. The first example of a manifold with almost positive curvature was given by P.Petersen and F.Wilhelm in [66], where it was shown that TI§4 has this property. In [82] it was shown that the Gromoll Meyer sphere 2:;7 = Sp(2)// Sp(l) admits a metric with almost positive curvature as well. See [29, 30] for a simpler proof for a slightly different metric on 2:;7. We now describe some remarkable examples of metrics with almost positive curvature due to Wilking [84]: THEOREM 5.1 (Wilking). Let M be one of the following manifolds:

(a) One of the projectivised tangent bundles PIRT(JRlpln), PcT(ClPln), or F\rnT(IHIlPln) . (b) The homogeneous space M:,~-I = U (n + 1) / H k,l with H k,l = {diag (zP, zq, A) Ilzl = 1, A E U(n - I)}, where pq < 0 and (p, q) = 1.

Then M carries a metric with almost positive curvature.

Here projectivised means that we identify a tangent vector v with ).V where). E JR, C or IHI respectively. Notice that in contrast to the known positively curved examples, these almost positively curved manifolds exist in arbitrarily high dimensions. Furthermore, in the case of n = 2 these are one of the known manifolds with positive curvature, namely PCT(Cp2) and F\rnT(IHIp2) are the flag manifolds W 6 and W I2 and M;,q is the Aloff Wallach space Wp,q' Notice that the unique Aloff Wallach space WI,O = WI,-I which does not admit a homogeneous metric with positive curvature, thus admits a metric with almost positive curvature. These examples also show that in general a metric with almost positive curvature cannot be deformed to positive curvature everywhere. Indeed, PIRT(JRIP'2n+l) is an odd dimensional non-orient able manifold and hence by Synge's theorem does not admit positive curvature. A particularly interesting special case is PIRT(JRIP'3) = JRp3 X JRp2 and ART(JRp7) = JRp7 X JR]p>6. If the manifold is compact and simply connected, it is not known whether an almost positively curved metric can be deformed to positive curvature. On the other hand, there either are obstructions, or the generalized Hopf conjecture on §3 x §2 is false. PROOF. We prove Theorem 5.1 in the simplest case of ART(JRp3) JRIP'3 x JRp2.

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 91

Define a left invariant metric 9 on G = S3 X S3 by scaling a biinvariant metric on 9 in the direction of the diagonal subgroup K = ~ S3 as in Lemma 4.2. Since G I K = S3 is a symmetric pair of rank one, and since (X, Y)e = !(X + Y, X + Y) and (X, Y)m = !(X - Y, -X + Y), Lemma 4.2 implies that a 0 curvature plane is spanned by vectors (u, 0), (0, u) with o =1= u E 1m 1HI. Here we regard S3 as the unit quaternions with Lie algebra ImlHI. G acts on Tl§3 = {(p, v) Ilpl = Ivl = 1, (u, v) = O} via (ql, q2)*(p, v) = (qlpq2"l,Qlvq2"l) and the isotropy group of (l,i) is equal to H = (e i9 ,ei9 ) and thus G I H = Tl§3. We can rewrite the homogeneous space G I H as a biquotient ~G\G x GI(l x H) since ~G\G x G = G. We now claim that the product metric 9 + 9 on G x G induces a metric with almost positive curvature on ~G\G x GI(l x H). For this, notice that each orbit of ~G acting on the left on G x G contains points of the form P = (a, b, 1, 1), a, b E S3. The vertical space, translated to the identity via left translation with (a, b, 1, 1), is equal to the direct sum of (Ad(a)v, Ad(a)w, v, w) with v, wE ImlHI and (0,0, i, i) . R If we set g(A, B) = Q(PA, B) where Q is a biinvariant metric on G, a horizontal vector is of the form (P- 1 (-Ad(a)v,-Ad(b)w),P- 1 (v,w)) with (v,w)..l(i,i). Since P clearly preserves 2-planes spanned by (v, 0), (0, v), a horizontal O-curvature plane is spanned by (P-l(-Ad(a)v,O), P- 1 (v,0)) and (p-l(O,-Ad(b)v), p-l(O,v)) with Ad(a)v = ±Ad(b)v. Thus ab either commutes or anticommutes with v E 1m 1HI and since also v..li, either ab..li or ab..ll. Hence points with O-curvature 2 planes lie in two hypersurfaces in GI H. Since the group L generated by (1, -1) and (j,j) normalizes H, and since the left invariant metric 9 is also right invariant under L, the quotient G I H . L inherits a metric with almost positive curvature and one easily sees that this quotient is -RRT(]R]p>3) = ]Rjp>3 X ]R]p>2. 0 See [84] for further examples. In [76] K. Tapp proved that the unit tangent bundles oU:]p>n, lHI]p>n and Ca]P>2, as well as the manifolds in Theorem 5.1 (b) with pq > 0, have quasi positive curvature. In [56] M. Kerin showed that all Eschenburg spaces have a metric with quasi positive curvature and that Eo, the unique cohomogeneity one Eschenburg space which does not admit a cohomogeneity metric with positive curvature, admits a metric with almost positive curvature. All known examples of almost positive curvature (and in fact all positively curved examples as well) can be described, after possibly enlarging the group, as so called normal biquotients, i.e., M = Gil H with metric on M induced by a biinvariant metric on G. B. Wilking showed in [84] that for such normal biquotients the exponential image of a O-curvature 2-plane is totally geodesic and flat. As was observed by K. Tapp, this remains true more generally for a Riemannian submersion G -+ M. It is a natural question to ask if the existence of an immersed flat 2-torus is sometimes an obstruction to deform a metric with non-negative curvature to one with positive curvature.

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In [84] one finds a number of natural open questions: • Can every quasi positively curved metric be deformed to almost positive curvature. • Can a quasi positively curved metric where the points with 0curvatures are contained in a contractible set be deformed to positive curvature. • Does an even dimensional almost positively curved manifold have positive Euler characteristic. Almost non-negative curvature

We say that a manifold M has almost non-negative curvature if there exists a sequence of metrics gi, normalized so that the diameter is at most 1, with sec(gi) ~ -I/i for all i E N. This includes the almost flat manifolds where sec(gi) ~ I/i as well. By Gromov's almost flat manifold theorem, the latter are finitely covered by a compact quotient of a nilpotent Lie group under a discrete subgroup. This is a much larger class of manifolds. Besides being invariant under taking products, it is also well behaved under fibrations. In [34], FukayaYamaguchi showed that: THEOREM 5.2 (Fukaya-Yamaguchi). The total space of a principal Gbundle with G compact over an almost non-negatively curved manifold is almost non-negatively curved as well. To see this, one puts a metric on the total space M such that the projection onto the base is a Riemannian submersion with totally geodesic fibers. Scaling the metric on M in the direction of the fibers then has the desired properties as the scale goes towards O. Thus every associated bundle P Xc F, where G acts isometrically on a non-negatively curved manifold F, has almost non-negative curvature as well. This applies in particular to all sphere bundles. As was shown in [70], this class also includes all cohomogeneity one manifolds: THEOREM 5.3 (Schwachh6fer-Tuschmann). Every compact cohomogeneity one manifold has almost non-negative curvature. PROOF. As was observed by B. Wilking, this follows easily by using a Cheeger deformation. If a compact group G acts by cohomogeneity one on M then M/G is either a circle or an interval. In the first case M carries a Ginvariant metric with non-negative curvature. In the second case we choose a G-invariant metric 9 on M which has non-negative curvature near the two singular orbits. This is clearly possible since a neighborhood of a singular orbit is a homogeneous disk bundle G XK D. We now claim that a Cheeger deformation gt with respect to the action of G on M has almost positive

EXAMPLES OF MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE 93

curvature as t -+ 00. Setting s = lit, the metric on M = M x GI!::1G is induced by a metric of the form 9 + sQ on M x G. Thus its diameter is clearly bounded as s -+ O. On the regular part, we can assume that our 2-plane is spanned by vectors X + OtT and Y, where X, Yare tangent to a principal orbit and T is a unit vector orthogonal to all principal orbits. As we saw in Section 2, the horizontal lift of these vectors to M x G under the Riemannian submersion M x G -+ M are of the form (sp-1 (sP- 1+Id) -1 X + OtT,-(sP- 1 +Id)-lX) and (sp-1(sp-1+Id)-lY,_(sp-1+Id)-ly). Only the first component can contribute to a negative curvature. The curvature tensor of this component goes to 0 with s, of order 2 if Ot i= 0 and of order 4 if Ot = O. On the other hand, I(X + OtT) 1\ YI~t goes to 0 with order 1 if ()( i= 0 and order 2 if Ot = O. Hence the negative part goes to 0 as s -+ O. 0 The main obstruction theorems for almost non-negative curvature are: • (Gromov) The Betti numbers are universally bounded in terms of the dimension. • (Fukaya-Yamaguchi [34]) The fundamental group contains a nilpotent subgroup of finite index. • (Kapovitch-Petrunin-Tuschmann [54]) A finite cover is a nilpotent space, i.e. the action of the fundamental group on its higher homotopy groups is nilpotent, Notice that a compact quotient of a nilpotent non-abelian Lie group has almost non-negative curvature, but does not admit a metric with nonnegative curvature. On the other hand, for simply connected manifolds there are no known obstructions which could distinguish between almost nonnegative and non-negative (or even positive) curvature. As was suggested by K. Grove, it is also natural to formulate the Bott conjecture more generally for almost non-negatively curved manifolds.

6. Where to look for new examples? There are two natural suggestions where one might look for new examples with positive sectional curvature. The first is given by a structure that almost all known examples share. They are the total space of a Riemannian submersion. If one considers the more general class where the base space of the submersion is allowed to be an orbifold, then all known examples share this property, see [33].

Fiber bundles A. Weinstein [87] considered fiber bundles 7r: M -+ B, where 7r is a Riemannian submersions with totally geodesic fibers. He called such a bundle fat if all vertizontal curvatures, i.e. the curvature of a 2-plane spanned by a horizontal and a vertical vector, are positive. This seemingly week

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assumption already places strong restrictions on the bundle. In fact, one has [21]:

6.1. (Derdzinski-Rigas) Every is a Hopf bundle. THEOREM

§3

bundle over

§4

which is fat

This negative result seems to have discouraged the study of fat fiber bundles until recently. On the other hand, as we saw in Proposition 4.3, most homogeneous examples of positive curvature are the total space of a fat bundle. 8ee [90] for a survey of what was known up to that point. In Theorem 6.7 we will see that there are infinitely many §3 orbifold bundles over §4 which are fat. It is thus natural and important to study fat bundles in this more general category. A natural class of metrics is given by a connection metric on a principal G-bundle 1l': P ---+ B, sometimes also called a Kaluza Klein metric. Here one chooses a principal connection (), a metric 9 on the base B, and a fixed biinvariant metric Q on G and defines:

gt(X, Y) = tQ((}(X), (}(Y))

+ g(1l'*(X), 1l'*(Y)).

The projection 1l' is then a Riemannian submersion with totally geodesic fibers isometric to (G, tQ). Weinstein observed that the fatness condition is equivalent to requiring that the curvature n of () has the property that nu = Q(n, u) is a symplectic 2-form on the horizontal space for every u E g. If G = 8 1 , this is equivalent to the base being symplectic. If one wants to achieve positive curvature on the total space, we need to assume, in addition to the base having positive curvature, that G = 8 1 , 8U(2) or 80(3). In [17] a necessary and sufficient condition for positive curvature of such metrics was given. The proof carries over immediately to the category of orbifold principal bundles, which includes the case where the G action on P has only finite isotropy groups. 6.2. (Chaves-Derdzinski-Rigas) A connection metric gt on an orbifold G-principal bundle with dim G :s; 3 has positive curvature, for t sufficiently small, if and only if THEOREM

(\7 xnu) (x, y)2 < li x n u I2kB(X, y), for all linearly independent horizontal vectors x, y and 0

i= u

E g.

Here kB(X, y) = g(RB(X, y)y, x) is the unnormalized sectional curvature and ixnu i= 0 is precisely the above fatness condition. We call a principal connection with this property hyperfat. The simplest examples of hyperfat principal connections are given by the Aloff-Wallach spaces Wk,l, considered as a circle bundle over 8U(3)jT2, since the fibers of a homogeneous fibration are totally geodesic. As mentioned earlier, Wl,l can also be considered as an 80(3) principal bundle over ClP2

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95

which thus carries an SO(3) hyperfat connection (a fact first observed in [16]). In the orbifold category one has many more examples. Recall that a metric is called 3-Sasakian if SU(2) acts isometrically and almost freely with totally geodesic orbits of curvature 1. Moreover, for U tangent to the SU(2) orbits and X perpendicular to them, X /\ U is required to be an eigenvector of the curvature operator R with eigenvalue 1. In particular the vertizontal curvatures are equal to 1, i.e., the bundle is fat. This gives rise to a large new class of fat orbifold principal bundles, see [13] for a survey. The dimension of the base is a multiple of 4, and its induced (orbifold) metric is quaternionic Kahler with positive scalar curvature. One easily sees that the condition on the curvature operator is equivalent to \7 xOu = 0 for all u E g. Hence we obtain the following Corollary, which was proved independently in [22]: COROLLARY 6.3. A 3-Sasakian manifold has positive sectional curvature, after the metric in the direction of the SU(2) orbits is scaled down sufficiently, if and only if the quaternionic Kahler quotient has positive sectional curvature. In [11] it was shown that a quaternionic Kahler manifold of dimension 4n > 4 has positive sectional curvature if and only if it is isometric to lliIlPn , which also holds for orbifolds. When the base is 4-dimensional, quaternionic Kahler is equivalent to being self dual Einstein and here an interesting new family of examples arises. In [14] it was shown that the cohomogeneity two Eschenburg spaces Ea,b,c = diag(za,zb,zc)\SU(3)/diag(I,I,za+b+C), with a, b, c positive and pairwise relatively prime, carry a 3-Sasakian metric with respect to the right action by SU(2). The quotients are weighted projective spaces and Dearricott examined their sectional curvatures in [23]: COROLLARY 6.4. (Dearricott) The principal connection for the 3-Sasakian manifold Ea,b,c with 0 < a ::; b ::; c is hyperfat if and only if c2 < abo Although the total space also carries an Eschenburg metric with positive curvature, the projection to the base in that case does not have totally geodesic fibers. Since many of the known examples are also the total space of sphere bundles, it is natural to study this category as well. A connection metric on a sphere bundle can be defined in terms of a metric connection \7 on the corresponding vector bundle. It induces a horizontal distribution on the sphere bundle and the fibers are endowed with a metric of constant curvature. An analogue of 6.2 for sphere bundles was proved in [75]: THEOREM 6.5 (Tapp). A connection metric on an orbifold sphere bundle E --+ B has positive curvature, for sufficiently small radius of the fibers, if and only if

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for all linearly independent x, y E TpB and u, VEEp, where we have set (R(u, v)x, y) = (R(x, y)u, v) Notice that R" (u, v)x =1= 0 for all u A v =1= 0, x =1= 0, means that (x, y) -+< R" (x, y)u, v > is nondegenerate for all u A v =1= o. But this is simply Weinstein's fatness condition for the sphere bundle. Known examples are the homogeneous Wallach flag manifolds W6, W 12 , W 24 • The Aloff-Wallach spaces Wp,q with p + q = 1 are hyperfat §3 bundles over C]p>2. Furthermore, the 3-Sasakian Eschenburg spaces Ea,b,c in Corollary 6.4, where one of a, b, c is even, are hyperfat §3 orbifold bundles and the associated §2 bundle Ea,b,c/ S1 -+ Ea,b,c/ SU(2) is hyperfat as well. If the base is a manifold, the condition \7 R = 0 is rather restrictive. For example, if the base is a symmetric space, it was shown in [45J that the bundle must be homogeneous. Homogeneous fat fiber bundles were classified in [8J. If the fiber dimension is larger than one, the base is symmetric. But if we assume in addition that the base has positive curvature, only the Wallach spaces remain. The fiber bundle structure for most of the Eschenburg spaces and Bazaikin spaces do not have totally geodesic fibers. The Berger space B7 = SO(5)/ SO(3) is also the total space of an SU(2) orbifold principal bundle over §4, but the fibers are again not totally geodesic. It is therefore also natural to examine warped connection metrics on the total space where the metric on the fiber is multiplied by a function on the base, see [75, 72J. But notice that the known fibrations of the Eschenburg and Bazaikin spaces, with fiber dimension bigger than one, are not of this form either. Connection metrics with non-negative curvature have also been studied in [73, 88J and [72J.

Cohomogeneity one manifolds A second natural class of manifolds where one can search for new examples, especially in light of Theorem 4.1, are manifolds with low cohomogeneity. Positively curved homogeneous spaces are classified, so cohomogeneity one manifolds are the natural next case to study. There are many cohomogeneity one actions on symmetric spaces of rank one. Among the examples of positive curvature discussed in Section 4, one has a number of other cohomogeneity one actions. As mentioned there, the positively curved Eschenburg spaces

E; = diag(z, z, zP)\ SU(3)/ diag(l, 1, Zp+2),p ~ 1, and the Bazaikin spaces B~3

= diag(z, z, z, z, z2p -1)\ SU(5)/ Sp(2) diag(l, 1, 1, 1, z2p +3),p ~ 1,

admit cohomogeneity one actions. Two further examples are the Wallach space Wl,1 = SU(3) SO(3)/ U(2) where SO(3) SO(3) acts by cohomogeneity

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one, and the Berger space B7 = SO(5)/ SO(3) with SO(4) c SO(5) acting by cohomogeneity one. In even dimensions, L. Verdiani [80] classified all positively curved cohomogeneity one manifolds. Here only rank one symmetric spaces arise. In odd dimensions, one finds a "preclassification" in [41]. In dimension seven, a new family of candidates arises: Two infinite families Pk, Qk, k ~ 1, and one isolated manifold R. The group diagram for Pk is similar to the one considered in Theorem 2.10:

whereas the one for Qk is given by

R is similar to Qk with slopes (3,1) on the left, and (1,2) on the right. THEOREM 6.6. (Verdiani, Grove- Wilking-Ziller) A simply connected cohomogeneity one manifold M with an invariant metric of positive sectional curvature is equivariantly diffeomorphic to one of the following: • An isometric action on a rank one symmetric space, • One 01 E7p' B13 p or B7 , • One of the 7-manifolds Pk, Qk, or R, with one of the actions described above.

The first in each sequence Pk, Qk admit an invariant metric with positive curvature since H = §7 and Q1 = wI 1 . Among the cohomogeneity one m'anifolds with co dimension 2 singular orbits, which all admit non-negative curvature by Theorem 2.5, are two families like the above Pk and Qk, but where the slopes for K± are arbitrary. It is striking that in positive curvature, with one exception, only the above slopes are allowed. The exception is given by the positively curved cohomogeneityone action on B 7 , where the isotropy groups are like those for P k with slopes (3, -1) and (-1,3). In some tantalizing sense then, the exceptional Berger manifold B7 is associated with the Pk family in an analogues way as the exceptional candidate R is associated with the Qk family. It is also surprising that all non-linear actions in the above Theorem, apart from the Bazaikin spaces B~3, are cohomogeneity one under a group locally isomorphic to S3 x S3. These candidates also have interesting topological properties. Qk has the same cohomology groups as Ek with H 4 (Qk,71) = 7lk. The manifolds Pk are all 2-connected with H 4 (Pk, 7l) = 7l 2k-1. Thus it is natural to ask: 7. Are any of the manifolds Qk, k > 1, diffeomorphic to a positively curved Eschenburg space? Are any of the manifolds Pk, k > 1, diffeomorphic to an §3 bundle over §4 ? PROBLEM

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Manifolds of this type are classified by their Kreck-Stolz invariants, but these can be very difficult to compute in concrete cases. A somewhat surprising property that these candidates also have is that they admit fat principal connections, in fact they admit 3-Sasakian metrics: THEOREM

6.7. Pk and Qk admit 3-Sasakian metrics which are orbifold

S3 -principal bundles over §4 respectively ClP2 . This follows [41] from the celebrated theorem due to Hitchin [48] that admits a family of self dual Einstein orbifold metrics invariant under the cohomogeneity one action by SO(3), one for each k ~ 1, which is smooth everywhere except normal to one of the singular orbits where it has angle 27r/k. One then shows that the induced 7-dimensional 3-Sasakian metric has no orbifold singularities, and by comparing the isotropy groups of the cohomogeneity one actions, it follows that they are equivariantly diffeomorphic to Pk and Qk' Unfortunately, the self dual Einstein metric on the base does not have positive curvature, unless k = 1, corresponding to the smooth metrics on §4 respectively ClP2 . So Corollary 6.3 does not easily yield the desired metrics of positive curvature on Pk and Qk. Hence these candidates surprisingly have both features, they admit cohomogeneity one actions, and are also the total space of an orbifold principal bundle. Both of these properties thus suggest concrete ways of finding new metrics with positive curvature. We thus end with our final problem. §4

PROBLEM 8. Do all manifolds Pk , Qkand R admit a cohomogeneity metric with positive curvature?

Notice that a positive answer for the manifolds Pk would give infinitely many homotopy types of positively curved 2-connected manifolds. Thus the pinching constants 8k , for any metric on Pk, would necessarily go to 0 as k -+ 00, and P k would be the first examples of this type. It is natural to suggest that the manifolds Ep, although not 2-connected, should have the same property since the pinching constant for the Eschenburg metric goes to 0 as p -+ 00.

References [1] A.V. Alekseevsy and D.V. Alekseevsy, G-manifolds with one dimensional or·bit space, Ad. in SOy. Math. 8 (1992), 1-31. [2] S. Aloff and N. Wallach, An infinite family of 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81(1975), 93-97. [3] Y. Bazaikin, On a family of 13-dimensional closed Riemannian manifolds of positive curvature, Siberian Math. J. 37 (1996),1068-1085. [4] Y. Bazaikin, A manifold with positive sectional cur"Vature and fundamental group Z3 EB Z3 , Siberian Math. J. 40 (1999), 834-836. [5] I. Belegradek, Vector bundles with infinitely many souls, Proc. Amer. Math. Soc. 131 (2003), 2217-2221.

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[6J I. Belegradek and V. Kapovitch, Topological obstructions to nonnegative curvature, Math. Ann. 320 (2001), 167-190. [7J I. Belegradek and V. Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory, J. Amer. Math. Soc. 16 (2003), 259-284 . [8J L. Berard-Bergery, Sur certaines fibrations d'espaces homogenes riemanniens, Compositio Math 30 (1975), 43-6l. [9J L. Berard Bergery, Les varietfs riemanniennes homogenes simplement connexes de dimension impaire d courbure strictement positive, J. Math. pure et appl. 55 (1976), 47-68. [lOJ M. Berger, Les varietfs riemanniennes homogenes normales simplement connexes d courbure strictement positive, Ann. Scuola Norm. Sup. Pisa 15 (1961), 179-246. [11J M. Berger, Trois remarques sur les varietf riemanniennes d courbure positive, C.R. Acad. Sci. Paris 263 (1966), 76-78. [12J R. Bock, Doppelquotienten ungerader dimension und positive Schnittkrumung, Dissertation, University of Augsburg, 1998. [13J C.P. Boyer and K. Galicki 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom. VI (1999), 123-184. [14J C.P. Boyer, K. Galicki and B. M. Mann The Geometry and Topology of 3-Sasakian Manifolds, J. fur die reine und angew. Math. 455 (1994), 183-220. [15J N. Brown, R. Finck, M. Spencer, K. Tapp and Z. Wu, Invariant metrics with nonnegative curvature on compact Lie groups, Can. Math. Bull., to appear. [16J L.M. Chaves, A theorem of finiteness for fat bundles, Topology 33 (1994),493-497. [17J L. Chaves, A. Derdzinski and A. Rigas A condition for positivity of curvature, Bol. Soc. Brasil. Mat. 23 (1992), 153-165. [18] J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Diff. Geom. 8 (1973), 623-628. [19J J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. [20J T. Chinburg, C. Escher, and W. Ziller, Topological properties of Eschenburg spaces and 3-Sasakian manifolds, Math. Ann. 339 (2007), 3-20. [21J A. Derdzinski and A. Rigas. Unftat connections in 3-sphere bundles over §4, Trans. of the AMS, 265 (1981),485-493. [22J O. Dearricott, Positive sectional curvature on 3-Sasakian manifolds, Ann. Global Anal. Geom. 25 (2004), 59-72. [23J O. Dearricott, Positively curved self-dual Einstein metrics on weighted projective spaces, Ann. Global Ana. Geom. 27 (2005), 79-86. [24J A. Dessai and W. Tuschmann, Nonnegative curvature and cobordism type, Math. Zeitschrift 257 (2007), 7-12. [25J W. Dickinson, Curvature properties of the positively curved Eschenburg spaces, Diff. Geom. and its Appl. 20 (2004), 101-124. [26J H.J. Eliasson, Die Krummung des Raumes Sp(2)/ SU(2) von Berger, Math. Ann. 164 (1966),317-323. [27J J.H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982), 469-480. [28J J.H. Eschenburg, Jilreie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrummten Orbitriiumen, Schriftenr. Math. Inst. Univ. Munster 32 (1984). [29] J.H. Eschenburg, Almost positive curvature on the Gromoll-Meyer 7-sphere, Proc. Amer. Math. Soc. 130 (2002), 1165-1167. [30J J.H. Eschenburg and M. Kerin, Almost positive curvature on the Gromoll-Meyer 7-sphere (corrected), Preprint 2007. [31J C. Escher and W. Ziller, On the topology of non-negatively curved manifolds, in preparation.

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[32] L.A. Florit and W. Ziller, On the topology of positively curved Bazaikin spaces, J. Europ. Math. Soc., to appear. [33] L.A. Florit and W. Ziller, Orbifold fibrations of Eschenburg spaces, Geom. Ded., to appear. [34] K. Fukaya and T. Yamaguchi, The fundamental group of almost nonnegatively curved manifolds, Ann. of Math. 136 (1992), 253-333. [35] S. Goette, N. Kitchloo and K. Shankar, Diffeom01phism type of the Berger space SO(5)/ SO(3), Amer. Math. J. 126 (2004), 395-416. [36] D. Gromoll and W. Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. (2) 100 (1974), 401-406. [37] K. Grove and S. Halperin, Contributions of rational homotopy theory to global problems in geometry, Publ. Math. I.H.E.S. 56 (1982), 171-177. [38] K. Grove and K. Shankar, Rank two fundamental groups of positively curved manifolds, J. Geometric Analysis, 10 (2000), 679-682. [39] K. Grove, R. Shankar and W. Ziller, Symmetries of Eschenburg spa.ces and the Chern problem, Special Issue in honor of S. S. Chern, Asian J. Math. 10 (2006), 647-662. [40] K. Grove, L. Verdiani, B. Wilking and W. Ziller, Non-negative curvature obstruction in cohomogeneity one and the Kervaire spheres, Ann. del. Scuola Norm. Sup. 5 (2006), 159-170. [41] K. Grove, B. Wilking and W. Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, J. Diff. Geom., to appear. [42] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. 152 (2000), 331-367. [43] K. Grove and W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Inv. Math. 149 (2002), 619-646. [44] K. Grove and W. Ziller, Lifting group actions and nonnegative curvature, Preprint 2007. [45] L. Guijarro, L. Sadun, and G. Walschap. Parallel connections over symmetric spaces, J. Geom. Anal. 11 (2001), 265-28l. [46] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255-306. [47] E. Heintze, The curvature of SU(5)/ Sp(2) S 1, Invent. Math. 13 (1971), 205-212. [48] N. Hitchin, A new family of Einstein metrics, Manifolds and geometry (Pisa, 1993), 190-222, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996. [49] C. Hoelscher, Cohomogeneity one manifolds in low dimensions, Ph.D. thesis, University of Pennsylvania, 2007. [50] H.-M. Huang, Some remarks on the pinching problems, Bull. Inst. Math. Acad. Sin. 9 (1981), 321-340. [51] J. Huizenga and K. Tapp, Invariant metrics with nonnegative curvature on SO(4) and other Lie groups, Mich. Math. J., to appear. [52] W.Y. Hsiang and B. Lawson, Minimal submanifolds of low cohomogeneity, J. Diff. Geom. 5 (1971), 1-38. [53] V. Kapovitch, A. Petrunin and W. 1\lschmann, Nonnegative pinching, moduli spaces and bundles with infinitely many souls, J. of Diff. Geometry 71 (2005), 365-383. [54] V. Kapovitch, A. Petrunin and W. Tuschmann, Nilpotency, almost nonnegative curvature and the gradient push, Ann. of Math., to appear. [55] V. Kapovitch-W. Ziller, Biquotients with singly generated rational cohomology, Geom. Dedicata 104 (2004), 149-160. [56] M. Kerin, Biquotients with almost positive curvature, Ph.D. thesis, University of Pennsylvania, in preparation. [57] B. Kruggel, Homeomorphism and diffeomorphism classification of Eschenburg spaces, Quart. J. Math. Oxford Ser. (2) 56 (2005), 553-577.

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[58] M. Kreck and S. Stolz, Some non diffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature, J. Diff. Geom. 33 (1991), 465-486. [59] M. Kreck and S. Stolz, Non connected moduli spaces of positive sectional curvature metrics, J. Amer. Math. Soc. 6 (1993), 825-850. [60] S. Lopez de Medrano, Involutions on Manifolds, Springer-Verlag, Berlin, 1971. [61] P. Miiter, Krummungserhohende Deformationen mittels Gruppenaktionen, Ph.D. thesis, Univerity of Miinster, 1987. [62] M. Ozaydin and G. Walschap, Vector bundles with no soul, Proc. Amer. Math. Soc. 120 (1994), 565-567. [63] R.S. Palais and T.E. Stewart, The cohomology of differentiable transformation groups, Am. J. Math. 83 (1961), 623-644. [64] J. Parker, 4-dimensional G-manifolds with 3-dimensional orbits, Pacific J. Math. 125 (1986), 187-204. [65] G.P. Paternain and J. Petean, Minimal entropy and collapsing with curvature bounded from below, Inv. Math. 151 (2003), 415-450. [66] P. Petersen and F. Wilhelm, Examples of Riemannian manifolds with positive curvature almost everywhere, Geom. Topol. 3 (1999), 331-367. [67] T. Piittmann, Optimal pinching constants of odd dimensional homogeneous spaces, Invent. math. 138, (1999),631-684. [68] A. Rigas, Geodesic spheres as generators of the homotopy groups of 0, BO, J. Differential Geom. 13 (1978), 527-545. [69] L. Schwachhofer, Lower curvature bounds and cohomogeneity one manifolds, Differential Geom. Appl. 17 (2002), 209-228. [70] L. Schwachhofer and W. Thschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), 59-91. [71] K. Shankar, On the fundamental groups of positively curved manifolds, J. Diff. Geom. 49 (1998), 179-182. [72] K. Shankar, K. Tapp, and W. Thschmann, Nonnegatively and positively curved metrics on circle bundles, Proc. of the AMS, 133 (2005), 2449-2459. [73] M. Strake and G. Walschap, Connection metrics of nonnegative curvature on vector bundles, Manuscr. Math., 66 (1990), 309-318. [74] E. Straume, Compact connected Lie transformation groups on spheres with low cohomogeneity. 1,11, Mem. Amer. Math. Soc. 119 (1996), no. 569; 125 (1997), no. 595. [75] K. Tapp, Conditions for nonnegative curvature on vector bundles and sphere bundles, Duke Math. J. 116 (2003), 77-101. [76] K. Tapp, Quasipositive curvature on homogeneous bundles, J. Diff. Geom. 65 (2003), 273-287. [77] B. Totaro, Cheeger manifolds and the classification of biquotients, J. Diff. Geom. 61 (2002), 397-451. [78] B. Totaro, Curvature, diameter, and quotient manifolds, Math. Res. Lett. 10 (2003), 191-203. [79] F.M. Valiev, Precise estimates for the sectional curvatures of homogeneous Riemannian metrics on Wallach spaces, Sib. Mat. Zhurn. 20 (1979), 248-262. [80] L. Verdiani, Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature, J. Diff. Geom. 68 (2004), 31-72. [81] L. Verdiani, W. Ziller, Positively curved homogeneous metrics on spheres, Preprint 2007. [82] F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), 519-560. [83] B. Wilking, The normal homogeneous space SU(3) SO(3)/ U(2) has positive sectional curvature, Proc. Amer. Soc. 127 (1999), 1191-1194. [84] B. Wilking, Manifolds with positive sectional curvature almost everywhere, Inv. Math. 148 (2002), 117-141.

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[85] B. Wilking, Positively curved manifolds with symmetry, Ann. of Math. 163 (2006), 607-668. appear. [86] N. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295. [87] A. Weinstein, Fat bundles and symplectic manifolds, Adv. in Math. 37 (1980), 239-250. [88] D. Yang, On complete metrics of nonnegative curvature on 2-plane bundles, Pacific J. Math. 171 (1995), 569-583. [89] W. Ziller, Homogeneous spaces, biquotients, and manifolds with positive curvature, Lecture Notes 1998, unpublished. [90] W. Ziller, Fatness revisited, Lecture Notes 2000, unpublished. UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, PA

E-mail address:

wziller~ath.upenn.edu

19104

Surveys in Differential Geometry XI

Perelman's Stability Theorem Vitali Kapovitch ABSTRACT. We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv ~ k Gromov-Hausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i.

1. Introduction

A fundamental observation of Gromov says that the class of complete n-dimensional Riemannian manifolds with fixed lower curvature and upper diameter bounds is precompact in Gromov-Hausdorff topology. The limit points of this class are Alexandrov spaces of dimension ~n with the same lower curvature and upper diameter bounds. Given a sequence of manifolds Mi in the above class converging to an Alexandrov space X it's interesting to know what can be said about the relationship between topology of the limit and elements of the sequence. The main purpose of this paper is to give a careful proof of the following Theorem of Perelman which answers this question in the situation when dimX = n. STABILITY THEOREM 1.1. Let xn be a compact n-dimensional Alexandrov space of curv ~ Ie Then there exists an E = E(X) > 0 such that for any n-dimensional Alexandrov space yn of curv;;:: /'i, with dC-H(X, Y) < E, Y is homeomorphic to X.

To prove the Stability Theorem, it's clearly enough to show that if Xr is a sequence of compact n-dimensional Alexandrov spaces with curv ;;:: /'i" diam ~ D converging in Gromov-Hausdorff topology to a space X of dimension n, then Hausdorff approximations Xi ~ X can be approximated by homeomorphisms for all large i. It is this statement that will be proved in the present paper. 2000 AMS Mathematics Subject Classification: 53C20. Keywords: nonnegative curvature, nilpotent. The author was supported in part by an NSF grant. ©2007 International Press 103

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A proof of the Stability Theorem was given in [16]. However, that paper is very hard to read and is not easily accessible. We aim to provide a comprehensive and hopefully readable reference for Perelman's result. Perelman also claims to have a proof of the Lipschitz version of the Stability Theorem which says that the stability homeomorphisms can be chosen to be bi-Lipschitz. However, the proof of that result has never been written down and the author has never seen it (although he very much wants to). It is also worth pointing out that the Stability Theorem in dimension 3 plays a key role in the classification of collapsing of 3-manifolds with a lower curvature bound by Shioya and Yamaguchi [26, 27] which in turn plays a role in Perelman's work on the geometrization conjecture. However, as was communicated to the author by Kleiner, for that particular application, if one traces through the proofs of [26, 27] carrying along the additional bounds arising from the Ricci flow, then one finds that in each instance when a 3-dimensional Alexandrov space arises as a Gromov-Hausdorff limit of smooth manifolds, it is in fact smooth, and after passing to an appropriate subsequence, the convergence will also be smooth to a large order. For such convergence the stability theorem is very well known and easily follows from Cheeger-Gromov compactness. Therefore Perelman's stability theorem is unnecessary for the application to geometrization. The Stability Theorem immediately implies the following finiteness theorem due to Grove-Petersen-Wu which was originally proved using controlled homotopy theory techniques. THEOREM 1.2. [7] The class of n-dimensional Riemannian manifolds (n =/: 3) with sectional curvature 2 k, diameter ~ D and volume 2 v has only finitely many topological (differentiable if n =/: 3,4) types of manifolds.

The restriction n =/: 3 in this theorem comes from the use of controlled homotopy results which do not hold in dimension 3. Using the Stability Theorem one gets that homeomorphism finiteness holds in all dimensions including dimension 3. Alternatively, the homeomorphism finiteness in dimension 3 follows from an earlier result of Grove-Petersen [6] that the above class has finitely many homotopy types and the fact that in dimension 3, a fixed homotopy type of closed manifolds contains only finitely many homeomorphism types (this a direct consequence of the geometrization conjecture). Let us mention that the proof of Stability Theorem 1.1 presented here is fundamentally the same as the one given in [16]. However, as was pointed out by Perelman in [17], the proof can be simplified using the constructions developed in [17] and [18]. We carry out these simplifications in the present paper. In [18] Perelman introduced the following notion: A function f: jRk -+ jR is called DC if it can be locally represented as a difference of two concave (or, equivalently, semiconcave) functions. It is easy

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to see that DC is an algebra and is DC if both f and g are DC and g is never zero. A map F: ]Rk -+]Rl is called DC if it has DC coordinates. It is also easy too see (see [18]) that if F: ]Rk -+]Rl and G: ]Rl -+ ]Rm are DC then so is G 0 F. This allows for an obvious definition of a DC geometric structure on a topological manifold. Despite the ease and naturality of its definition, to the best of the author's knowledge, this type of geometric structure has never been studied. In particular, the relationship between DC structures and classical geometric structures such as TOP, PL, smooth or Lipschitz is not at all understood. It is easy to see that a PL-manifold is DC and a DC manifold is Lipschitz but that's all one can say at the moment. The notion of DC functions and DC homeomorphisms also makes sense on Alexandrov spaces since Alexandrov spaces admit an abundance of semiconcave (and hence DC) functions coming from distance functions. In [18J, Perelman showed that the set of regular points of an Alexandrov space has a natural structure of a DC manifold. The above discussion naturally leads to the following question. QUESTION 1.3. Let Xi be a non collapsing sequence of Alexandrov spaces with curv ~ r;" diam ~ D Gromov-Hausdorff converging to an Alexandrov space xn. Is it true that Xi is DC-homeomorphic to X for all large i ? Or more weakly. Suppose Mi is a non collapsing sequence of Riemannian manifolds with sec ~ r;" diam ~ D Gromov-Hausdorff converging to an Alexandrov space xn. Is it true that Mi are DC homeomorphic to each other for all large i ?

Let us say a few words about the proof of the stability theorem. One of the main ingredients is the Morse theory for functions on Alexandrov spaces. The starting point is based on the following simple observation: Given k + 1 nonzero vectors in ]Rn with pairwise angles bigger than 1["/2, any k of them are linearly independent. Motivated by this, we'll say that a map f = (iI, .. ·, fk) (with coordinates given by distance functions fi = d(·, ai)) from an Alexandrov space to ]Rk is regular at a point p if there exists a point ao such that the comparison angles at p for the triangles D.aipaj are bigger than 1["/2 for all O~i=l=j~k.

The fibration theorem [16, 17J shows that just like for regular points of smooth functions on differentiable manifolds, a map from an Alexandrov space X to ]Rk is a topological submersion on the set of its regular points. In particular, if k = dimX then it's a local homeomorphism. It's well known that for any point p in an Alexandrov space, d(·,p) has no critical points in a sufficiently small punctured ball B (p, E) \ {p} . Therefore, by the fibration theorem, B(p, E) is homeomorphic to the cone over the metric sphere S (p, E) . The fibration theorem (and, perhaps, even more importantly, its proof) plays a key role in the proof of the stability theorem. In particular it implies

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that Alexandrov spaces are stratified topological manifolds. The k-dimensional strata of X is basically equal to the set of points P in X such there exist k + 1 (but not k + 2) vectors in TpX with pairwise angles bigger than 7r /2. Another important tool in the proof of stability is the gluing theorem derived from deformations of homeomorphisms results of Siebenmann [24]. Given a noncollapsing sequence of Alexandrov spaces Xi -+ X, it says that one can glue Hausdorff close stability homeomorphisms defined on a fixed open cover of the limit space X to make a global nearby homeomorphism, Le. it reduces the stability theorem to the local situation. To construct local stability homeomorphisms near a point P E X one argues by reverse induction on the dimension k of the strata containing p. The base of induction follows from the fact mentioned above that a map f: xn -+ jRn is a local homeomorphism near a regular point. The general induction step is quite involved and can not be readily described in an introduction. To show some of its flavor, we will however try to say a few words about the last step of induction from k = 1 to k = O. Given a point P in the zero strata (Le. with the diameter of the space of directions at P at most 7r /2), look at a ball B(p, r) where again r is so small that d(·, p) is regular in B (p, r) \ {p}. One can show that the same holds true for appropriately chosen lifts Pi E Xi of p for all large i provided r is sufficiently small. This means that both B(p,r)\{p} and B(Pi,r)\{Pi} consist of points lying in the union of strata of dimensions ~ 1. Fixing a small 8 «: r, the induction assumption implies the existence of homeomorphisms of the annuli B(p, r)\B(p, 8) and B(pi' r)\B(pi' 8) close to the original Hausdorff approximations. In particular, metric spheres S(p, 8) and S(pi,8) are homeomorphic. On the other hand, the fibration theorem implies that B(p,8) is homeomorphic to the cone over S(p,8) and B(pi,8) is homeomorphic to the cone over S(P'i' 8) for all large i. Gluing these homeomorphisms we obtain homeomorphisms B(p, r) onto B(Pi, r) which will be close to the original Hausdorff approximations since 8 «: r. The general induction step is a rather nontrivial fibered version of the above argument. It is carried out in Local Stability Lemma 7.9. with the main geometric ingredient provided by Lemma 6.9. For technical reasons, one has to work with more general semiconcave functions than just distance functions. An important role here is played by a technical construction from [17] of strictly concave functions obtained by manipulating distance functions. Let us briefly describe the structure of this paper. In section 3 we give a simplified proof of the stability theorem in the special case of limits of closed Riemannian manifolds using controlled homotopy theory techniques. In section 4 we give the background on necessary topological results on stratified spaces and some geometric constructions on Alexandrov spaces. In sections 5 and 6 we define admissible maps and their regular points and show that they satisfy similar properties to regular

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points of smooth maps between manifolds. Section 7 contains the proof of the stability theorem. In section 8 we use the Stability Theorem to prove a finiteness theorem for Riemannian submersions. In section 9 we generalize the Stability Theorem to show that stability homeomorphisms can be chosen to respect stratification of Alexandrov spaces into extremal subsets. Throughout this paper we will assume some knowledge of Alexandrov geometry (see [2] as a basic reference). For a more recent treatment we strongly recommend [20] in the same volume. We will also rely on the Morse theory results and constructions from [17] in particular the fibration theorem. Perelman later generalized these results to a larger class of Morse functions in the unpublished preprint [18] (cf. also [20]). However, these generalizations are not needed for the proof of the Stability theorem presented here. Various references to [18] throughout this paper are only given for bibliographical completeness. Acknowledgements: The author is profoundly grateful to Anton Petrunin and Fred Wilhelm for numerous conversations and suggestions regarding the preparation of this paper. The author would like to thank Alexander Lytchak for discussions and suggestions regarding the controlled homotopy theory proof of the stability theorem for manifolds.

2. Notations

For an open subset U in a space X and a subset A c U we'll write U if A c U. We'll denote by Alexn(D, 1'1" v) the class of n-dimensional Alexandrov spaces of curv ~ 1'1" diam :::;; D, vol ~ v. Similarly, we'll denote by Alexn(D, 1'1,) the class of n-dimensional Alexandrov spaces of curv ~ 1'1" diam :::;; D and by Alexn(K) the class of n-dimensional Alexandrov spaces of curv ~ K. For a point v = (VI, ... , Vk) E ]Rk and r > 0 we'll denote by I:(v) the cube {x E]Rk I IXi - vii:::;; r for all i}. For an Alexandrov space E of curv ~ 1 we'll often refer to its points as vectors and to distances between its points as angles. For a metric space X with diam :::;; 7r, we'll denote by ex the Euclidean cone over X. Let be the vertex of ex. For u E ex we'll denote lui = d(o, u). We'll call a function h: ex -+ ]R 1-homogeneous if h(t· x) = t· h(x) for any t E ]R, x EX. For any space X we'll denote by KX the open cone on X and by Rx the closed cone on X (i.e. the join of X and a point). For two points p, q in an Alexandrov space, we'll denote by an element of EpX tangent to a shortest geodesic connecting p to q. We'll denote by 1tZ the set of all such directions. For three points x,p, y in an Alexandrov space X of curv ~ 1'1" we'll denote by L.xpy the comparison angle at p, i.e the angle Lipfj in the triangle A

(S

°

tZ

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xfrfj in the complete simply connected space of constant curvature

K with d(x,y) = d(x,y), d(p,y) = d(p,y), d(x,p) = d(x,p) , We will also often use the following convention. In the proofs of various theorems we'll denote by c or C various constants depending on the dimension and the lower curvature bound present and which sometimes will depend on additional parameters present. When important this dependence will be clearly indicated. We will denote by x various continuous increasing functions x: R+ ---+ lR+ satisfying x( 0) = O. By o( i) we will denote various positive functions on Z+ such that o(i) .--+0. ~-+oo We'll write o(ilc) to indicate a function which depends on an extra parameter c and satisfies o(ilc) .--+0 for any fixed c. Sometimes we'll use ~-+oo

the same convention for x(t5lc).

3. Simplified proof of the Stability Theorem for limits of manifolds The author is grateful to A. Lytchak for bringing to his attention the fact, that using todays knowledge of the local structure of Alexandrov spaces, a simple proof of the stability theorem for n ;;:=: 4 can be given for the special case of limits of Riemannian manifolds. The proof uses controlled homotopy theory techniques employed in [7]. Let us briefly describe the argument the general outline of which was suggested to the author by A. Lytchak. It is now well-known that the class Alexn(D, K, v) has a common contractibility function (see e.g. [21] or [17]). Therefore, by [7, Lemma 1.3], if Xi~-If xn is a convergent sequence in Alexn(D, K, v), then ~-+oo

Xi

is o(i)-

homotopy equivelent to X for all large i (see [7]). This means that there are homotopy equivalences Ii: Xi ---+ X with homotopy inverses hi: X ---+ Xi such that Ii 0 hi ~ Idx and hi 0 Ii ~ Idxi through homotopies Fi: X x [0, 1] ---+ X and Gi: Xi x [0, 1] ---+ Xi such that all the point trajectories of Fi and Ii 0 Gi have o(i)-small diameters in X. In fact, in our case, one can make Ii, hi to be o(i)-Hausdorff approximations. It is relatively easy to show (see [7]) that if Mi E Alexn(D, K, v) is a sequence of closed Riemannian manifolds Gromov-Hausdorff converging to a space X, then X is a homology manifold. However, at the time of the writing of [7] the local structure of Alexandrov spaces was not well understood and it was therefore not known if X is an actual manifold. This made the application of controlled topology techniques employed in [7] fairly tricky. We first show that X is a manifold. We'll need the following result from [11] (cf. [9]).

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LEMMA 3.1. Suppose Mi E Alexn(D, 1>;, v) is a sequence of Riemannian manifolds Gromov-Hausdorff converging to a space X. Let p E X be any point. Then there exists a 15 > 0, a 1- Lipschitz function h: X -+ R, strictly convex on B(p,c5) such that h(p) = 0 is a strict local minimum of h and a sequence of smooth 1-Lipschitz functions hi: Mi -+ R uniformly converging to h such that hi is strictly convex on B(pi'c5) (where Pi E Mi converges to p) for all large i. This lemma together with the fibration theorem of Perelman [17] easily implies that a noncollapsing limit of Riemannian manifolds with lower sectional curvature bound is a topological manifold. LEMMA 3.2. Let Mi E Alexn(D, 1>;, v) be a sequence of closed Riemannian manifolds converging to an Alexandrov space X. Then X is a topological manifold. PROOF. We argue by induction on dimension. The cases of n ~ 2 are easy and are left to the reader as an exercise. Suppose n ) 3. Let p E X be any point and let h be the function provided by Lemma 3.1. Then for some small f > 0 the set {h ~ f} is a compact convex subset of X and {hi ~ f} is compact convex in Mi for large i. Obviously, {hi ~ f} ~-¥ {h ~ f}. By [19, Theorem 1.2], we have t-+oo

that {hi = f} ~-¥ {h

= f} with respect to their induced inner metrics. By t-+oo the Gauss Formula, sec( {hi = f}) ) I>; for all large i and hence {h = f} is an (n - 1) -dimensional Alexandrov space of curv ) I>; which is a manifold by induction assumption. Moreover, since hi is smooth, strictly convex with unique minimum, {hi = f} is diffeomorphic to sn-l for large i and hence {h = f} is a homotopy (n - 1) -sphere. Since h is strictly convex in {h ~ f}, it has no critical points in {h ~ f}\{p}. Therefore, by [17, Theorem 1.4], {h < f} is homeomorphic to the open cone over {h = f} . If n = 3 then {h = f} is obviously homeomorphic to S2 which means that {h < f} is homeomorphic to R3. If n = 4 then {h = f} is a homotopy 3-sphere and a manifold. By the work of Freedman [5, Corollary 1.3], this implies that the cone over {h = f} is homeomorphic to R4. If n ) 5 then {h = f} is a homotopy sn-l and a manifold and hence is homeomorphic to sn-l by the Poincare conjecture. 0 REMARK 3.3. As was suggested to the author by A. Lytchak, a different proof of Lemma 3.2 can be given for n ) 5 by verifying that X satisfies the disjoint disk property and thus is a manifold by a result of Edwards [4]. This method is used in [14] to prove a version of stability for limits of manifolds with curvature bounded above. However, the author prefers his own argument given above.

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Now that we know that X is a manifold and Ii: Mi -+ X are o(i)homotopy equivalences, we can apply the results from [3] for n ~ 5 and [23] together with [3] for n = 4, which say that under these conditions lis can be o( i) - approximated by homeomorphisms for all large i. The same holds in dimension 3 by [10] but only modulo the Poincare conjecture. However, it is possible (and would certainly be a lot more preferable) that one can use the fact that all the subsets {hi ~ €} are actually topological balls and not merely contractible to give a proof in dimension 3 which does not rely on the Poincare conjecture. REMARK 3.4. By using relative versions of controlled homotopy theory results from [23] and [3] mentioned above, it should be possible to generalize the above proof to the case of pointed Gromov-Hausdorff convergence of manifolds. This would amount to the manifold case of Theorem 7.11 below. Alternatively, one can handle the pointed case as follows. Suppose we have a pointed convergence (Mlt,qn) -+ (Xn,q) where M? are (possibly noncompact) Riemannian manifolds of sec ~ K,. The proof of Lemma 3.2 is obviously local and hence X is a topological manifold. Let p be any point in X. Let h and hi be the functions constructed in the proof of Lemma 3.2. Let Yi be the double of {hi ~ €} and Y be

the double of

sn

{h ~ €}.

Obviously Yi~-1fY and ~-+oo

Yi

is homeomorphic to

for large i. While the metric on Yi is not smooth along the boundary of {hi ~ €}, it's easy to see that the proof of Lemma 3.2 still works for the convergence Yi -+ Y and hence Y is a closed topological manifold. By the same controlled homotopy theory results used earlier, we conclude that Hausdorff approximations Y -+ Yi can be o( i) -approximated by homeomorphisms 9i: Y -+ Yi for all large i. Restricting 9i to {h < €/2} we obtain an open embedding of {h < €/2} into Mi, which is o(i)-close to the original Hausdorff approximation (M, q) -+ (Mi' qi). Finally, by using topological gluing theorem 4.11 below, for any fixed R > 0 we can glue finitely many such local homeomorphisms to get an open embedding B( q, R) -+ Yi, which is o( i) -Hausdorff close to the original Hausdorff approximation (M, q) -+ (Mi' qd.

4. Background 4.1. Stratified spaces. Most of the material of this section is taken with almost no changes from [16] as no significant simplifications or improvements of the exposition seem to be possible. DEFINITION 4.1. A metrizable space X is called an MSC-space (space with multiple conic singularities) of dimension n if every point x E X has a neighborhood pointed homeomorphic to an open cone over a compact (n - 1) dimensional MCS space. Here we assume the empty set to be the unique (-1) -dimensional MCS-space.

PERELMAN'S STABILITY THEOREM

III

REMARK 4.2. A compact O-dimensional MCS-space is a finite collection of points with discrete topology. REMARK 4.3. An open conical neighborhood of a point in an MCS-space is unique up to pointed homeomorphism [13]. It easily follows from the definition that an MCS space has a natural topological stratification. We say that a point p E X belongs to the l-dimensional strata Xl if l is the maximal number m such that the conical neighbourhood of p is pointed homeomorphic to R m x K(S) for some MCS-space S. It is clear that Xl is an l-dimensional topological manifold. We will need two general topological results which hold for spaces more general than Alexandrov spaces and follow from the general theory of deformations of homeomorphisms developed by Siebenmann [24]. THEOREM 4.4. [24, Theorem 5.4, Corollary 6.14, 6.9] Let X be a metric space and f: X -+ IRk be a continuous, open, proper map such that for each x E X we have (1) f-l(J(x)) is a compact MCS-space; (2) x admits a product neighborhood with respect to f, i.e there exists an open neighbourhood Ux of x and a homeomorphism Fx: Ux -+ Ux n f-l(J(x)) x f(Ux ) such that fx = P2 0 Fx where P2: Ux n f-l(J(X)) x f(Ux ) -+ IRk is the coordinate projection onto the second factor. Then f is a locally trivial fiber bundle. Moreover, suppose we have in addition that f(Ux ) = J k . Let K c Ux be a compact subset. Then there exists a homeomorphism '(J: f-l(Ik) -+ f- l (J (x)) X Jk respecting f {i. e. such that f = P2 O'{J. and such that '{JI K = FxIK' The next gluing theorem is the key topological ingredient in the proof of the Stability Theorem. It says that for MCS spaces close local homeomorphisms given on a finite open covering can be glued to a nearby global homeomorphism under some mild (but important!) geometric assumptions. First we need a technical definition. DEFINITION 4.5. A metric space X is called x-connected if for any two points Xl, X2 E X there exists a curve connecting Xl and X2 of diam ~ X(d(Xl' X2)). GLUING THOREM 4.6. Let X be a compact MCS-space, {UQ}QEQ( be a finite covering of X. Given a function xo, there exists x = x(X, {UQ}QE21. xo) such that the following holds: Given a Xo -connected MCS-space X, an open cover of X {UQ}QE2l, a 8 -Hausdorff approximation '{J: X -+ X and a family of homeomorphisms

V. KAPOVITCH

112

O.

f

~

d(a(to), (3(to)) exp(A(tl - to)) for all tl

~

to

is concave then its gradient flow 'Pt is I-Lipschitz for

5. Admissible functions and their derivatives

5.1. Let X be an Alexandrov space of curv ~ /'i,. Let f: X-+ ~ have the form f = La Aa'Pa (d(·, Aa,)) where each Aa is a closed subset of X, Aa ~ 0, La Aa ~ 1 and each 'Pa: ~ -+ ~ is a twice differentiable function with 0 ~ 'P~ ~ 1. We say that such f is admissible on U = X\ U a Aa. DEFINITION

With a slight abuse of notations we'll sometimes simply say that f: X-+ IR of the above form is admissible. It's obvious from the definition that an admissible function is 1- Lipschitz and semiconcave on U. More precisely, f is A-concave near p E U where A depends on /'i, and d(p, UiAi).

V. KAPOVITCH

116

By the first variation formula, dfp:

~pX

-+ lR has the form

dfp = L -aa cos(d(·, 11":)) where aa = Aacp~(d., Aa)). a

Note that aa ~ 0, ~aaa ~ 1. In view of this, following [17], we call a function h: curv~ ~ 1 a function of class DER if it has the form

L -aa cos(d(·, Aa)) where ao: ~ 0,

~

-+ lR where

Laa ~ l.

a

a

for some finite collection {Aa}aE!2l of subsets of ~. DER functions are spherically concave on ~ and concave when radially extended to C~. We define the scalar product of two functions in DER by the formula: for

(5.1)

h = L -aa cos(d(·, Aa)),

9 = L -bj3 cos(d(·, Bj3)) j3

a

put

a,j3 Note that this definition depends on the representations of h, 9 given by (5.1) and not just the values of f,g at every point. It is shown in [17] that the scalar product satisfies the following properties: (i) (dph,dpg) ~ (h,g) - h(p)g(p) for any p E~; (ii) (h, h) ~ (infgEDER(~) (h, g))2 ~ 0; (iii) For any h E DER(~) there is a point A E ~,o ~ a ~ 1 such that for h = -acos(d(.,A)) we have (h,g) ~ (h,g). Notice that an admissible function f on an Alexandrov space X can be naturally lilfted to a nearby Alexandrov space X by lifting the sets Ao: from the definition of f to nearby sets in X and defining j by the same formula as f. We will need the following simple lemma: LEMMA

G-H t--+oo

be admissible at p EX. Let Xi :3 Pi --+p. Then

h

~

and let f, g: X -+ lR gi: Xi -+ lR be natural lifts oj f, g. Let

5.2. Let Xi.--+ X where curvXi

f'i,

i--+oo

(dpJ, dpg)

~

limsup(dpih dpigi ). i

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117

By linearity it's easy to see that it's enough to prove the lemma for f,g of the form f = d(·,A),g = d(·,B). For functions of this form the statement easily follows from Toponogov comparison by an argument by contradiction. D PROOF.

6. Admissible maps and their regular points

6.1. A map g: X --+

called admissible on an open set U C X if it admits a representation 9 = G 0 f where all the components of f: U --+ jRk are admissible on U and G is a bi-Lipschitz homeomorphism between open sets in jRk . DEFINITION

jRk is

REMARK 6.2. The inclusion of the bi-Lipschitz map G in the definition of an admissible map might seem strange at this point. However, it will significantly simplify certain steps in the proof of the Stability theorem as well as prove useful in applications. To unburden the exposition we will employ the following convention. If an admissible map is denoted by f (with any indices) we'll automatically assume that in the above definition G == Id.

6.3. An admissible map g: U --+ jRk is called E-regular at p E U if for some representation 9 = G 0 f of 9 the following holds: DEFINITION

(1) mini#j -(dpfi, dpfj) > E; (2) There exists v E I:pX such that fI (v) > E for all i. We say that g is regular at p if it's E -regular at p for some

E

> O.

It is easy to see that f = (II, ... , /k) is E-regular at p iff there is a point q such that for fo = d(·, q) we have (dpfi, dp/j) < -E for all 0 ~ i

i= j

~ k.

EXAMPLE 6.4. As was mentioned in the introduction, a basic example of a regular map is as follows. Suppose {Ii = d(·, Ai)h=o, ... ,k satisfy

L

1I: 1l: > i

j

7r /2

for all i

i= j.

Then f = (II,· .. , fk) is regular at p. This example shows that regular points of admissible maps naturally generalize regular points of smooth maps because of the following simple observation: Given k + 1 non-zero vectors in jRn with all pairwise angles > 7r /2, any k of them are linearly independent. As an obvious corollary of Lemma 5.2 we obtain

6.5. Let f: X --+ jRk be admissible and E -regular at p EX. < E. Suppose Xi ~-If X where Xi E Alexn(D,~). Let Xi :3

COROLLARY

Let 0


O. Then

(1) k ~ n; (2) There exists WEI:: such that fi (w) > € for all i =1= O. (3) There exists v E I:: such that fo(v) > €, h(v) < -€ and h(v) = 0 for i = 2, ... , k.

Let g: X ~ Rk be an admissible map. Let X~ey(g) eX (Xreg(g) eX) be the set of €-regular (regular) points of g. Then the following properties hold [17, 18] a) X~eg(g) (and hence also (Xreg(g)) is open for any € > 0 (see Corollary 6.5). b) If 9 is €-regular on an open set U C X then 9 is €-co-Lipschitz on U. (This is an easy corollary of Lemma 6.7. See [17,18] for details.) c) g: Xreg(g) ~ Rk is open. d) k ~ dimX if Xreg(g) =1= 0). This immediately follows from part (1) of Lemma 6.7. THEOREM 6.8 (Local Fibration Theorem). [17, 18] Let g: X ~ Rk be an admissible map. Then glxreg(g): Xreg(g) ~ Rk is locally a topological bundle map. This means that any point p E Xreg(g) possesses an open product neighborhood with respect to 9 with an M SC -space as a fiber.

PERELMAN'S STABILITY THEOREM

119

We will use the original Local Fibration Theorem without a proof. However, we will prove a more general version of it (see Theorem 9.7) in Section 9 in order to prove the Relative Stability Theorem. We will need the following technical Lemma due to Perelman which plays a key role in the proofs of both the Stability Theorem and the Fibration Theorem above. KEY LEMMA 6.9. [17, Section 3J Let p be a regular point of f: X -t Suppose f is incomplementable at p, i. e. for any admissible function /1: X -t ]R, the point p is not regular for (f, /1): X -t ]Rk+ 1 . Then there exists an admissible function h: X -t ]R with the following properties

]Rk.

(i) h(p) = O. (ii) h is strictly concave on B(p, R) for some R > O. (iii) There are r > 0, A > 0 such that h < A on f- 1 (lk(f(p), r)) and f- 1 (lk(f(p) , r)) n {h;;:: -A} is compact in B(p, R). (iv) h has a unique maximum in B(p, R) n f-l(v) for all v E lk(f(p) , r). Let S denote the set of such maximum points. (v) (f, h) is regular on (lk(f(P), r) n B(p, R)) \S. As an immediate corollary we obtain COROLLARY 6.10. Let f: xn -t]Rn be an admissible map. Then f is locally bi-Lipschitz on the set of its regular points Xreg(f). PROOF OF COROLLARY 6.10. Let p be a regular point of f. By part a) of Lemma 6.7, f is incomplementable at p. Let h be the function in B(p,R) provided By lemma 6.9. By part (v) of Lemma 6.9, (f, h): B(p, R)\S -t ]Rn+1 is regular. However, as was just mentioned, by part (1) of Lemma 6.7, a map from xn to ]Rn+l can not have any regular points. Therefore B(p, R)\ S = 0. By part (iv) of Lemma 6.9, this is equivalent to saying that B(p, R) n f- 1 ( v) consists of a single point for all v E lk (f (p), r) which means that f is injective near p. Finally, recall that being regular, f is both Lipschitz and co-Lipschitz near p, which together with local injectivity means that it's locally bi-Lipschitz on Xreg(f). 0 PROOF OF KEY LEMMA 6.9. Since a complete proof is given in [17J we don't include as many details. For simplicity we assume that all components fj of f are actually concave near p (the proof can be easily adapted to the general case of semiconcave fj s). Since f is E - regular at p there is a point q near p such that h(q) > h(p) + Ed(p,q) for all j = I, ... ,k. Since h's are I-Lipschitz, for all r near p and all x E B(q, Ed(p, q)/4) we have (6.1)

h(x) > h(r)

d(x, r)

+ E4-

V. KAPOVITCH

120

Fix a small positive ~ « Ed(p, q). Choose a maximal ~-net {qa}aE2( in S(q, Ed(p, q)/4). A standard volume comparison argument shows that 1211 = N ~ c~l-n where n = dim X . Consider the function h = N-1Eaha where ha = <Pa(d(·, qa)) and <Pa: lR -t lR is the unique continuous function satisfying the following properties (1) 0 and > 0 for all j. Combined with (6.2) this means that (j, h) is regular at x. 0 fj(~)

V. KAPOVITCH

122

REMARK 6.13. Key Lemma 6.9 in conjunction with Theorem 4.4 easily yields Fibration Theorem 6.8 (see [17] for details). We will prove a more general version of it in Section 9. We will need the following strengthened version of the Key Lemma which is the main geometric ingredient in the proof of the Stability Theorem. LEMMA 6.14. Under the conditions of Lemma 6.9 suppose we have a noncollpasing converging sequence xr~-1f xn and admissible functions Ii on Xi converging to

f

t-too

Let Pi E Xi satisfy Pi.---+ p. t-too

Then there exist admissible lifts hi of the function h provided by Lemma 6.9 such that for all large i, the functions fi' hi satisfy the properties (i)-(v) of Lemma 6.9.

PROOF. Part (i) is obvious by construction of h and hi where we might have to shift hi by a small constant to make it zero at Pi. The fact that the natural lifts hi of h are strictly concave in B(pi, R) is basically the same as the proof of the concavity of h itself (see [17, Lemma 3.6]). It is carried out in full detail in [11, Lemma 4.2]. This proves part (ii). For the proof of (iii) we can not use Sublemma 6.11 as we did in the proof of Lemma 6.9 because Ii might be complement able everywhere near Pi. Nevertheless, part (iii) is obvious because it holds for (I, h) by Lemma 6.9 and Ii ---t f, hi ---t h, Pi ---t p. Parts (iv) and (v) are proved in exactly the same way as in the proof of Lemma 6.9. 0 LEMMA 6.15. Let F = (h, ... , fk): X ---t]Rk be f.-regular near p. Then the level set H = {F = F(p)} is locally x-connected near P for a linear function x. PROOF. In what follows all constants C will depend on f.. Denote F(p) = v = (VI, ... , Vk) and let H_ be the set ni{1i ~ Vi}' By definition of a regular point, there exists q near P such that for ~ = tZ we have ff(~) > f. for all i ( and the same holds for all z near p). Let '" = "Vpd(·,q). Then", is polar to ~ by (4.2). Therefore dfi(~) +dfi("') ~ 0 for all i by (4.1) and hence (6.8)

dli(",) ~

-f.

for all i.

Let x, y E B(p, r) with r « d(p, q) be two close points on the level set H. Let 'Y be a shortest geodesic connecting x and y. Consider the gradient flow 0 for all large i and the level sets {Ii = Ii(Pi)} are x-connected near Pi for all large i with the same x(t) = C(E,K,A,X)· t.

7. Proof of the stability theorem DEFINITION

7.1. Let p be a point in an Alexandrov space X. Let g =

G 0 f = (gl, ... , gk): X --+ jRk be regular at p. Then an open product neighborhood of p with respect to such g is called a product neighborhood of a p of rank k. DEFINITION 7.2. We'll call a subset H of jRn a generalized quadrant if it has the following form

where I, J are some (possibly empty) subsets of {I, ... , n} and (Xl, ... ,xn ) are the standard coordinates on jRn . DEFINITION 7.3. A compact subset P in an Alexandrov space X is called k-framed if P can be covered by a finite collection of open sets Ua such

124

V. KAPOVITCH

that each Ua is a product neighborhood of rank ka ~ k for some Pa E P with respect to some ga=Gaofa: X-+]Rka and P n Ua = g;;I(Ha) n Ua where Ha is a generalized quadrant in Rka. We say that a framing {Ua ,Ja,Ha }aE21 respects a map f: X -+]Rl if the first l functions of fa coincide with f for all a. More generally, we'll say that a framing {Ua,ga = Ga 0 fa,H a }aE21 respects a representation of an admissible map 9 9 = G 0 f: X -+ ]Rl if the first l functions of each fa coincide with f and Ga has the form Ga(x,y) = (G(x), Ta(x,y)) where x E ]Rl,y E jRka-l, i.e the first l coordinates of Ga are equal to G. (In particular, the first l functions of ga also coincide with g).

To simplify notations, we'll often simply say that a framing respects a map 9 to mean that it respects the representation 9 = G 0 f. EXAMPLE

7.4. Any compact Alexandrov space has a zero framing.

7.5. For any point p E X there exists E > 0 such that d(·,p) has no critical points in B(p, E) \ {p}. Hence, for any positive r < R < E, the annulus A(r, R,p) = f3(p, R)\B(p.r) is I-framed respecting f = d(·,p). EXAMPLE

Suppose Xi -+ xn is a converging noncollapsing sequence of compact Alexandrov spaces with curv ~ K,. Let (Ji: X -+ Xi be a sequence of o( i)Hausdorff approximations. Let P c X be a k-framed compact subset of X. We define the corresponding k-framed subsets Pi C Xi as follows. Let ga = G a 0 fa be a representation of ga given by the definition of an admissible map. We lift the defining functions fa and f to fi and f a,i in the natural way. Suppose Ga,i is a sequence of uniformly bi-Lipschitz homeomorphisms of open sets in jRk converging to G. Put ga,i = Ga,i 0 fa,i' Then ga,i will still be admissible and regular on the corresponding subsets of Xi by Corollary 6.5. In particular we get product neighborhoods Ua,i with respect to ga,i' We'll say that a compact set Pi C Xi is a lifting of P if Pi n Ua,i = g~J(Ha) for all a. REMARK 7.6. Note that a lifting of P need not exists! However, if it does, it is automatically k-framed. Moreover, if P is k-framed with respect to f: X -+ jRl and la, Ja ~ {I, ... ,l} for all a then the lifting exists for all large i. In particular, if X is a compact Alexandrov space and Xi -+ xn with curv ~ K, then the lifting of X with respect to a zero framing exists for any large i and is equal to Xi.

PERELMAN'S STABILITY THEOREM

125

LEMMA 7.7. The set's Pi are x-connected for all large i and the same x(t) = Ct. PROOF. Let x, y be two close points in P n Ua for some a. Since ga is L-Lipschitz, Iga(x) -ga(y)1 ~ L·d(x, y). Since Ha is convex, the straight line segment connecting ga (x) and ga (y) lies in H a. Since fa (and hence ga) is f-co-Lipschitz, by Lemma 4.13 we can lift it to a curve 1'1: [0,1] -+ Ua n P of length ~ ~d(x, y) with 1'1 (0) = x. Observe that 1'1 (1) and y lie in the same fiber of ga (and hence of fa) and d(')'l (1), y) ~ C(L, f)d(x, y) by the triangle inequality. By Lemma 6.15 we can connect 1'1(1) and y by a curve 1'2 inside the fiber {ga = ga(Y)} of length ~ 6· d(')'l(l),y). The concatenation of 1'1 and 1'2 provides a curve l' in P n Ua connecting x to y with L(')') ~ C· d(x, y). As was observed in Remark 6.16 the constant 6 in the above argument can be chosen to be the same for all fa,i and hence, since all Ga,i are uniformly bi-Lipschitz, all Pi are x-connected for all large i for the same x(t) = Ct. 0 The proof of the stability theorem proceeds by reverse induction in framing and, in fact, it requires us to to prove the following stronger version of it:

Xr

xn

THEOREM 7.8 (Parameterized Stability Theorem). Suppose -+ is a converging noncollapsing sequence of Alexandrov spaces with curvature bounded below and diameter bounded above. Let (}i: X -+ Xi be a sequence o(i) -Hausdorff approximations. Let P c X be a k -framed compact subset of X whose framing respects g: X -+ Rl. Let K c P be a compact subset such that the framing of P respects 9 on P and (g, h) on K for some g: X -+ Rl and h: X -+ R. Then for all large i there exist homeomorphisms ()~: P -+ Pi such that (}i is o( i) -close to (}i and respects 9 on P and (g, h) on K. PROOF. We proceed by reverse induction in k. If k = n then the locally defined maps ga,i: Ua,i -+ R n , ga: Ua -+ Rn are bi-Lipschitz homeomorphisms. By construction, the maps (}a,i = g~) 0 ga: Ua -+ Ua,i are homeomorphisms Hausdorff close to (}i. Moreover, by construction, (}a,i sends P n Ua onto Pi n Ua,i. Thus the statement of the theorem follows from Strong Gluing Theorem 4.10 and Lemma 7.7. Induction step. Suppose the theorem is proved for k + 1 ~ n and we need to prove it for k. Let P be k-framed and let p lie in P. Then p E Ua for some a. Let ga = G a 0 fa: Ua -+ Rk be the admissible map regular at p coming from the definition of a k-framed set.

126

V. KAPOVITCH

To simplify the notations we will assume that G a = Id and fa = ga' The proof in the general case easily follows from this one with obvious modifications. Let Pi = (h(p), By possibly adding more components to fa we can construct an admissible map fp: X --+ ]Rkp where kp ~ k which is incomplementable at p. Let h: B(p, R) --+ R be a strictly concave function provided by Lemma 6.9. By choosing a sufficiently small r, A > 0 we can assume that the set Up = f;1(Ikp(Jp(p), r) n {h ~ -A} is compact. By reducing r further we can assume that Ih(x)1 ::;; a «: A for xES n Up. We will call Up a special neighborhood of p. By Lemma 6.9 we have that Up is a k-framed compact subset of X. Since h and all the coordinates of fp are admissible, they have natural admissible lifts hi and fp,i which define corresponding neighborhoods UPi of Pi. The proof of Stability Theorem 7.8 will easily follow from the following LOCAL STABILITY LEMMA 7.9. For all large i there exist homeomorphisms Op,i: Up --+ UPi respecting fp and o(i) -close to the Hausdorff approximation Oi. Let us first explain how to finish the proof of theorem 7.8 given Lemma 7.9. Choose a finite cover of P by the interiors of the special neighborhoods UP{3' For all large i, Lemma 7.9 provides homeomorphisms 0p{3,i: UP{3 --+ Up{3,i respecting fp{3 and o( i) -close to Oi. Observe that since each 0p{3,i respects f p{3' it sends P n Up{3 onto Pi n Up{3,i. Taking into account Lemma 7.7 we can apply Gluing Theorem 4.10 to obtain the desired homeomorphism O~: P --+ Pi. PROOF OF LOCAL STABILITY LEMMA 7.9. All throughout the proof of the Lemma we will work only with points in Up in X and UPi in Xi. If kp > k then the statement follows directly from the induction hypothesis. Let's suppose kp = k. First we change the function h to an auxiliary function it by shifting h by a constant on each fiber of f to make it identically zero on S. More precisely, let it(x) = h(x) - h(S n f-1(J(x))). Recall that by Lemma 6.9(iii), S n f-1(J(x))) consists of a single point so that this definition makes sense. Also by Lemma 6.9(iii), we have h(S n f-1(J(x))) = maxyEupn/-1(f(x)) h(y) and therefore h- = h - H 0 f where H:]R k --+]R is given by H(v) = maxxEupn/-1(v) h(x). Since f is co-Lipschitz and h is Lipschitz, using Lemma 4.13 we easily conclude that H is Lipschitz. In particular, (J, it) = fI 0 (J, h) where fI is a bi-Lipshitz homeomorphism of some open domains in ]Rk+1 given by fI(a, b) = (a, b - H(a)).

PERELMAN'S STABILITY THEOREM

127

Therefore, we still have that (f, h) is regular on Up\S and hence it's locally a bundle map on Up \S by Theorem 6.8. In addition, by construction, h = 0 on Sand h < 0 on Up \S. We define hi, Hi and Hi in a similar fashion using Ii, hi. We obviously have that hi -; h, Hi -; H, Hi -; H. Moreover, since Ii are uniformly co-lipschitz, all Hi are uniformly Lipschitz and hence all Hi are uniformly bi-Lipschitz. Then we again have that hi = 0 on Si and h < 0 on UPi \Si. By Lemma 6.14 we also have that (Ii, hi) is regular on UPi \Si. Fix a small 8 « A. Then the set {h ::;; -8} is (k + I)-framed with the corresponding sets in UPi given by {hi::;; -8} and therefore, by induction assumption, for large i there exist homeomorphisms 05,i: {h::;; -8} -; {hi::;; -8} respecting (f, h) and o(iI8)-dose to Oi. In particular, the fiber Fi of (fi, hi) is homeomorphic to the fiber F of (f, h) for all large i. Next consider the set {h> -38/2} and consider the map (f, h): {h> -38/2}\S -; [k(f(p),r) x (-38/2,0). By Lemma 6.9, this map is regular and proper. Therefore, by Theorem 6.8 and Theorem 4.4, it is a fiber bundle. Hence, {h> -38/2}\S is homeomorphic to [k(f(p),r) x (-38/2,0) xF with the first two coordinates given by (f, h). By restriction this gives a homeomorphism {h ~ 8}\S to [k(f(p), r) x [-8,0) x F with the first two coordinates still given by (f, h). By Lemma 6.9 and construction of h, h has a unique max ( equal to zero) on 1-1 (v) for any v E jk (f (p), r). Therefore, the above homeomorphism can be uniquely extended to a homeomorphism 0 (i.e if f is both I-Lipschitz and 1-co-Lipschitz). It is obvious that a Riemannian submersion between complete Riemannian manifolds is a submetry. Moreover, converse is also true according to [1]. It is also clear that if a compact group G acts on a Riemannian manifold M by isometries, then the projection M -+ MIG is a submetry. Submetries enjoy many properties of Riemannian submersions. In particular, one can talk about horizontal and vertical tangent vectors and curves. Also, it's easy to see [2] that submetries increase Alexandrov curvature, that is, if curv X ~ K and f: X -+ Y is a submetry, then curv Y ~ K. For more basic information on submetries see [15]. Suppose 11": X -+ B is a submetry between compact Alexandrov spaces. It trivially follows from the definition that if f: B -+ Rk is admissible then f 0 11" is admissible on M. Moreover, f is regular at p E B iff f 0 11" is regular at any y E 1I"-I(p). In particular, if B is a Riemannian manifold (or more generally, if B is everywhere n-strained) then 1r is a fiber bundle. Thus the above notion of equivalence of submersions naturally extends to submetries. Theorem 8.1 generalizes a theorem of J. Y. Wu, [29] which proved the same result under a strong extra assumption that fibers of the submersions are totally geodesic. The proof of Theorem 8.1 relies on the proof ofWu's theorem which just as the proof of Grove-Petersen-Wu finiteness Theorem 1.2 uses techniques of controlled homotopy theory. This explains the assumption k ~ 4 in Theorem 8.1. However, this assumption is, in fact, unnecessary as this result follows from the Parameterized Stability Theorem which does not require any dimensional restrictions.

V. KAPOVITCH

130

THEOREM 8.2. Given k E Z+, V, D, E ~+, >., K, E JR, there are at most finitely many equivalence classes of submetries xn+k -+ Bn where X n+k E Alexn +k (D, K" v) and B n is a closed Riemannian manifold satisfying vol(B) ;;::: V, Isec(B)1 ::;;; >.. PROOF. We first give a proof in case of fixed B. Let 7r: X -+ B be a submetry where X is an Alexandrov space of curvature bounded below. Let P E B be any point. Choose n + 1 unit vectors VO, •.. , Vn E TpB with pairwise angles bigger than 7r /2. Then for all sufficiently small R > 0 the points Pi = expp(Rvi) define an admissible map f: B -+ JRn given by y H (d(Y,Pl), ... ,d(Y,Pn))' This map is obviously regular on B(p,r) for r « R and it gives a bi-Lipschitz open embedding ( in fact a smooth one) f: B(p,R) -+ JR n . Let Fi = 7r- l (Pi). Let X -+ JRn be given by i(x) = (d(x, Fl ), ... , d(x, Fn)). Since 7r is a submetry we obviously have that d(x, Fi ) = d(7r(X),Pi) for any i and any x EX. Therefore == f 0 7r. It is also obvious that is regular at x E X iff f is regular at 7r( x). In particular, i is regular on the r-neighborhood of 7r- l (7r(p)). Thus, up to a bi-Lipschitz change of coordinates on the target, when restricted to Ur (7r- l (7r(p))) , we can write 7r as a proper regular map to JRn. Let's cover B by finitely many coordinate neighborhoods Ua = B(Pa, ra) as above and let fa: Ua -+ JRn be the corresponding coordinate projections. Since all fa are bijections we obviously have that for any x, y E 7r-l(Ua nUb), ia(x) = ia(Y) iff i{3(x) = i{3(Y)' Therefore Parameterized Stability Theorem 7.8 can be easily amended to include the case when a framing on X respects a submetry to a fixed manifold as all the arguments can be made local on B where instead of the submetry 7r one can work with regular maps ia. The case of variable base easily follows given the fact that by CheegerGromov compactness the class of manifolds {Bn I vol(B);;::: V, Isec(B) I ::;;; >., diam(B) ::;;; D} is precompact in Lipschitz topology and its limit points are Cl,a-Riemannian manifolds (see e.g. [8]). 0

1:

i

i

i

REMARK 8.3. It's interesting to see whether Theorem 8.2 remains true if one removes the assumption about the uniform upper bound on the curvature of B. 9. Stability with extremal subsets The results proved in this section are new. The notion of an extremal subset in an Alexandrov space was introduced in [21]. DEFINITION 9.1. A closed subset E in an Alexandrov space X is called extremal if for any q E X\E and f = d(·, E) the following holds:

PERELMAN'S STABILITY THEOREM

131

If pEE is a point of local minimum of fiE then it's a critical point of maximum type of f on X, i.e.

Alternatively, it was shown in [20] that E is extremal iff it's invariant under gradient flows of all semiconcave functions on X. An extremal subset is called primitive if it doesn't contain any proper extremal subsets with nonempty relative interiors. We refer to [21, 20] for basic properties of extremal subsets. It is easy to see [21] that closures of topological strata in an Alexandrov space X are extremal. Therefore stratification into extremal subsets can be considered as a geometric refinement of the topological stratification of X. It is of course obvious that a homeomorphism between two Alexandrov spaces has to preserve topological strata. The goal of this section is to generalize the Stability Theorem by showing that the stability homeomorphisms can be chosen to preserve extremal subsets. Namely we will prove the following THEOREM 9.2 (Relative Stability Thorem). Let

Xi G-H .--+xn

be a noncol-

~-+oo

lapsing sequence of compact Alexandrov spaces in Alexn(D, K). Let ()i: X -+ Xi be a sequence o(i) -Hausdorff approximations. Let Ei C Xi be a sequence of extremal subsets converging to an extremal subset E in X. Then for all large i there exist homeomorphisms ()~: (X, E) -+ (Xi, Ei), o(i) -close to ()i.

In order to prove this theorem we'll need to generalize all the machinery used in the proof of the regular Stability Theorem to its relative version respecting extremal subsets. This is fairly straightforward and only minor modifications of the proofs are required. In particular we'll have to prove the relative version of Local Fibration Theorem 6.8. Along the way we'll obtain some new topological information about the way a general extremal subset is embedded into an ambient Alexandrov space. It was shown in [21] that just as Alexandrov spaces, extremal subsets are naturally stratified in the sense of the following definition.

-------

DEFINITION 9.3. A metrizable space X is called an M se -space of dimension ~ n if every point x E X has a neighborhood pointed homeomorphic to an open cone over a compact Me S -space of dimension ~ n - 1 . As for MCS-spaces we assume the empty set to be the unique MeS -space of dim ~ -1.

-------

-------

We will also need a relative version of the above definition.

-------

DEFINITION 9.4. Let X be an M se -space of dim ~ n. A subset E C X is called a stratified subspace of X of dimension ~ k if every x E X has

132

V. KAPOVITCH

a pointed neighborhood (U, p) such that (U, U

n E, p)

is homeomorphic to

(KE,KE',o) where E is a compact MSc-space of dim ::;; n-1 and E' c E is a compact stratified subspace of E of dimension ::;; k - 1. As usual, the only stratified subspace of dim::;; -1 is the empty set. It is obvious from the definition that a stratified subspace in X of dim ::;; k is an M C S -space of dimension ::;; k.

REMARK 9.5. It is easy to see that a connected MSC-space is an AfCSspace iff its local topological dimension is constant. It was shown in [21] that a primitive extremal subset is equal to the closure of its top dimensional strata and therefore is an MCS-space by above. In the process of proving the Relative Stability Theorem we'll obtain the following result which clarifies the relative topology of extremal subsets with respect to their ambient spaces. THEOREM 9.6 (Relative Stratification Theorem). Let X be an Alexandrov space and let E c X be an extremal subset. Then E is a stratified subspace of x. Just as in the non-relative case, this theorem is a Corollary of the following local fibration theorem applied to the natural map X -* ]R.0. THEOREM 9.7 (Relative Local Fibration Theorem). Let f: X -*]R.k be regular at pEE where E c X is an extremal subset. Then there exists an open neighborhood U of p, an M C S -space A, a stratified subspace B c A and a homeomorphism cp: (U, E n U) -* (A, B) x]R.k such that 7r2 0 cp = f. It was shown in [21] by Perelman and Petrunin that the intrinsic metric on an extremal subset of an Alexandrov space is locally bi-Lipschitz to the ambient metric. On closer examination their proof actually gives the following somewhat stronger statement which we'll need for the proof of the relative stability theorem:

LEMMA 9.8. There exists € = €(n.D, K" v) > 0 such that if X E Alexn (D, K" v) and E C X is extremal, then for any p, q E E with d(p, q) ::;; € there exists a curve in E connecting p and q of length::;; c 1 d(p, q). PROOF. Because the argument is very easy we give it here for reader's convenience. It is well-known (see [21] or [6]) that for the class Alexn(D, K" v) there exists an € > 0 such that the following holds: If X E Alexn(D, K" v) and p, q E X with d(p, q) < €2 then

(9.1)

jVpd(-,q)j >

€

or jVqd(·,p)j >

€

Suppose the first alternative holds. Then there exists x near p such that L.xpq;;:: 7r/2+€. Then Vpd(·,x) is polar to 1t~ so that L.Vpd(·,x) tZ::;;

PERELMAN'S STABILITY THEOREM

133

71" /2-E. This means that moving p along the gradient flow of d(·, x) decreases d(p, q) in the first order (with the derivative at 0 at least E). Since E is extremal, the flow through p remains in E. N ow a standard argument shows that we can construct a curve in E connecting p and q of length::::; c 1 d(p, q). 0

We will need the following generalization of Lemma 6.7 proved in [21]: LEMMA 9.9. [21, 20] Suppose En - 1 has curv ;;:;: 1 and let 10, ... , Ik: E--+ ]R be functions 01 class DER such that E = mini;tj -(Ii, fJ) > O. Let E c E be an extremal subset. Then (1) There exists wEE such that Ii (w) > E for all i =1= O. (2) There exists vEE such that fo(v) > E, h(v) < -E and h(v) = 0 for i = 2, ... , k. Just as in the case of regular functions on Alexandrov spaces, this lemma implies that if I: X --+ ]Rk is regular at pEE where E c X is extremal then fiE is co-lipschitz near p. The main geometric ingredient in the proof of the Relative Local Fibration Theorem and the Relative Stratification Theorem is the following relative analogue of Lemma 6.9. LEMMA 9.10. Let pEE be a regular point of I: X --+]Rk where E is an extremal subset of X. Suppose I is incomplementable at pEE. Then there exists an admissible function h: X --+ ]R with the following properties (i) h(p) =0. (ii) h is strictly concave on B(p, R) for some R > O. (iii) There are r > 0, A > 0 such that h < A on f- 1 (Ik(I(p), r)) and 1-1 (Ik(I(p), r)) n {h;;:;: -A} is compact in B(p, R). (iv) h has a unique maximum in B(p, R) n f-l(v) for all v E Ik(I(p), r). Let S denote the set of such maximum points. (v) (I, h) is regular on f- 1 (Ik(l(p) , r)) n B(p, R)\S. (vi) SeE. PROOF. The proof is identical to the proof of Lemma 6.9 except for part (vi) which is new. Let x be a point of max of h on En f-l(v). Let z be the unique point of maximum of h on 1- 1 (v). If x =1= z then (I,h) is regular at x by Lemma 6.9. However, by Lemma 9.9, a point on E is regular for F: X --+]Rm iff it's regular for FIE. Thus, (I, h)IE is regular at x. Therefore, (I, h)IE is co-lipschitz near x and hence x is not a point of 0 maximum of h on En 1- 1 (v). This Lemma easily implies that if I: X --+ ]Rk is regular at pEE where E C X is extremal then the local dimension of E near p is ;;:;:k and the equality is only possible if FIE is locally bi-Lipschitz near p (cf. Corollary 6.10).

134

V. KAPOVITCH

Lemma 9.10 also yields the relative local fibration theorem in exactly the same way as Lemma 6.9 yields the absolute local fibration theorem. PROOF OF THEOREM 9.7. Since the proof is almost identical to the proof of the fibration theorem in [17] we only give a sketch. We argue by reverse induction in k. Since the base of induction is clear we only have to consider the induction step from k + 1 ~ n to k. Let f: X --+ Rk be regular at pEE where E c X is extremal. If f is complement able at p the statement follows by induction assumption. Suppose f is incomplementable at p. Let h be the function provided by Lemma 9.10. Suppose for simplicity that h is identically zero on S. Let U = f- 1 ([k(f(p),r)) n {-A < h < O} n B(p,R) and W = f- 1 ([k(f(p),r)) n {-A < h ~ O} n B(p,R). Then U = W\S and (f, h) is regular on U. Therefore the relative local fibration theorem holds for (f, h) on U by induction assumption. By Lemma 9.10, (f, h) is proper on U and hence, (f, h): (U, Un E) --+ [k(f(p), r) x (-A, 0) is a relative bundle map by [24, 6.10]. This means that (U,U n E) is homeomorphic to (F,B) x [k(f(p),r) x (-A,O) respecting (f, h) where F is an MCS-space of dim = n - k - 1 and B is a stratified subspace in F. By Lemma 9.10, we can extend this homeomorphism to a homeomorphism (W, W n U) --+ (KF, KB) x [k(f(p) , r) which proves the induction step. The general case when h is not constant on S is handled in exactly the same way as in [17] and the proof of Key Lemma 7.9 by constructing an auxiliary function h obtained by shifting h by constants on each of the fiber of f to make it identically zero on S. 0 Before we can start with the proof of the Relative Stability Theorem we first need to observe that by [24] the corresponding version of Theorem 4.4 and Strong Gluing Theorem 4.10 hold in relative category for pairs of MCS spaces and their stratified subspaces. The relative version of Theorem 4.4 follows from [24, Complement 6.10 to Union Lemma 6.9] by the same argument as in the proof of the absolute version of Theorem 4.4 given by [24, Corollary 6.14]. The relative version of the Strong Gluing Theorem still follows from the same deformation of homeomorphism result [24, Theorem 6.1] which also covers relative homeomorphisms. Here we'll need the following definition DEFINITION 9.11. A pair of metric spaces (X, E) is called x-connected if both X and E (taken with the restricted ambient metric) are x-connected. Let us state the Relative Gluing Theorem. For simplicity we only state the unparameterized version. RELATIVE GLUING THOREM 9.12. Let (X, E) be a stratified pair, {Ua }aE21 be a finite covering of X. Given a function xo, there exists x = x((X, E), {Ua }aE21, xo) such that the following holds:

PERELMAN'S STABILITY THEOREM

135

Given a Xo -connected stratified pair (.i, E), an open cover 0/ .i {Ua }aE2l, a 8-Hausdorffapproximation 'P: (X,E) -+ (.i,E) and a/amily o/homeomorphisms 'Pa:(Ua,UanE) -+ (Ua,UanE), 8-close to 'P, then there exists a homeomorphism ({J: (X, E) -+ (.i,E), x(8)-close to 'P. Observe that under the assumptions of the Relative Stability Theorem, all the elements of the sequence (Xi, E i ) and the limit (X, E) are x-connected by Lemma 9.8. Furthermore, Lemma 6.15 and Remark 6.16 hold for regular level sets of admissible functions on extremal subsets. The proof is exactly the same as the proof of Lemma 6.15 modulo Lemma 9.8 and the fact that extremal subsets are invariant under all gradient flows. We are now ready to prove the Relative Stability Theorem. PROOF OF THEOREM 9.2. The proof proceeds by reverse induction on the framing and is, in fact, exactly the same as the proof of the usual stability theorem except we make all the arguments relative. Everywhere in the proof substitute U (with various subindices) by (U, Un E). In the proof of the relative version of Key Lemma 7.9, use Lemma 9.10 instead of Lemma 6.9 whenever necessary. Also, use the Relative Local Fibration Theorem instead of Local Fibration Theorem 4.4 and the Relative Gluing Theorem instead of the Gluing Theorem whenever called for. 0 REMARK 9.13. The relative stability theorem trivially implies the following hitherto unobserved fact. Under the assumptions of the Relative Stability Theorem, Ei:---7 E without collapse. Then dim E = dim Ei for all l-tOO

large i. A fairly simple direct proof of this statement can be given without using the Relative Stability Theorem. However, we chose not to present it here because this indeed obviously follows from the Relative Stability Theorem. References [1] V. N. Berestovskii and L. Guijarro. A metric characterization of Riemannian submersions. Ann. Global Anal. Geom., 18(6):577-588, 2000. [2] Yu. Burago, M. Gromov, and G. Perel'man. A. D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk, 47(2(284)):3-51, 222, 1992. [3] T. A. Chapman and S. Ferry. Approximating homotopy equivalences by homeomorphisms. Amer. J. Math., 101(3):583-607, 1979. [4] J. F. Davis. Manifold aspects of the Novikov conjecture. In Surveys on surgery theory, Vol. 1, volume 145 of Ann. of Math. Stud., pages 195-224. Princeton Univ. Press, Princeton, NJ, 2000. [5] M. H. Freedman. The topology of four-dimensional manifolds. J. Differential Geom., 17(3):357-453, 1982. [6] K. Grove and P. Petersen. Bounding homotopy types by geometry. Ann. of Math. (2), 128(1):195-206, 1988. [7] K. Grove, P. Peterson, and J.-Y. Wu. Geometric finiteness theorems via controlled topology. Invent. Math., 99:205-213, 1991.

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[8] R. E. Greene and H. Wu. Lipschitz convergence of Riemannian manifolds. Pacific J. Math., 131(1):119-141, 1988. [9] K. Grove and F. Wilhelm. Metric constraints on exotic spheres via Alexandrov geometry. J. Reine Angew. Math., 487:201-217, 1997. [10] W. Jakobsche. Approximating homotopy equivalences of 3-manifolds by homeomorphisms. Fund. Math., 130(3):157-168, 1988. [11] V. Kapovitch. Regularity of limits of noncollapsing sequences of manifolds. Geom. Funct. Anal., 12(1):121-137, 2002. [12] V. Kapovitch, A. Petrunin, and W. 'IUschmann. Nilpotency, almost nonnegative curvature and the gradient push. to appear in Annals of Mathematics, http://arxiv.org/abs/math.DG/0506273, 2005. [13] K.W. Kwun. Uniqueness of the open cone neighborhood. Proc. Amer. Math. Soc., 15:476-479, 1964. [14] A. Lytchak and K. Nagano. Topological regularity of spaces with curvature bounded above. in preparation. [15] A. Lytchak. Allgemeine Theorie der Submetrien und verwandte mathematische Probleme. Bonner Mathematische Schriften [Bonn Mathematical Publications], 347. Universitat Bonn Mathematisches Institut, Bonn, 2002. Dissertation, Rheinische Friedrich-Wilhelms-Universitat Bonn, Bonn, 2001. [16] G. Perelman. Alexandrov spaces with curvatures bounded from below II. preprint, 1991. [17] G. Perelman. Elements of Morse theory on Aleksandrov spaces. St. Petersbg. Math. J., 5(1):205-213, 1993. [18] G. Perelman. DC structure on Alexandrov space with curvature bounded below. preprint, http://www.math.psu.edu/petrunin/papers/papers.html, 1995. [19] A. Petrunin. Applications of quasigeodesics and gradient curves. In Grove, Karsten (ed.) et al., Comparison geometry. Cambridge: Cambridge University. Math. Sci. Res. Inst. Publ. 30, 203-219. 1997. [20] A. Petrunin. Semiconcave functions in alexandrov geometry. Anton's paper in the same volume, 2006. [21] G. Perelman and A. M. Petrunin. Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem. Algebra i Analiz, 5(1):242-256, 1993. [22] G. Perelman and A. Petrunin. Quasigeodesics and gradient curves in alexandrov spaces. preprint, http://www.math.psu.edu/petrunin/papers/papers.html, 1996. [23] F. Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geom., 17(3):503521, 1982. [24] L. C. Siebenmann. Deformation of homeomorphisms on stratified sets. I, II. Comment. Math. Helv., 47:123-136; ibid. 47 (1972), 137-163, 1972. [25] D. Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 543-555. Academic Press, New York,1979. [26] T. Shioya and T. Yamaguchi. Collapsing three-manifolds under a lower curvature bound. J. Differential Geom., 56(1):1-66, 2000. [27] T. Shioya and T. Yamaguchi. Volume collapsed three-manifolds with a lower curvature bound. Math. Ann., 333(1):131-155, 2005. [28] K. Tapp. Finiteness theorems for submersions and souls. Proc. Amer. Math. Soc., 130(6):1809-1817 (electronic), 2002. [29] J.-Y. Wu. A parametrized geometric finiteness theorem. Indiana Univ. Math. J., 45(2):511-528, 1996. VITALI KAPOVITCH, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, ONTARIO, CANADA, M5S2E4 E-mail address: vtk~ath. toronto. edu

Surveys in Differential Geometry XI

Semiconcave Functions in Alexandrov's Geometry Anton Petrunin *

ABSTRACT. The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.

Introduction This paper is not about results, it is about available techniques in Alexandrov's geometry which are linked to semiconcave functions. We consider only spaces with lower curvature bound, but most techniques described here also work for upper curvature bound and even in more general settings. Many proofs are omitted, I include only those which necessary for a continuous story and some easy ones. The proof of the existence of quasigeodesics is included in appendix A (otherwise it would never be published). I did not bother with rewriting basics of Alexandrov's geometry but I did change notation, so it does not fit exactly in any introduction. I tried to make it possible to read starting from any place. As a result the dependence of statements is not linear, some results in the very beginning depend on those in the very end and vice versa (but there should not be any cycle). Here is a list of available introductions to Alexandrov's geometry: • [BGP] and its extension [Perelman 1991] is the first introduction to Alexandrov's geometry. I use it as the main reference. Some parts of it are not easy to read. In the English translation of [BGP] there were invented some militaristic terms, which no one ever used, mainly burst point should be strained point and explosion should be collection of strainers. • [Shiohama] intoduction to Alexandrov's geometry, designed to be reader friendly. * Supported in part by the National Science Foundation under grant # DMS-0406482. ©Copyright (c) All rights reserved. Redistribution and modification are permitted provided that the following conditions are met: 1. If modifications contain at least half of the original text, they must retain the above copyright notice and this list of conditions. 2. In a modification, authorship must be changed. 137

A. PETRUNIN

138

• [Plaut 2002] A survey in Alexandrov's geometry written for topologists. The first 8 sections can be used as an introduction. The material covered in my paper is closely related to sections 7-10 of this survey. • [BBI, Chapter 10] is yet an other reader friendly introduction. I want to thank Karsten Grove for making me write this paper, Stephanie Alexander, Richard Bishop, Sergei Buyalo, Vitali Kapovitch, Alexander Lytchak and Conrad Plaut for many useful discussions during its preparation and correction of mistakes, Irina Pugach for correcting my English. CONTENTS

1. 2. 3. 4. 5. 6.

Semi-concave functions Gradient curves Gradient exponent Extremal subsets Quasigeodesics Simple functions 7. Controlled concavity 8. Tight maps 9. Please deform an Alexandrov's space A. Existence of quasigeodesics References

139 144 151

161 168 172 174

179 186 188 199

Notation and conventions • By ALexm(lI;) we will denote the class of m-dimensional Alexandrov's spaces with curvature ~II;. In this notation we may omit II; and m, but if not stated otherwise we assume that dimension is finite. • Gromov-Hausdorff convergence is understood with fixed sequence of approximations. I.e. once we write Xn ~ X that means that we fixed a sequence of Hausdorff approximations in : Xn ~ X (or equivalently

9n: X

•

• • • •

~

Xn).

This makes possible to talk about limit points in X for a sequence Xn E X n , limit of functions in : Xn ~ lR, Hausdorff limit of subsets Sn c Xn as well as weak limit of measures J.Ln on X n . regular fiber - see page 167 .i.xyz - angle at y in a geodesic triangle l:::.xyz c A .i.(~, 'T}) - an angle between two directions ~,'T} E ~p LK,xyz - a comparison angle, i.e. angle of the model triangle 6xyz in JIK, at y. LK,(a, b, c) - an angle opposite b of a triangle in JIK, with sides a, band c. In case a + b < c or b + c < a we assume LK,(a, b, c) = o. a direction at p of a minimazing geodesic from p to q

• tZ -

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

• • • • •

• • • • • • •

• • • • •

• • • •

•

11~

139

-

the set of all directions at p of minimazing geodesics from p to q A - usually an Alexandrov's space argmax - see page 184 8A - boundary of A distx(Y) = ixyi - distance between x and y dpJ - differential of J at p, see page 140 gexpp - see section 3 gexpp(/\':; v) - see section 3.2 'Y± - right/left tangent vector, see 2.1 JI" - model plane see page 140 JI;t - model halfplane see page 156 JI~ - model m-space see page 174 logp - see page 141 V PJ - gradient of J at p, see definition 1.3.2 p" - see page 140 1:(X) - the spherical suspension over X see [BGP, 4.3.1], it is called spherical cone, see [Plaut 2002, 89] and [Berestovskii]. (7" see footnote 15 on page 156 Tp = TpA - tangent cone at pEA, see page 140 TpE - see page 164 1:p = 1:p A - see footnote 4 on page 141 1:p E - see page 164 J± - see page 145 1. Semi-concave functions 1.1. Definitions.

1.1.1. Let A E ALex, and n c A be an open subset. A locally Lipschitz function J: n -+ R is called A-concave if for any unit-speed geodesic 'Y in n, the function DEFINITION FOR A SPACE WITHOUT BOUNDARY

8A =

0

is concave. If A is an Alexandrov's space with non-empty boundary1, then its doubling2 A is also an Alexandrov's space (see [Perelman 1991, 5.2]) and

8.4=0.

Set p : A -+ A to be the canonical map. DEFINITION FOR A SPACE WITHOUT BOUNDARY

8A

i- 0

and

nc A

1.1.2. Let A

be an open subset.

1Boundary of Alexandrov's space is defined in [BGP, 7.19]. 2i.e. two copies of A glued along their boundaries.

E ALex,

140

A. PETRUNIN

A locally Lipschitz function f: 0 -+ R is called A-concave if fop is A-concave in p-l(O) cA. Note that the restriction of a linear function on R n to a ball is not O-concave in this sense. REMARK.

1.2. Variations of definition. A function f : A -+ R is called semiconcave if for any point x E A there is a neighborhood Ox :3 x and A E R such that the restriction flnx is A-concave. Let cp : R -+ R be a continuous function. A function f : A -+ R is called cp(f) -concave if for any point x E A and any c > 0 there is a neighborhood Ox :3 x such that flnx is (cp 0 f(x) + c)-concave. For the Alexandrov's spaces with curvature ~ "', it is natural to consider the class of (1 - ",f)-concave functions. The advantage of such functions comes from the fact that on the model space3 JIlt, one can construct model (1- ",f)-concave functions which are equally concave in all directions at any fixed point. The most important example of (1 - ",f)-concave function is Pit 0 dist x , where distx(Y) = Ixyl denotes distance function from x to y and

PIt(x) = [

~(1 - cos(xfi))

",>0

x 2 /2

",=0

if if ~(ch(xv'-"') - 1) if

k 0, for any homotopy

Ft : K ---* A, t E [0, 1] one can construct an

E -close

gradient homotopy

Gt : K ---* A such that Go

==

Fo .

3. Gradient exponent One of the technical difficulties in Alexandrov's geometry comes from nonextendability of geodesics. In particular, the exponential map, expp : Tp ---* A, if defined the usual way, can be undefined in an arbitrary small neighborhood of origin. Here we construct its analog, the gradient exponential map gexpp : Tp ---* A, which practically solves this problem. It has many important properties of the ordinary exponential map, and is even "better" in certain respects. Let A be an Alexandrov's space and pEA, consider the function f = dist; /2. Recall that is : sA ---* A denotes canonical maps (see page 140). Consider the one parameter family of maps ../2.

l1i.e. maps with Lipschitz constant l. 12For general lower curvature bound, f is only (1+0(r 2))-concave in the ball Br(P). Therefore 0, the level sets Lx = dist8~ (x) c A, x > 0 are strictly concave hypersurfaces. THEOREM

PROOF.

tion

f

We have to show that for any unit-speed geodesic" the func-

0, is (-",f)-concave; i.e. for any to,

in a barrier sense17. Without loss of generality we can assume to

= o.

Direct calculations show that the statement is true for A = JI;t, the halfspace of the model space JI Ii • Let p E 8A be a closest point to ,(0) and a = L(r+(O), t~(O))' 15al< : ~ -+ ~ is defined by

16Note that by definition 1.1.2, f is not semiconcave in A. 17For a continuous function f, fl/(to) ~ c in a bamer sense means that there is a smooth function such that f ~ f (to) = J( to) and 1" (to) ~ c.

J

J,

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

157

Consider the following picture in the model halfspace JIt: Take a point fJ E 8JIt and consider the geodesic i in JIt such that 1,(O)pl = li(O)fJl = li(0)8JI~ I, so fJ is the closest point to i(O) on the boundary 18 and

Then it is enough to show that distaA ,(T) ~ distaJI;t i(T)

+ 0(T2).

Set

(3(T) = L,(O) p,(T) and

j3(T) = Li(O) pi(T). From the comparison inequalities

IP'Y(T) I ~ In(T)1 and

19(T) = max {O, j3(T) - (3(T) } = O(T). Note that the tangent cone at p splits: TpA = lR+ x Tp8A.19 Therefore we can represent v = 10gp,(T) E TpA as v = (s,w) E lR+ x Tp8A. Let ii = ii(T) E 8JIK, be the closest point to i(T), so

L(t;(T) , w) = ~ - (3(T) ~ ~ - j3(T) -19(T) = Li(T)pii + O(T).

fwr)

Set q = gexpp ( K; Ipiil .20 Since gradient curves preserve extremal subsets q E 8A (see property 3.1.1 on page 152). Clearly IfJiil = O(T), therefore applying the comparison from section 3.2 (or Corollary 3.1.3 if K = 0) together with (*), we get distaA ,(T) ~ Iq,(T)1 ~ liii(T)I

+0

(Ipiil . 19(T))

= distaJI;t i(T) + 0(T2). D

The following corollary implies that the Lipschitz condition in the definition of convex function 1.1.2-1.1.1 can be relaxed to usual continuity.

2:r..,

18In case '" > 0 it is possible only if h'(O)pj ~ but this is always the case since otherwise any small variation of p in 8A decreases distance j')'(O)pj. 19This follows from the fact that p lies on a shortest path between two preimages of ')'(0) in the doubling A of A, see [BGP, 7.15]. 20 Alternatively, one can set q = ')'(jPiij) , where,), is a quasigeodesic in 8A starting at p in direction 1:1 E ~p (it exists by second part of property 4 on page 169).

A. PETRUNIN

158

COROLLARY 3.3.2. Let A E AleX, aA = 0, A E JR and n c A be open. Assume f : n --+ JR is a continuous function such that for any unit-speed geodesic 'Y in n we have that the function

t

f---t

f

0

'Y - At2/2

is concave; then f is locally Lipschitz. In particular, f is A-concave in the sense of definition 1.1.2. PROOF. Assume f is not Lipschitz at pEn. Without loss of generality we can assume that n is convex21 and A < 0 22 . Then, since f is continuous, sub-graph Xf = {(x,y) x JRly ~ f(x)} is closed convex subset of Ax JR, therefore it forms an Alexandrov's space. Since f is not Lipschitz at p, there is a sequence of pairs of points (Pn, qn) in A, such that

En

Pn, qn --+ P and Consider a sequence of radial curves an in X f which extend shortest paths from (Pn,f(Pn)) to (qn,f(qn))' Since the boundary aXf C Xf is an extremal subset, we have an(t) E aXf for all

Clearly, the function h : X f --+ JR, h : (x, y) f---t Y is concave. Therefore, from 3.1.2, there is a sequence tn > in, so an(tn) --+ (p,f(p) - 1). Therefore, (p, f(p) - 1) E aXf thus P E aA, i.e. aA i= 0, a contradiction. 0 COROLLARY 3.3.3. Let A E ALeXm(K) , m ~ 2 and 'Y be a unit-speed curve in A which has a convex K -developing with respect to any point. Then 'Y is a quasigeodesic, i. e. for any A -concave function f, function f 0 'Y is A-concave. PROOF. Let us first note that in the proof of theorem 3.3.1 we used only two properties of curve 'Y: I'Y± I = 1 and the convexity of the K-development of'Y with respect to p. Assume K = A = a then sub-graph of f Xf

= {(x,y)

E A x JR I y ~ f(x)}

is a closed convex subset, therefore it forms an Alexandrov's space. Applying the above remark, we get that if 'Y is a unit-speed curve in Xf\aXf with convex a-developing with respect to any point then distaxf 0 'Y 210therwise, pass to a small convex neighborhood of p which exists by by corollary 7.1.2. 220t herwise, add a very concave (Lipschitz) function which exists by theorem 7.1.1.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

159

is concave. Hence, for any c > a, the function fE' which has the level set dist f (c) c ffi. x A like the graph, has a concave restriction to any curve 'Y in A with a convex a-developing with respect to any point in A \ 'Y. Clearly, fE --+ f as c --+ a, hence f 0 'Y is concave. For A-concave function the set X f is no longer convex, but it becomes convex if one changes metric on A x ffi. to parabolic cone 23 and then one can repeat the same arguments. 0

81

REMARK One can also get this corollary from the following lemma: LEMMA 3.3.4. Let A E ALexm(K) , n be an open subset of A and f : be a A -concave L -Lipschitz function. Then function

n --+ ffi.

is (A + 8) -concave in the domain of dejinition24 for some25 8 = 8(L, A, K, c), 8 --+ a as c --+ a. Moreover, if m ~ 2 and 'Y is a unit-speed curve in A with K-convex developing with respect to any point then fE 0'Y is also (A + 8) -concave. PROOF. It is analogous to theorem 3.3.1. We only indicate it in the simplest case, K = A = a. In this case 8 can be taken to be a. Let 'Y be a unit-speed geodesic (or it satisfies the last condition in the lemma). It is enough to show that for any to

(fE 0 'Y)"(to) ::;;

a

in a barrier sense. Let Y = 'Y(to) and x E n be a point for which fE(Y) = f(x)+~lxYI2. The tangent cone Tx splits in direction t~, i.e. there is an isometry Tx --+ ffi. x Cone such that t~f---t (1,0), where 0 E Cone is its origin. Let logx 'Y(t)

= (a(t), v(t)) E ffi. x Cone = Tx.

Consider vector

w(t) = (a(t) -lxyl, v(t)) E ffi. x Cone = Tx. Clearly Iw(t)1 ~ IX'Y(t)l. Set x(t) = gexpy(w(t)) then lemma 3.1.2 gives an estimate for f 0 x(t) while corollary 3.1.3 gives an estimate for 1'Y(t)x(t)12. Hence the result. 0 23i.e. warped-product lR xexp(Constt) A, which is an Alexandrov's space, see [BGP, 4.3.3], [Alexander-Bishop 2004]. 24i.e. at the set where the minimum is defined. 25This function 8(L,.A, K, c) is achieved for the model space A".

A. PETRUNIN

160

Here is yet another illustration for the use of gradient exponents. At first sight it seems very simple, but the proof is not quite obvious. In fact, I did not find any proof of this without applying the gradient exponent. LYTCHAK'S PROBLEM 3.3.5. Let A E Atexm (1). Show that VOlm - l

vA ~

VOlm - l

sm-l

where vA denotes the boundary oj A and sm-l the unit (m - 1) -sphere.

The problem would have followed from conjecture 9.1.1 (that boundary of an Alexandrov's space is an Alexandrov's space), but before this conjecture has been proven, any partial result is of some interest. Among other corollaries of conjecture 9.1.1, it is expected that if A E ALex(l) then vA, equipped with induced intrinsic metric, admits a noncontracting map to sm-l. In particular, its intrinsic diameter is at most 7r, and perimeter of any triangle in vA is at most 27r. This does not follow from the proof below, since in general gexpA1; vB7r / 2 (oz)) ct. vA, i.e. gexpA1; vB7r / 2 (oz)) might have some creases left inside of A, which might be used as a shortcut for curves with ends in vA. Let us first prepare a proposition: PROPOSITION 3.3.6. The inverse oj the gradient exponential map gexpp1 (/'i,; *) is uniquely defined inside any minimizing geodesic starting at p. PROOF. Let '"'( : [0, to] -+ A be a unit-speed minimizing geodesic, '"'((0) = p, '"'((to) = q. From the angle comparison we get that IV' x distp I ~ -cosL.r;.pxq. Therefore, for any ( we have

Therefore, L.r;.p q Ct( (t) is nondecreasing in t, hence the result.

o

Proof oj 3.3.5. Let z E A be the point at maximal distance from vA, in particular it realizes maximum of J = 0"1 0 distaA = sin 0 distaA. From theorem 3.3.1, J is (-f)-concave and J(z) ~ l. Note that A c B7r / 2 (z) , otherwise if yEA with Iyzl > 7r /2, then since J is (-f)-concave and J(y) ~ 0, we have dJ(t~) > 0, i.e. z is not a maximum of J. From this it follows that gradient exponent

is a short onto map. Moreover,

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

161

Indeed, gexp gives a homotopy equivalence 8 B7r /2 (oz) -+ A \z. Clearly, Ez = 8(B7r / 2(oz),s) has no boundary, therefore Hm -l(8A,Z2) =1= 0, see [Grove-Petersen 1993, lemma 1]. Hence for any point x E 8A, any minimizing geodesic zx must have a point of the image gexp(1; 8B7r/2(O)) but, as it is shown in proposition 3.3.6, it can only be its end x. Now since gexpA1; *) : (B7r / 2 (oz),s) -+ A is short and (8B7r / 2 (o),s) is isometric to EzA we get vo18A ~ volEzA and clearly, vol EzA ~ vol 8 m - I . D 4. Extremal subsets

Imagine that you want to move a heavy box inside an empty room by pushing it around. If the box is located in the middle of the room, you can push it in any direction. But once it is pushed against a wall you can not push it back to the center; and once it is pushed into a corner you cannot push it anywhere anymore. The same is true if one tries to move a point in an Alexandrov's space by pushing it along a gradient flow, but the role of walls and corners is played by extremal subsets. Extremal subsets first appeared in the study of their special case - the boundary of an Alexandrov's space (see [Perelman-Petrunin 1993]' and [Petrunin 1997], [Perelman 1997]). An Alexandrov's space without extremal subsets resembles a very nonsmooth Riemannian manifold. The presence of extremal subsets makes it behave as something new and maybe intersting; it gives an interesting additional combinatoric structure which reflects geometry and topology of the space itself, as well as of nearby spaces. 4.1. Definition and properties. It is best to define extremal subsets as "ideals" of the gradient flow, i.e.

4.1.1. Let A E AI.eX. E c A is an extremal subset, if for any semiconcave function t ~ 0 and x E E, we have } (x) E E. DEFINITION

f

on A,

Recall that } denotes the f -gradient flow for time t, see 2.2. Here is a quick corollary of this definition: (1) Extremal subsets are closed. Moreover: (i) For any point pEA, there is an E > 0, such that if an extremal subset intersects E-neighborhood of p then it contains p. (ii) On each extremal subset the intrinsic metric is locally finite. These properties follow from the fact that the gradient flow for a A-concave function with dpflEp < 0 pushes a small ball Bc:(p) to p in time proportionate to E.

162

A. PETRUNIN

Examples.

(i) An Alexandrov's space itself, as well as the empty set, forms an extremal subsets. (ii) A point pEA forms a one-point extremal subset if its space of directions Ep has a diameter ~ 1r12 (iii) If one takes a subset of points of an Alexandrov's space with tangent cones homeomorphic 26 to each other then its closure27 forms an extremal subset. In particular, if in this construction we take points with tangent cone homeomorphic to ~+ x IRm-1 then we get the boundary of an Alexandrov's space. This follows from theorem 4.1.2 and the Morse lemma (property 7 page 181). (iv) Let AIG be a factor of an Alexandrov's space by an isometry group, and SH C A be the set of points with stabilizer conjugate to a subgroup H c G (or its connected component). Then the closure of the projection of SH in AIG forms an extremal subset. For example: A cube can be presented as a quotient of a flat torus by a discrete isometry group, and each face of the cube forms an extremal subset. The following theorem gives an equivalence of our definition of extremal subset and the definition given in [Perelman-Petrunin 1993]: 4.1.2. A closed subset E in an Alexandrov's space A is extremal if and only if for any q E A \E, the following condition is fulfilled: If distq has a local minimum on E at a point p, then p is a critical point of distq on A, i. e., V p distq = op. THEOREM

PROOF. For the "only if" part, note that if pEE is not a critical point of distq, then one can find a point x close to p so that t~ is uniquely defined and close to the direction of V p distq, so dp distq(t~) > o. Since V p dist x is polar to ~ (see page 143) we get

dp distq(V p dist x ) < 0, see inequality 1.3 on page 144. Hence, the gradient flow ~istx pushes the point p closer to q, which contradicts the fact that p is a minimum point distq on E. To prove the "if" part, it is enough to show that if F c A satisfies the condition of the theorem, then for any p E F, and any semiconcave function f, either V pf = op or I~:~I E EpF. If so, an f -gradient curve 26Equivalently, with homeomorphic small spherical neigborhoods. The equivalence follows from Perelman's stability theorem. 27As well as the closure of its connected component.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

163

can be obtained as a limit of broken lines with vertexes on F, and from uniqueness, any gradient curve which starts at F lives in F. Let us use induction on dim A. Note that if F c A satisfies the condition, then the same is true for L,pF C L,p, for any p E F. Then using the inductive hypothesis we get that L,pF C L,p is an extremal subset. If p is isolated, then clearly diam L,p ~ 7r /2 and therefore V pf = 0, so we can assume L,pF =I- 0. Note that dpf is (-dpf)-concave on L,p (see 1.2, page 140). Take ~ =

I~:~I ' so ~ E L,p is the maximal point of dpf. Let

E L,pF be a direction closest to ~, then L.(~, 1]) ~ 7r /2; otherwise F would not satisfy the condition in the theorem for a point q with ~ ~. Hence, since L,pF C L,p is an extremal subset, V TJdpf E L,TJL,pF and therefore 1]

tZ

dTJdpf(~) ~ (VTJdpf, ~) ~ O. Hence, dpf(1]) ~ dpf(~), and therefore ~

= 1],

i.e. I~:~I E L,pF.

D

From this theorem it follows that in the definition of extremal subset (4.1.1), one has to check only squares of distance functions. Namely: Let A E ALex, then E C A is an extremal subset, if for any point pEA, and any x E E, we have CPtd' t 2 (x) E E for any t ~ O. IS P

In particular, applying lemma 2.1.5 we get LEMMA 4.1.3. The limit of extremal subsets is an extremal subset.

Namely, if An E ALeXm(K) , An ~ A and En C An is a sequence of extremal subsets such that En -+ E c A then E is an extremal subset of A. The following is yet another important technical lemma: LEMMA 4.1.4. [Perelman-Petrunin 1993, 3.1(2)] Let A E ALex be compact, then there is e > 0 such that distE has no critical values in (0, e). Moreover, IV x distE I > e if 0 < distE(x) < e. For a non-compact A, the same is true for the restriction distE In to any bounded open n cA. PROOF. Follows from lemma 4.1.5 and theorem 4.1.2.

D

LEMMA ABOUT AN OBTUSE ANGLE 4.1.5. Given v > 0, r > 0, K E ~ and mEN, there is e = e(v, r, K, m) > 0 such that if A E ALexffi(K), pEA, volm Br(P) > v, then for any two points x, y E Br(P) , IxYI < e there is point z E Br (p) such that L.zxy > 7r /2 + e or L.zyx > 7r /2 + e. The proof i.s based on a volume comparison for logx : A -+ Tx similar to [Grove-Petersen 1988, lemma 1.3].

164

A. PETRUNIN

Note that the tangent cone TpE of an extremal subset E c A is well defined; i.e. for any pEE, subsets sE in (sA, p) converge to a sub cone of TpE C TpA as s --+ 00. Indeed, assume E C A is an extremal subset and pEE. For any ~ E L. pE28, the radial curve gexp(t .~) lies in E .29 In particular, there is a curve which goes in any tangent direction of E. Therefore, as s --+ 00, (sE c sA, p) converges to a sub cone TpE C TpA, which is simply cone over L.pE (see also [Perelman-Petrunin 1993,3.3]) Next we list some properties of tangent cones of extremal subsets: (2) A closed subset E C A is extremal if and only if the following condition is fulfilled: • At any point pEE, its tangent cone TpE C TpA is well defined, and it is an extremal subset of the tangent cone TpA. (compare [Perelman-Petrunin 1993, 1.4]) (Here is an equivalent formulation in terms of the space of directions: For any pEE, either (a) L.pE = 0 and diam L.p ~ 7r /2 or (b) L.pE = {~} is one point extremal subset and B7r/2(~) = L.p or (c) L.pE is extremal subset of L.p with at least two points.) TpE is extremal as a limit of extremal subsets, see lemma 4.1.3. On the other hand for any semiconcave function I and pEE, the differential dpl : Tp --+ lR is concave and since TpE C Tp is extremal we have V' pi E TpE. I.e. gradient curves can be approximated by broken geodesics with vertices on E, see page 145. (3) [Perelman-Petrunin 1993,3.4-5] If E and F are extremal subsets then so are (i) EnF and for any p E EnF we have Tp(EUF) = TpEUL.pF (ii) EUF and for any p E EUF we have Tp(EnF) = TpEnL.pF (iii) E\F and for any p E E\F we have Tp(E\F) = TpE\TpF In particular, if TpE = TpF then E and F coincide in a neighborhood of p. The properties (i) and (ii) are obvious. The property (iii) follows from property 2 and lemma 4.1.4. We continue with properties of the intrinsic metric of extremal subsets: (4) [Perelman-Petrunin 1993,3.2(3)] Let A E Atexm(K.) and E C A be an extremal subset. Then the induced metric of E is locally biLipschitz equivalent to its induced intrinsic metric. Moreover, the local Lipschitz constant at point pEE can be expressed in terms of m, K. and volume of a ball v = vol Br(P) for some (any) r > O. From lemma 4.1.5, it follows that for two sufficiently close points x, y E E near p there is a point z so that (V' x dist z , t~) > c or (V' y dist z , t~) > c. Then, for the corresponding point, say x, the 28For a closed subset X C A, and p EX, EpX C Ep denotes the set of tangent directions to X at p, i.e. the set of limits of t~n for qn -t p, qn EX. 29That follows from the fact that the curves t t-t gexp( t . t~n) starting with qn belong to E and their converge to gexp(t· E).

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

165

gradient curve t -+ cI>~istJX) lies in E, it is I-Lipschitz and the distance IcI>~istJ x ) y I is decreasing with the speed of at least €. Hence the result. (5) Let An E ALexffi(K) , An ~ A without collapse (i.e. dimA = m) and En C An be extremal subsets. Assume En -+ E c A as subsets. Then (i) [Kapovitch 2007,9.1] For all large n, there is a homeomorphism of pairs (An, En) -+ (A, E). In particular, for all large n, En is homeomorphic to E, (ii) [Petrunin 1997,1.2] En ~ E as length metric spaces (with the intrinsic metrics induced from An and A). The first property is a coproduct of the proof of Perelman's stability theorem. The proof of the second is an application of quasigeodesics . (6) [Petrunin 1997, 1.4] The first variation formula. Assume A E ALex and E C A is an extremal subset, let us denote by I**IE its intrinsic metric. Let p, q E E and a(t) be a curve in E starting from p in direction a+(O) E L,pE. Then

la(t) qlE = IpqlE - cos cp. t

+ o(t).

where cp is the minimal (intrinsic) distance in L,pE between a+(O) and a direction of a shortest path in E from p to q (if cp > 1f, we assume coscp = -1). (7) Generalized Lieberman's Lemma. Any minimizing geodesic for the induced intrinsic metric on an extremal subset is a quasigeodesic in the ambient space. See 2.3.1 for the proof and discussion. Let us denote by Ext(x) the minimal extremal subset which contains a point x EA. Extremal subsets which can be obtained this way will be called primitive. Set ExtO(x)

= {y

E

AI Ext(y) = Ext(x)};

let us call ExtO(x) the main part of Ext(x). ExtO(x) is the same as Ext(x) with its proper extremal subsets removed. From property 3iii on page 162, ExtO(x) is open and everywhere dense in Ext(x). Clearly the main parts of primitive extremal subsets form a disjoint covering of M. (8) [Perelman-Petrunin 1993,3.8] Stratification. The main part of a primitive extremal subset is a topological manifold. In particular, the main parts of primitive extremal subsets stratify Alexandrov's space into topological manifolds. This follows from theorem 4.1.2 and the Morse lemma (property 7 page 181); see also example iii, page 162.

A. PETRUNIN

166

4.2. Applications. The notion of extremal subsets is used to make more precise formulations. Here is the simplest example, a version of the radius sphere theorem: THEOREM 4.2.1. Let A E Atexm (l), diamA > 7r/2 and A have no extremal subsets. Then A is homeomorphic to a sphere. From lemma 5.2.1 and theorem 4.1.2, we have A E Atex(l) , radA > 7r /2 implies that A has no extremal subsets. I.e. this theorem does indeed generalize the radius sphere theorem 5.2.2(ii). PROOF. Assume p, q E A realize the diameter of A. Since A has no extremal subsets, from example iii, page 162, it follows that a small spherical neighborhood of pEA is homeomorphic to ]Rm. From angle comparison, distp has only two critical points p and q. Therefore, this theorem follows from the Morse lemma (property 7 page 181) applied to distp. 0 The main result of such type is the result in [Perelman 1997]. It roughly states that a collapsing to a compact space without proper extremal subsets carries a natural Serre bundle structure. This theorem is analogous to the following: FIBRATION THEOREM 4.2.2. [Yamaguchi].Let An E Atexn(K) and An GH) M, M be a Riemannian manifold. Then there is a sequence of locally trivial fiber bundles Un : An -+ M. Moreover, Un can be chosen to be almost submetries30 and the diameters of its fibers converge to o. The conclusion in Perelman's theorem is weaker, but on the other hand it is just as good for practical purposes. In addition it is sharp, i.e. there are examples of a collapse to spaces with extremal subsets which do not have the homotopy lifting property. Here is a source of examples: take a compact Riemannian manifold M with an isometric and non-free action by a compact connected Lie group G, then (M x cG)/G GH) M/G as c -+ 0 and since the curvature of G is non-negative, by O'Naill's formula, we get that the curvature of (M x cG)/G is uniformly bounded below. HOMOTOPY LIFTING THEOREM 4.2.3. Let An ~ A, An E Atexm(K), A be compact without proper extremal subsets and K be a finite simplicial complex. Then, given a homotopy

Ft : K -+ A, t E [0, 1] 30Le.

Lipshitz and co-Lipschitz with constants almost 1.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

and a sequence of maps GO;n : K -+ An such that GO,n -+ Fo as n -+ one can extend GO;n by homotopies

167

00

Gt;n: K -+ A

such that Gt;n -+ F t as n -+

00.

An alternative proof is based on Lemma 2.3.4. REMARK 4.2.4. As a corollary of this theorem one obtains that for all large n it is possible to write a homotopy exact sequence:

where the space Fn can be obtained the following way: Take a point pEA, and fix e > 0 so that distp : A -+ lR. has no critical values in the interval (0, 2e). Consider a sequence of points An :1 Pn -+ P and take Fn = Bc(Pn) C An. In particular, if P is a regular point then for large n, Fn is homotopy equivalent to a regular fiber over p31. Next we give two corollaries of the above remark. The last assertion of the following theorem was conjectured in [Shioya] and was proved in [Mendonc;a]. THEOREM 4.2.5. [Perelman 1991,3.1]. Let M be a complete noncompact Riemannian manifold of nonnegative sectional curvature. Assume that its asymptotic cone Cone oo (M) has no proper extremal subsets, then M splits isometrically into the product L x N, where L is a compact Riemannian manifold and N is a non-compact Riemannian manifold of the same dimension as Cone oo (M) . In particular, the same conclusion holds if radius of the ideal boundary of M is at least 7f /2. The proof is a direct application of theorem 4.2.3 and remark 4.2.3 for collapsing GH eM ---+ Coneoo(M), as e -+ O. GH THEOREM 4.2.6. [Perelman 1991,3.2]. Let An E Alexm (l), An ---+ A be a collapsing sequence (i. e. m > dim A), then Cone( A) contains proper extremal subsets. In particular, rad A ~ 7f /2. 31 It is constructed the following way: take a distance chart G : B2. (p) ---+ JRk, k = dimA around pEA and lift it to An. It defines a map Gn : B.(Pn) ---+ JR k . Then take Fn = G;;l 0 G(p) for large n. If An are Riemannian then Fn are manifolds and they do not depend on p up to a homeomorphism. Moreover, Fn are almost non-negatively curved in a generalized sense; see [KPT, definition 1.4].

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The last assertion of this theorem (in a stronger form) has been proven in [Grove-Petersen 1993, 3(3)]. The proof is a direct application of theorem 4.2.3 and remark 4.2.3 for collapsing of spherical suspensions

5. Quasigeodesics The class of quasigeodesics32 generalizes the class of geodesics to nonsmooth metric spaces. It was first introduced in [Alexandrov 1945] for 2-dimensional convex hypersurfaces in the Euclidean space, as the curves which "turn" right and left simultaneously. This type of curves was studied further in [Alexandrov-Burago], [Pogorelov], [Milka 1971]. They were generalized to surfaces with bounded integral curvature [Alexandrov 1949], to multidimensional polyhedral spaces [Milka 1968]' [Milka 1969] and to multidimensional Alexandrov's spaces [Perelman-Petrunin QG]. In Alexandrov's spaces, quasigeodesics behave more naturally than geodesics, mainly: • There is a quasigeodesic starting in any direction from any point; • The limit of quasigeodesics is a quasigeodesic. Quasigeodesics have beauty on their own, but also due to the generalized Lieberman lemma (2.3.1), they are very useful in the study of intrinsic metric of extremal subsets, in particular the boundary of Alexandrov's space. Since quasigeodesics behave almost as geodesics, they are often used instead of geodesics in the situations when there is no geodesic in a given direction. In most of these applications one can instead use the radial curves of gradient exponent, see section 3; a good example is the proof of theorem 3.3.1, see footnote 20, page 157. In this type of argument, radial curves could be considered as a simpler and superior tool since they can be defined in a more general setting, in particular, for infinitely dimensional Alexandrov's spaces. 5.1. Definition and properties. In section 1, we defined A-concave functions as those locally Lipschitz functions whose restriction to any unitspeed minimizing geodesic is A-concave. Now consider a curve , in an Alexandrov's space such that restriction of any A-concave function to , is A-concave. It is easy to see that for any Riemannian manifold, has to be a unit-speed geodesic. In a general Alexandrov's space , should only be a quasigeodesic. 32It should be noted that the class of quasigeodesics described here has nothing to do with the Gromov's quasigeodesics in 6-hyperbolic spaces.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

169

5.1.1. A curve I in an Alexandrov's space is called quasigeodesic if for any A E R, given a A-concave function f we have that f 0 I is A-concave. DEFINITION

Although this definition works for any metric space, it is only reasonable to apply it for the spaces where we have A-concave functions, but not all functions are A-concave, and Alexandrov's spaces seem to be the perfect choice. The following is a list of corollaries from this definition:

(1) Quasigeodesics are unit-speed curves. i.e., if I(t) is a quasigeodesic then for any to we have lim I/(t)J(to) I = 1. t-+to

It -- tol

To prove that quasigeodesic I is I-Lipschitz at some t = to, it is enough to apply the definition for f = dist;(to) and use the fact that in any Alexandrov's space dist~ is (2 + O( r2) ) -concave in Br(P)' The lower bound is more complicated, see theorem 7.3.3. (2) For any quasigeodesic the right and left tangent vectors 1+, I are uniquely defined unit vectors. E T-y(to) for To prove, take a partial limits

e±

log-y(to) I(tO ± T) ---'-'-"-'-----, as

T

-+ 0+

T

It exists since quasigeodesics are I-Lipschitz (see the previous property). For any semiconcave function f, (f 0/)± are well defined, therefore

Taking f = dist~ for different q E A, one can see that E± is defined uniquely by this identity, and therefore I±(to) = (3) Generalized Lieberman's Lemma. Any unit-speed geodesic for the induced intrinsic metric on an extremal subset is a quasigeodesic in the ambient Alexandrov's space. See 2.3.1 for the proof and discussion. (4) For any point x E A, and any direction E ~x there is a quasigeodesic I : R -+ A such that 1(0) = x and 1+(0) = E. Moreover, if E c A is an extremal subset and x E E, EE ~xE, then I can be chosen to lie completely in E. The proof is quite long, it is given in appendix A.

e± .

e

Applying the definition locally, we get that if f is a (1 -- K,f)-concave function then f 0 I is (1 -- K,f 0 I)-concave (see section 1.2). In particular,

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A. PETRUNIN

if A is an Alexandrov's space with curvature ~ K, pEA and hp(t) = p,. 0 distp O"Y(t)33 then we have the following inequality in the barrier sense

h; :::;; 1 - Khp. This inequality can be reformulated in an equivalent way: Let A E AI.eXm(K) , pEA and "Y be a quasigeodesic, then function

t t-+ L,.(b(O)pl, 1"Y(t)pl, t) is decreasing for any t > 0 (if K > 0 then one has to assume t :::;; 7r / y'K,). In particular,

for any t > 0 (if K > 0 then in addition t :::;; 7r / y'K,). It also can be reformulated more geometrically using the notion of developing (see below): Any quasigeodesic in an Alexandrov's space with curvature ~ K, has a convex K -developing with respect to any point. DEFINITION OF DEVELOPING 5.1.2. [Alexandrov 1957] Fix a real K. Let X be a metric space, "Y : [a, b] -+ X be a I-Lipschitz curve and p E X\"Y. If K > 0, assume in addition that In(t)1 < 7r/y'K, for all t E [a,b]. Then there exists a unique (up to rotation) curve i : [a, b] -+ JI,., parametrized by the arclength, and such that 10i(t)1 = In(t)1 for all t and some fixed 0 E JI,., and the segment oi(t) turns clockwise as t increases (this is easy to prove). Such a curve i is called the K-development of"Y with respect to p. The development i is called convex if for every t E (a, b) , for sufficiently small T > 0 the curvilinear triangle, bounded by the segments 01'( t ± T) and the arc ilt-r,t+r, is convex.

In [Milka 1971]' it has been proven that the developing of a quasigeodesic on a convex surface is convex. (5) Let A E ALe:z:m(K), m > 134. A curve "Y in A is a quasigeodesic if and only if it is parametrized by arc-length and one of the following properties is fulfilled: (i) For any point p E A\"Y the K-developing of "Y with respect to p is convex. (ii) For any point pEA, if hp(t) = p,. 0 distp O"Y(t) , then we have the following inequality in a barrier sense

h;:::;; 1- Khp. 33Function PI< : lR -+ lR is defined on page 140. 34This condition is only needed to ensure that the set A \ "I is everywhere dense.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

171

(iii) Function t

H

L,.(b(O)p!, b(t)p!, t)

is decreasing for any t > 0 (if

t

~

7r /

/'i,

> 0 then in addition

..fo,).

(iv) The inequality

holds for all small t > O. The "only if" part has already been proven above, and the "if" part follows from corollary 3.3.3 (6) A pointwise limit of quasigeodesics is a quasigeodesic. More generally: Assume An ~ A, An E Atexm{/'i,) , dim A = m (i.e. it is not

a collapse). Let "Yn : [a, b] ---+ An be a sequence of quasigeodesics which converges pointwise to a curve "Y : [a, b] ---+ A. Then "Y is a quasigeodesic. As it follows from lemma 7.2.3, the statement in the definition is correct for any A-concave function f which has controlled convexity type (A, /'i,). I.e. "Y satisfies the property 7.3.4. In particular, the /'i,developing of "Y with respect to any point pEA is convex, and as it is noted in remark 7.3.5, "Y is a unit-speed curve. Therefore, from corollary 3.3.3 we get that it is a quasigeodesic. Here is a list of open problems on quasigeodesics: (i) Is there an analog of the Liouvile theorem for "quasigeodesic flow"? (ii) Is it true that any finite quasigeodesic has bounded variation of turn? or Is it possible to approximate any finite quasigeodesic by sequence of broken lines with bounded variation of turn? (iii) Is it true that in an Alexandrov's space without boundary there is an infinitely long geodesic? As it was noted by A. Lytchak, the first and last questions can be reduced to the following: Assume A is a compact Alexandrov's m-space without boundary. Let us set V{r) = fA volm(Br{x)) , then

V{r) = volm{A)wmrm

+ o{rm+1).

The technique of tight maps makes it possible to prove only that V{r) = + o (rm+1 ). Note that if A is a Riemannian manifold with boundary then

volm(A)wmrm

V(r) = volm(A)wmrm

+ VOl m_l(8A)w:nrm+1 + o(rm+1).

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5.2. Applications. The quasigeodesics is the main technical tool in the questions linked to the intrinsic metric of extremal subsets, in particular the boundary of Alexandrov's space. The main examples are the proofs of convergence of intrinsic metric of extremal subsets and the first variation formula (see properties 5ii and 6, on page 165). Below we give a couple of simpler examples: LEMMA 5.2.1. Let A E ALexm (l) and radA > 7r/2. Then for any pEA the space of directions ~p has radius >7r /2. PROOF. Assume that ~p has radius '5:.7r/2, and let ~ E ~p be a direction, such that fh,(7r/2) = ~p. Consider a quasigeodesic 'Y starting at p in direction ~. Then for q = 'Y(7r/2) we have Bq (7r/2) = A. Indeed, for any point x E A we have L(~, t~) ~ 7r/2. Therefore, by the comparison inequality (property 5iv, page 171), Ixql ~ 7r /2. This contradicts our assumption that radA > 7r/2. 0 COROLLARY 5.2.2. Let A E ALeXm(l) and radA > 7r/2 then (i) A has no extremal subsets. (ii) [Grove-Petersen 1993] (radius sphere theorem) A is homeomorphic to an m -sphere. Yet another proof of the radius sphere theorem follows immediately from [Perelman-Petrunin 1993, 1.2, 1.4.1]; theorem 4.2.1 gives a slight generalization. PROOF. Part (i) is obvious. Part (ii): From lemma 5.2.1, rad~p > 7r/2. Since dim~p < m, by the induction hypothesis we have ~p ~ sm-l. Now the Morse lemma (see property 7, page 181) for distp : A -+ jR gives that A ~ ~(~p) ~ sm, here ~(~p) denotes a spherical suspension over ~p. 0 6. Simple functions This is a short technical section. Here we introduce simple functions, a subclass of semiconcave functions which on one hand includes all functions we need and in addition is liftable; i.e. for any such function one can construct a nearby function on a nearby space with "similar" properties. Our definition of simple function is a modification of two different definitions of so called "admissible functions" given in [Perelman 1993, 3.2] and [Kapovitch 2007, 5.1]. DEFINITION 6.1.1. Let A E ALex, a function f : A -+ jR is called simple if there is a finite set of points {qd~1 and a semiconcave function 8 : jRN -+ jR which is non-decreasing in each argument such that

f(x) = 8(dist~1,dist~2'··· ,dist~N)

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

173

It is straightforward to check that simple functions are semiconcave. Class of simple functions is closed under summation, multiplication by a positive constant 35 and taking the minimum. In addition this class is liftable; i.e. given a converging sequence of Alexandrov's spaces An ~ A and a simple function I : A -+ lR. there is a way to construct a sequence of functions In: An -+ lR. such that In -+ I. Namely, for each qi take a sequence An :3 qi,n -+ qi E A and consider function In: An -+ lR. defined by

In = 8(dist~1,n,dist~2,n'··· ,dist~N,J· 6.2. Smoothing trick. Here we present a trick which is very useful for doing local analysis in Alexandrov's spaces, it was introduced in [Otsu-Shioya, section 5]. Consider function

diStp =

f

dist x dx.

Be(P)

In this nota~, we do not specify c assuming it to be very small. It is easy to see that distp is semiconcave. Note that dydistp =

f

dy dist x dx.

Be(P)

If yEA is regular, i.e. Ty is isometric to Euclidean space, then for almost all ~ Be (p) the differential dy dist x : Ty -+ lR. is a linear function. Therefore distp is differentiable at every regular point, i.e.

is a linear function for any regular YEA. The same trick can be applied to any simple function

This way we obtain function

f(x) =

1

8(dist;1,dist;2, ... ,dist;N)dx1dx2···dxN,

JBe(ql) XBe(Q2) x··· XBe(QN)

which is differentiable at every regular point, i.e. if Ty is isometric to the Euclidean space then is a linear function. 35As well as multiplication by positive simple functions.

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7. Controlled concavity In this and the next sections we introduce a couple of techniques which use comparison of m-dimensional Alexandrov's space with a model space of the same dimension J1~ (i.e. simply connected Riemannian manifold with constant curvature K,). These techniques were introduced in [Perelman 1993] and [Perelman DC]. We start with the local existence of a strictly concave function on an Alexandrov's space. THEOREM 7.1.1. [Perelman 1993,3.6]. Let A E ALex. For any point pEA there is a strictly concave function f defined in an open neighborhood of p. Moreover, given v E Tp, the differential, dpf(x) , can be chosen arbitrarily close to x H -(v, x) PROOF. Consider the real function

({Jr,c(x) = (x - r) - c(x - r)2Ir, so we have

({Jr,c(r)

= 0,

({J~,c(r)

=1

({J~,c(r)

= -2clr.

Let "I be a unit-speed geodesic, fix a point q and set

If r > 0 is sufficiently small and Iq'Y(t)1 is sufficiently close to r, then direct calculations show that )"() 3 - ccos2 a(t) . (({Jr c ° dlst t ~ . q 0"1 , r

q

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

175

Now, assume {qd, i = {1, .. , N} is a finite set of points such that Ipqi I = r for any i. For x E A and ~x E ~x, set O:i(~x) = L(~x,t~i). Assume we have a collection {qi} such that for any x E Be (p) and ~x E I: x we have maxi IO:i(~x) - 7r/21 ;:: c > O. Then taking in the above inequality c > 3NI cos 2 c , we get that the function

f

=

L 0 there is a function f of controlled concavity type (-2 + E, -E) on a neighborhood of E Tp such that

,±

o

Applying the property above we get I,±(O)I ~ 1.

REMARK 7.3.5. Note that we have proven a slightly stronger statement; namely, if a curve , satisfies the property 7.3.4 then it is a unit-speed curve.

7.3.6. Is it true that for any point pEA and any E > 0, there is a (-2 + E)-concave function fp defined in a neighborhood of p, such that fp(p) = 0 and fp ~ - dist~? QUESTION

Existence of a such function would be a useful technical tool. In particular, it would allow for an easier proof of the above theorem. 8. Tight maps The tight maps considered in this section give a more flexible version of distance charts. Similar maps (so called regular maps) were used in [Perelman 1991, Perelman 1993]; in [Perelman Dej, they were modified to nearly this form. This technique is also useful for Alexandrov's spaces with upper curvature bound, see [Lytchak-Nagano]. 8.1.1. Let A E ALexm and n c A be an open subset. A collection of semiconcave functions fo, h, .. . ,fi on A is called tight in n if DEFINITION

In this case the map F : n -+ RHl, F: x

M

(Jo(x), h(x), ... , !£(x))

is called tight. A point x E n is called a critical point of F if mini dxfi ::;; 0, otherwise the point x is called regular. 43Setting v

= ")'± (0) E Tp and f

= A( cpr,e

0

w

= 2")'± (0), this function can be presented as a sum

dist o +cpr,e 0 distw)

+B L

CPr' ,e'

0

dist q;,

i

for appropriately chosen positive reals A, B, r, r', c, c' and a collection of points qi such that, L.opqi = L.OOpqi = 7r /2, Ipqi I = r.

A. PETRUNIN

180

MAIN EXAMPLE

8.1.2. If A

E ALexm(K)

and aO,al, ... ,ae,P

E

A such

that then the map x

H

(Iaoxl, lalxl, ... , laexl) is tight in a neighborhood of p.

The inequality in the definition follows from inequality (**) on page 144 and a subsequent to it example (ii). This example can be made slightly more general. Let fo,!I, ... , fe be a collection of simple functions

fi = 8i( dist~l ., dist~2 x'···, dist~ nt,' . ·x) ,t

and the sets of points Ki

,t

= {ak,d satisfy the following inequality

L",xpY>7r/2 for any xEKi, yEKj , i=/=j. Then the map x H (fo(x), !I(x), ... , fe(x)) is tight in a neighborhood of p. We will call such a map a simple tight map. Yet further generalization is given in the property 1 below. The maps described in this example have an important property, they are liftable and their lifts are tight. Namely, given a converging sequence An ~ A, An E ALexm(K) and a simple tight map F : A -+ JRHI around pEA, the construction in section 6 gives simple tight maps Fn : An -+ JRe for large n, Fn -+ F. I was unable to prove that tightness is a stable property in a sense formulated in the question below. It is not really important for the theory since all maps which appear naturally are simple (or, in the worst case they are as in the generalization and as in the property 1). However, for the beauty of the theory it would be nice to have a positive answer to the following question. QUESTION 8.1.3. Assume An ~ A, An E ALexm(K) , f,g: A -+ JR is a tight collection around p and fn,gn : An -+ JR, fn -+ f, gn -+ 9 are two sequences of A-concave functions and An :3 Pn -+ pEA. Is it true that for all large n, the collection fn, gn must be tight around Pn? If not, can one modify the definition of tightness so that (i) it would be stable in the above sense, (ii) the definition would make sense for all semiconcave functions (iii) the maps described in the main example above are tight?

Let us list some properties of tight maps with sketches of proofs: (1) Let x H (fo(x) , !I(x), ... , fe(x)) be a tight map in an open subset n c A, then there is c > 0 such that if go, gl, ... ,gn is a collection of E-Lipschitz semiconcave functions in n then the map

+ 90(X), !I(x) + gl(X), ... , fe(x) + ge(x)) is also tight in n. x

H

(fo(x)

181

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

(2) The set of regular points of a tight map is open. Indeed, let x E 0 be a regular point of tight map F = (fa, II, ... , if). Take real>. so that all Ii are >.-concave in a neighborhood of x. Take a point p sufficiently close to x such that dxli(t~) > o and moreover Ii(p) - Ii(x) > >'lxpI2/2 for each i. Then, from >.-concavity of Ii, there is a small neighborhood Ox :3 x such that for any y E Ox and i we have dyIi(t~) ~ c for some fixed c > O. (3) If one removes one function from a tight collection (in 0) then (for the corresponding map) all points of 0 become regular. In other words, the projection of a tight map F to any coordinate hyperplane is a tight map with all regular points (in 0). This follows from the property 3 on page 149 applied to the flow for the removed Ii. (4) The converse also holds, i.e. if F is regular at x then one can find a semiconcave function 9 such that map z f--t (F(z),g(z)) is tight in a neighborhood of x. Moreover, 9 can be chosen to have an arbitrary controlled concavity type. Indeed, one can take 9 = distp , where p as in the property 2. Then we have

dxg(v) = -

max(~, ~E11'~

v)

and therefore

On the other hand, from inequality (**) on page 144 and example (ii) subsequent to it, we have

The last statement follows from the construction in theorem 7.1.1. (5) A tight map is open and even co-Lipschitz 44 in a neighborhood of any regular point. This follows from lemma 8.1.4. (6) Let A E ALeX, 0 c A be an open subset. If F : 0 -+ RH1 is tight then e ~ dimA. Follows from the properties 3 and 5. (7) Morse lemma. A tight map admits a local splitting in a neighborhood of its regular point, and a proper everywhere regular tight map is a locally trivial fiber bundle. Namely 44A map F: X -+ Y between metric spaces is called L-co-Lipschitz in any ball Br(x) C n we have F(Br(x)) :) Br/dF(x)) in Y.

nc

X if for

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A. PETRUNIN

(i) If F : 0 ~ Rl+l is a tight map and p E 0 is a regular point, then there is a neighborhood 0 :::::) Op :3 P and homeomorphism such that F 0 h coincides with the projection to the second coordinate Y x F(Op) ~ F(Op). (ii) If F: 0 ~ ~ C Rl+l is a proper tight map and all points in ~ C Rl+l are regular values of F, then F is a locally trivial fiber bundle. The proof is a backward induction on f, see [Perelman 1993, 1.4], [Perelman 1991, 1.4.1] or [Kapovitch 2001, 6.7]. The following lemma is an analog of lemmas [Perelman 1993,2.3] and [Perelman DC, 2.2]. LEMMA

8.1.4. Let x be a regular point of a tight map

F: x

f-t

(fo(x) , h(x), ... , ft(x)).

Then there is e > 0 and a neighborhood Ox :3 x such that for any y E Ox and i E {O, 1, ... ,f} there is a unit vector Wi E ~x such that dxfi(Wi) ~ e and dxf;(Wi) = 0 for all j t- i. Moreover, if E c A is an extremal subset and y E E then Wi can be chosen in ~yE. PROOF. Take p as in the property 2 page 181. Then we can find a neighborhood Ox :3 x and e > 0 so that for any y E Ox

(i) dyfi(t~)

> e for each

i;

(ii) -dyfi('l yf;) > e. for all i

t- j.

Note that if a(t) is an h-gradient curve in Ox then

(fi

0

a)+ > 0 and (f;

0

a)+ ~ -e for any j

t= i.

Applying lemma 2.1.5 for (sA, y) GH) Ty , s[h - h(y)] ~ dyfi' we get the same inequalities for dyfi-gradient curves on T y , i.e. if f3(t) is an dyhgradient curve in Ty then

Moreover, dyfi(V) > 0 implies ('lvdyfi, t~) < 0, therefore in this case 1f3(t)l+ > O. Take Wo E Ty to be a maximum point for dyfo on the set

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

183

Then Assume for some j

t=

dylo(wo) ~ dylo(t~) > c. 0 we have Ij (wo) > O. Then

where the function v is defined by v : v t--+ -Ivl; this is a concave function on Ty . Therefore, if (3j(t) is a dyfJ-gradient curve with an end 45 point at wo, then moving along {3j from Wo backwards decreases only dyfJ, and increases the other dyli and v in the first order; this is a contradiction. To prove the last statement it is enough to show that Wo E TyE, which follows since TyE C Ty is an extremal subset (see property 2 on page 164).

o MAIN THEOREM 8.1.5. Let A compact convex subset, and

E ALexffl(K) ,

0 C A be the interior of a

F: 0 -+ jRH1, F: x t--+ Uo(x),h(x), ... ,JR.(x))

be a tight map. Assume all Ii are strictly concave. Then (i) the set of critical points of F in 0 forms an f-submanifold M (ii) F: M -+ jRHl is an embedding. (iii) F(M) c jRHl is a convex hypersurface which lies in the boundary of

F(O)46. REMARK 8.1.6. The condition that all Ii are strictly concave seems to be very restrictive, but that is not really so; if x is a regular point of a tight map F then, using properties 1 and 4 on page 180, one can find c > 0 and 9 such that

F' : y t--+ Uo(y)

+ cg(y), ... , JR.(y) + cg(y), g(y))

is tight in a small neighborhood of x and all its coordinate functions are strictly concave. In particular, in a neighborhood of x we have

F where L : jRH2 -+

jRHl

= LoF'

is linear.

COROLLARY 8.1.7. In the assumptions of theorem 8.1.5, if in addition m = f then M = 0, F(O) is a convex hypersurface in jRffl+1 and F: 0 -+ jRffl+1 is a locally bi-Lipschitz embedding. Moreover, each projection of F to a coordinate hyperplane is a locally bi-Lipschitz homeomorphism.

45It does exist by property 3 on page 149. 46In fact F(M) = 8F(0.) n F(0.).

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A. PETRUNIN

8.1.5. Let, : [0, s]-+ A be a minimal unit-speed geodesic connecting x, YEO, so s = IxYI. Consider a straight segment i connecting F(x) and F(y): PROOF OF THEOREM

i: [0, s] -+ ~Hl, i(t) = F(x) Each function

Ii

0,

+ ~ [F(y) - F(x)].

is concave, therefore all coordinates of F o,(t) - i(t)

are non-negative. This implies that the Minkowski sum47

Q = F(O) + (~_ )Hl is a convex set. Let Xo E 0 be a critical point of F. Since mini dxofi ~ 0, at least one of coordinates of F(x) is smaller than the corresponding coordinate of F(xo) for any x E O. In particular, F sends its critical point to the boundary of Q. Consider map

where argmax{J} denotes a maximum point of f. The function mindli-Yi} is strictly concave; therefore argmax{mindfi - Yi}} is uniquely defined and G is continuous in the domain of definition. 48 The image of G coincides with the set of critical points of F and moreover Go FIM = idM. Therefore FIM is a homeomorphism49 . 0 PROOF OF COROLLARY 8.1.7. It only remains to show that F is locally bi-Lipschitz. Note that for any point x E 0, one can find € > and a neighborhood Ox 3 x, so that for any direction ~ E ~Y' Y E Ox one can choose Ii, i E {O, 1, ... ,m}, such that dxfi(~) ~ -€. Otherwise, by a slight perturbation 5o of collection {Jd we get a map F : Am -+ ~m+l regular at y, which contradicts property 5.

°

47Equivalently Q = {(xo, Xl, ... ,Xl) E Rl+ 1 13(yo, Yl,· .. ,Yl) E F(n)Vi Xi :::;; y;}. 48We do not need it, but clearly

for any hER. 49In general, G is not Lipschitz (even on F(M)); even in the case when all functions hare (-1) -concave it is only possible to prove that G is Holder continuous of class Co;~ . (In fact the statement in [Perelman 1991], page 20, lines 23-25 is wrong but the proposition 3.5 is still OK.) 50 As in the property 1 on page 180.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

Therefore applying it for ~ =t~ and t~, such that

Z,

185

yEn, we get two values i, j

Therefore F is bi-Lipschits. Clearly i =1= j and therefore at least one of them is not zero. Hence the projection map F' : x H (ft(x) , ... , fm{x)) is also locally bi-Lipschitz. D

8.2. Applications. One series of applications of tight maps is Morse theory for Alexandrov's spaces, it is based on the main theorem 8.1.5. It includes Morse lemma (property 7 page 181) and • Local structure theorem [Perelman 1993]. Any small spherical neighborhood of a point in an Alexandrov's space is homeomorphic to a cone over its boundary. • Stability theorem [Perelman 1991]. For any compact A E Atern(K) there is c > 0 such that if A' E Atexm (K) is c -close to A then A and A' are homeomorphic. The other series is the regularity results on an Alexandrov's space. These results obtained in [Perelman DC] are improvements of earlier results in [Otsu-Shioya], [Otsu]. It use mainly the corollary 8.1.7 and the smoothing trick; see subsection 6.2. • Components of metric tensor of an Alexandrov's space in a chart are continuous at each regular point 51 . Moreover they have bounded variation and are differentiable almost everywhere. • The Christoffel symbols in a chart are well defined as signed Radon measures. • Hessian of a semiconcave function on an Alexandrov's space is defined almost everywhere. I.e. if f : n -t jR is a semiconcave function, then for almost any Xo E n there is a symmetric bi-linear form Hessf such that

f(x) = f(xo)

+ dxof(v) + Hessf(v, v) + o(lvI 2 ),

where v = logxo x. Moreover, Hess f can be calculated using standard formulas in the above chart. Here is yet another, completely Riemannian application. This statement has been proven by Perelman, a sketch of its proof is included in an appendix to [Petrunin 2003]. The proof is based on the following observation: if n is an open subset of a Riemannian manifold and P : n -t jRl+l is a tight map with strictly concave coordinate functions, then its level sets p-l(x) inherit the lower curvature bound. • Continuity of the integral of scalar curvature. Given a compact Riemannian manifold M, let us define F(M) = fM Sc. Then F is continuous on 51 Le. at each point with Euclidean tangent space.

A.

186

PETRUNIN

the space of Riemannian m-dimensional manifolds with uniform lower curvature and upper diameter bounds. 52 9. Please deform an Alexandrov's space

In this section we discuss a number of related open problems. They seem to be very hard, but I think it is worth to write them down just to indicate the border between known and unknown things. The main problem in Alexandrov's geometry is to find a way to vary Alexandrov's space, or simply to find a nearby Alexandrov's space to a given Alexandrov's space. Lack of such variation procedure makes it impossible to use Alexandrov's geometry in the way it was designed to be used: For example, assume you want to solve the Hopf conjecture 53. Assume it is wrong, then there is a volume maximizing Alexandrov's metrics d on 8 2 x 8 2 with curvature ~ 154 . Provided we have a procedure to vary d while keeping its curvature ~ 1, we could find some special properties of d and in ideal situation show that d does not exist. Unfortunately, at the moment, except for boring rescaling, there is no variation procedure available. The following conjecture (if true) would give such a procedure. Although it will not be sufficient to solve the Hopf conjecture, it will give some nontrivial information about the critical Alexandrov's metric. CONJECTURE 9.1.1. The boundary of an Alexandrov's space equipped with induced intrinsic metric is an Alexandrov's space with the same lower curvature bound. This also can be reformulated as: CONJECTURE 9.1.1~ Let A be an Alexandrov space without boundary. Then a convex hypersurface in A equipped with induced intrinsic metric is an Alexandrov's space with the same lower curvature bound. This conjecture, if true, would give a variation procedure. For example if ~ is concave

A is a non-negatively curved Alexandrov's space and f : A -+ (so A is necessarily open) then for any t the graph At

= {(x, tf(x))

E A x ~}

with induced intrinsic metric would be an Alexandrov's space. Clearly GH At --+ A as t -+ O. An analogous construction exists for semiconcave 52In fact :F is also bounded on the set of Riemannian m-dimensional manifolds with uniform lower curvature, this is proved in [Petrunin 2007] by a similar method. 53i.e. you want to find ont if S2 x S2 carries a metric with positive sectional curvature. 54There is no reason to believe that this metric d is Riemannian, but from Gromov's compactness theorem such Alexandrov's metric should exist.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

187

functions on closed manifolds, but one has to take a parabolic cone 55 instead of the product. It seems to be hopeless to attack this problem with purely synthetic methods. In fact, so far, even for a convex hypersurface in a Riemannian manifold, there is only one proof available (see [Buyalo]56) which uses smoothing and the Gauss formula. There is one beautiful synthetic proof (see [Milka 1979]) for a convex surface in the Euclidian space, but this proof heavily relies on Euclidean structure and it seems impossible to generalize it even to the Riemannian case. There is a chance of attacking this problem by proving a type of the Gauss formula for Alexandrov's spaces. One has to start with defining a curvature tensor of Alexandrov's spaces (it should be a measure-valued tensor field), then prove that the constructed tensor is really responsible for the geometry of the space. Such things were already done in the two-dimensional case and for spaces with bilaterly bounded curvature, see [Reshetnyak] and [Nikolaev] respectively. So far the best results in this direction are given in [Perelman DC], see also section 8.2 for more details. This approach, if works, would give something really new in the area. Almost everything that is known so far about the intrinsic metric of a boundary is also known for the intrinsic metric of a general extremal subset. In [Perelman-Petrunin 1993], it was conjectured that an analog of conjecture 9.1.1 is true for any primitive extremal subset, but it turned out to be wrong; a simple example was constructed in [Petrunin 1997]. All such examples appear when co dimension of extremal subset is ~ 3. So it still might be true that CONJECTURE 9.1.2. Let A E ALex(K) , E c A be a primitive extremal subset and codim E = 2 then E equipped with induced intrinsic metric belongs to ALex( K)

The following question is closely related to conjecture 9.1.1. QUESTION 9.1.3. Assume An ~ A, An E ALeXm(K) , dim A = m (Le. it is not a collapse). Let I be a A-concave function of an Alexandrov's space A. Is it always possible to find a sequence of A-concave functions In : An -+ ~ which converges to I : A -+ ~ ?

Here is an equivalent formulation: QUESTION 9.1.3~

GH

Assume An --+ A, An it is not a collapse) and 8A = 0.

E ALeXm(K) ,

dim A = m (Le.

55see footnote 23 on page 159. 56In fact in this paper the curvature bound is not optimal, but the statement follows from nearly the same idea; see [AKP].

188

A.

PETRUNIN

Let SeA be a convex hypersurface. Is it always possible to find a sequence of convex hypersurfaces Sn C An which converges to S? If true, this would give a proof of conjecture 9.1.1 for the case of a smoothable Alexandrov's space (see page 177). In most of (possible) applications, Alexandrov's spaces appear as limits of Riemannian manifolds of the same dimension. Therefore, even in this reduced generality, a positive answer would mean enough. The question of whether an Alexandrov space is smoothable is also far from being solved. From Perelamn's stability theorem, if an Alexandrov's space has topological singularities then it is not smoothable. Moreover, from [Kapovitch 2002] one has that any space of directions of a smoothable Alexandrov's space is homeomorphic to the sphere. Except for the 2dimensional case, it is only known that any polyhedral metric of non-negative curvature on a 3-manifold is smoothable (see [Matveev-Shevchishin]). There is yet no procedure of smoothing an Alexandrov's space even in a neighborhood of a regular point. Maybe a more interesting question is whether smoothing is unique up to a diffeomorphism. If the answer is positive it would imply in particular that any Riemannian manifold with curvature ~ 1 and diam > 7r /2 is diffeomorphic(!) to the standard sphere, see [Grove-Wilhelm] for details. Again, from Perelman's stability theorem ([Perelman 1991]), it follows that any two smoothings must be homeomorphic. In fact it seems likely that any two smoothings are PL-homeomorphic; see [Kapovitch 2007, question 1.3] and discussion right before it. It seems that today there is no technique which might approach the general uniqueness problem (so maybe one should try to construct a counterexample). One may also ask similar questions in the collapsing case. In [PWZ] there were constructed Alexandrov's spaces with curvature ~ 1 which can not be presented as a limit of an (even collapsing) sequence of Riemannian manifolds with curvature ~ '" > 1/4. In [Kapovitch 2005] there were found some lower bounds for codimension of collapse with arbitrary lower curvature bound to some special Alexandrov's spaces, see section 7.3 for more discussion. It is expected that the same spaces (for example, the spherical suspension over the Cayley plane) can not be approximated by sequence of Riemannian manifolds of any fixed dimension and any fixed lower curvature bound, but so far this question remains open.

A. Existence of quasigeodesics This appendix is devoted to the proof of property 4 on page 169, i.e. EXISTENCE THEOREM A.O.1. Let A E ALexm , then for any point x E A, and any direction ~ E Ex there is a quasigeodesic , : lR ---+ A such that ,(0) = x and ,+(0) = ~.

SEMI CONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY Moreover if E c A is an extremal subset and x E E, can be chosen to lie completely in E.

~ E ~xE

189

then 'Y

The proof is quite long; it was obtained by Perelman around 1992; here we present a simplified proof similar to [Perelman-Petrunin QG] which is based on the gradient flow technique. We include a complete proof here, since otherwise it would never be published. Quasigeodesics will be constructed in three big steps. A.2: Monotonic curves --t convex curves. A.3: Convex curves --t pre-quasigeodesics. A.4: Pre-quasigeodesics --t quasigeodesics. In each step, we construct a better type of curves from a given type of curves by an extending-and-chopping procedure and then passing to a limit. The last part is most complicated. The second part of the theorem is proved in the subsection A.5.

A.I. Step 1: Monotonic curves. As a starting point we use radial curves, which do exist for any initial data (see section 3), and by lemma 3.1.2 are monotonic in the sense of the following definition: DEFINITION A.1.1. A curve a(t) in an Alexandrov's space A is called monotonic with respect to a parameter value to iffor any >.-concave function f, >. ~ 0, we have that function

t t--+

f 0 a(t + to) - f 0 a(to) - >.t2 /2 t

is non-increasing for t > O. Here is a construction which gives a new monotonic curve out of two. It will be used in the next section to construct convex curves. EXTENTION A.1.2. Let A E A~ex, a1[a, (0) -+ A and a2 : [b, (0) -+ A be two monotonic curves with respect to a and b respectively. Assume

Then its joint

/3 : [a, (0) -+ A, /3(t)

a (t)

1 = [a2(t)

if t.t2/2

A. PETRUNIN

190

is non-increasing for t ;;:: b - a. By simple algebra, it follows from the following two facts: • a2 is monotonic and therefore

is non-increasing for t > o. • From monotonicity of al ,

(J 0 (2)+(b) = d01 (b)f(at(b)) = (J 0 ad+(b) ~ ~

f

0

a1(b)

+f

0

al(a) - A(b - a)2j2 . b-a D

A.2. Step 2: Convex curves. In this step we construct convex curves with arbitrary initial data. DEFINITION A.2.1. A curve /3 : [0,00) -+ A is called convex if for any A-concave function f, A ;;:: 0, we have that function

is concave. Properties of convex curves. Convex curves have the following properties; the proofs are either trivial or the same as for quasigeodesics: (1) A curve is convex if and only if it is monotonic with respect to any value of parameter. (2) Convex curves are I-Lipschitz. (3) Convex curves have uniquely defined right and left tangent vectors. (4) A limit of convex curves is convex and the natural parameter converges to the natural parmeter of the limit curves (the proof the last statement is based on the same idea as theorem 7.3.3). The next is a construction similar to A.1.2 which gives a new convex curve out of two. It will be used in the next section to construct prequasigeodesics. A.2.2. Let A E ALex, /31 : [a, 00) -+ A and /32 : [b,oo) -+ A be two convex curves. Assume EXTENTION

then its joint

is a convex curve.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

PROOF.

Follows immidetely from A.1.2 and property 1 above.

191

0

EXISTENCE A.2.3. Let A E ALex, x E A and ~ E Ex. Then there is a convex curve (3~ : [0,00) -+ A such that (3~(0) = x and (3{(0) = ~. PROOF.

For v

E

TxA, consider the radial curve

O:v(t) = gexpx(tv) According to lemma 3.1.2 if Ivl = 1 then O:v is I-Lipschitz and monotonic. Moreover, straightforward calculations show that the same is true for

Ivl :::; 1.

Fix e > O. Given a direction ~ E Ex, let us consider the following recursively defined sequence of radial curves O:Vn (t) such that Vo = ~ and Vn = O:;;n_l (e). Then consider their joint (3~,e(t) = O:vlt/eJ (t -

elt/eJ).

Applying an extension procedure A.1.2 we get that (3~,e : [0,00) -+ A is monotonic with respect to any t = ne. By property 1 on page 190, passing to a partial limit (3~,e -+ (3~ as e -+ 0 we get a convex curve (3~ : [0, 00) -+ A. It only remains to show that (3{(0) = ~. Since (3~ is convex, its right tangent vector is well defined and 1(3{(0)I :::; 157. On the other hand, since (3~,e are monotonic with respect to 0, for any semiconcave function f we have

Substituting in this inequality

f = dist y with

L(t~,~)

< e, we get

((3{(0), t~) > 1 - e for any e > implies that

o.

Together with 1(3{(0)I :::; 1 (property 2 on page 190), it

o A.3. Step 3: Pre-quasigeodesics. In this step we construct a prequasigeodesic with arbitrary initial data. DEFINITION A.3.1. A convex curve 'Y : [a, b) -+ A is called a prequasigeodesic if for any s E [a,b) such that 1'Y+(s)1 > 0, the curve 'Y s defined by

57See properties 3 and 2, page 190.

A. PETRUNIN

192

is convex for t ;;:: 0, and if h'+(s)1

= 0 then 'Y(t) = 'Y(s) for all t;;:: s.

Let us first define entropy of pre-quasigeodesic, which measures "how far" a given pre-quasigeodesic is from being a quasigeodesic. DEFINITION

A.3.2. Let l' be a pre-quasigeodesic in an Alexandrov's

space. The entropy of 1', I-l-y is the measure on the set of parameters defined by

Here are its main properties:

(1) The entropy of a pre-quasigeodesic l' is zero if and only if l' is a quasigedesic. (2) For a converging sequence of pre-quasigeodesics 'Yn --+ 1', the entropy of the limit is a weak limit of entropies, I-l-Yn --->. I-l-y. It follows from property 4 on page 190. The next statement is similar to A.1.2 and A.2.2; it makes a new prequasigeodesic out of two. It will be used in the next section to construct quasigeodesics. EXTENT ION A.3.3. Let A E ALex, be two pre-quasigeodesics. Assume

a ~ b, 'Y1(b)

= 1'2 (b) ,

1'1 (b)

1'1 :

[a, (0) --+ A and

1'2 :

[b, (0) --+ A

is polar to 'Y;t(b) and 1'Y;t(b) 1~

hl(b)1

then its joint

.

l' . [a, (0)

_ [1'1 (t ) if t ~ b --+ A, 'Y(t) - 'Y2(t) if t;;:: b

is a pre-quasigeodesic. Moreover, its entropy is defined by

PROOF. The same as for A.1.2.

o

A.3.4. Let A E ALex, x E A and ~ E Ex. Then there is a pre-quasigeodesic l' : [0, (0) --+ A such that 1'(0) = x and 1'+(0) = ~. EXISTENCE

PROOF. Let us choose for each point x E A and each direction ~ E Ex a convex curve f3~: [0,(0) --+ A such that f3~(0) = x, f3t(O) =~. If v = r~, then set

Clearly f3v is convex if 0

~

r

~

1.

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

193

Let us construct a convex curve '"Ye : [0,00) -+ M such that there is a representation of [0, 00) as a countable union of disjoint half-open intervals [ai, ad, such that lai - ail ~ c and for any t E [ai, ai) we have

Moreover, for each i, the curve '"Y~i : [0, 00) -+ A,

is also convex. Assume we already can construct '"Ye in the interval [0, t max ) , and cannot do it any further. Since '"Ye is I-Lipschitz, we can extend it continuously to [0, t max ]. Use lemma 1.3.9 to construct a vector v* polar to '"Y;(tmax ) with Iv* I ~ b; (t max ) I· Consider the joint of '"Ye with a short half-open segment of !3v, a longer curve with the desired property. This is a contradiction. Let '"Y be a partial limit of '"Ye as c -+ 0. From property 4 on page 190, we get that for almost all t we have b+(t)1 = lim l'"Yi;, (t)l. Combining this with inequality (*) shows that for any a ~

°

is convex.

D

A.4. Step 4: Quasigeodesics. We will construct quasigeodesics in an m-dimensional Alexandrov's space, assuming we already have such a construction in all dimensions <m. This construction is much easier for the case of an Alexandrov's space with only &-strained points; in this case we construct a sequence of special pre-quasigeodesics only by extending/chopping procedures (see below) and then pass to the limit. In a general Alexandrov's space we argue by contradiction, we assume that 0 is a maximal open set such that for any initial data one can construct an O-quasigeodesic (Le. a pre-quasigeodesic with zero entropy on 0, see A.3.2), and arrive at a contradiction with the assumption 0 =1= A. The following extention and chopping procedures are essential in the construction: A.4.1. Given a pre-quasigeodesic '"Y : [0, t max ) -+ A we can extend it as a pre-quasigeodesic '"Y : [0,00) -+ A so that EXTENTION

PROCEDURE

A. PETRUNIN

194

Let us set ,(tmax ) to be the limit of ,(t) as t -~ t max (it exists since pre-quasigeodesics are Lipschitz). From Milka's lemma A.4.2, we can construct a vector ,+(tmax ) which is polar to ,-(tmax ) and such that 1,+ (t max )1= 1,-(tmax)l. Then extend, by a pre-quasigeodesic in the direction ,+ (tmax ). By A.3.3, we get PROOF.

D MILKA'S LEMMA A.4.2.

vector

~

(existence of the polar direction). For any unit C, i. e. C E Ep such that

E Ep there is a polar unit vector (~,v)

+ (~*,v)

~ 0

for any v E Tp. The proof is taken from [Milka 1968]. That is the only instance where we use existence of quasigeodesics in lower dimensional spaces. PROOF. Since Ep is an Alexandrov's (m - 1) -space with curvature ~ 1, given ~ E Ep we can construct a quasigeodesic in Ep of length 71', starting at ~; the comparison inequality (theorem 5(5iv)) implies that the second endpoint C of this quasigeodesic satisfies

which is equivalent to the statement that CHOPPING PROCEDURE

A, for any t ~ 0 and

£

~

and

C

D

A.4.3. Given a pre-quasigeodesic , : [0, 00) -+

> 0 there is f> t such that

ILl ((t, f))

< £[19 + f - t], f - t < £, 19 < £,

where

PROOF.

are polar in Tp.

For all sufficiently small r

> 0 we have

19(t,t+r) 0 such that there is a finite collection of points {qk} which satisfy the following

196

A. PETRUNIN

property: for any x E BT (p) and ~ E ~x there is qk such that L (~, t~k) < c. Moreover, we can assume dist qk is A-concave in BT(p) for some A> O. Note that for any convex curve "( : [0, T) -+ BT(p) C A, the measures Xk on [0, T), defined by

Xk((a,b)) = (dist qk o"()-(b) - (dist qk o"()+(a)

+ A(b -

a),

are positive and their total mass is bounded by AT + 2 (this follows from the fact that dist qk is A-concave and I-Lipschitz). Let x E BT(p), and c5 be small enough. Then for any two directions ~,v E ~x there is qk which satisfies the following property:

Substituting in this inequality

v

-t"((iii) - "((ad'

and applying lemma A.4.5, we get

n

Therefore

L 19(ai' ai) ~ ION(AT + 2), i

where N is the number of points in the collection {qd.

o

LEMMA A.4.5. Let A E ALex, "( : [0, tj -+ A be a convex curve 1"(+(0)1 = I and f be a A-concave function, A ~ O. Set P = "((0), q = "((t), ~ = ("()+(O) and v =t~. Then

dpf(~) - dpf(v) ~ (f

provided that dpf(v)

~

P

PROOF.

Clearly,

0

"()+(O) - (f 0 "()-(t) + At,

O.

q

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

197

On the other hand,

f(p) ::;; f(q) - (f 0 ')')-(t)t + )..t 2 /2. Clearly, dpf(~)

= (f 0

o

')')+(0), whence the result.

What to do now? We have just finished the proof for the case, where all points of A are 8 -strained. From this proof it follows that if we denote by no the subset of all 8-strained points of A (which is an open everywhere dense set, see [BGP, 5.9]), then for any initial data one can construct a prequasigeodesic ')' such that J.Ll'b- l (n o)) = O. Assume A has no boundary; set It = A \ In this case it seems unlikely that we hit It by shooting a prequasigeodesic in a generic direction. If we could prove that it almost never happens, then we obtain existence of quasigeodesics in all directions as the limits of quasigeodesics in generic directions (see property 6 on page 171) and passing to doubling in case 8A =10. Unfortunately, we do not have any tools so far to prove such a thing58 . Instead we generalize inequality (*).

no.

THE (*) INEQUALITY A.4.6. Let A E ALeXm(K) and It c A be a closed subset. Let pElt be a point with 8 -maximal VOlm - l ~p, i. e.

VOlm - l ~p + 8 > inf VOlm - l ~P" xEI!

Then, if 8 is small enough, there is a finite set of points {qi} and c > 0, such that for any x E It n Be:(p) and any pair of directions ~ E ~x1t59 and 1/ E ~x we can choose qi so that 110 Lx (~, 1/) ::;; dx dist q; (~)

-

dx dist q; (1/)

and dx dist qk (1/)

~ O.

We can choose c > 0 so small that for any x E Be: (p), ~x is almost bigger than ~p .60 Since VOl m - l ~p is almost maximal we get that for any x E It n Be: (P), ~x is almost isometric to ~p. In particular, if one takes a set {qi} so that directions tZi form a sufficiently dense set and Lqipqj ~ LKqipqj, then directions t~; will form a sufficiently dense set in ~x for all x E It n Be:(p). Note that for any x E It n Be:(p) and ~ E ~xlt, there is an almost isometry ~x --+ ~(~~~x) such that ~ goes to north pole of the spherical suspension ~(~~~x) = ~~Tx .61 PROOF.

q;

Using these two properties, we can find qi so that tr~t~x in ~v~xA and L(~, t~i) > 1f/2, hence the statement follows. 0 58It might be possible if we would have an analog of the Liouvile theorem for "prequasigeodesic flow" . 59~xe: is defined on page 164. 60Le. for small 8> 0 there is a map f: ~p -+ ~x such that If(x)f(y)1 > Ixyl- 8. 610therwise, taking a point y E c, close to x in direction ~ we would get that VOlm-l ~y is essentially bigger than VOlm-l ~x, which is impossible since both are almost equal to VOlm-l ~p.

A. PETRUNIN

198

Now we are ready to finish construction in the general case. Let us define a subtype of pre-quasigeodesics: A.4.7. Let A E ALex and 0 c A be an open subset. A pre-quasigeodesic 'Y : [0, T) --* A is called O-quasigeodesic if its entropy vanishes on 0, i.e. DEFINITION

From property 2 on page 192, it follows that the limit of O-quasigeodesics is a O-quasigeodesic. Moreover, if for any initial data we can construct an O-quasigeodesic and an 0' -quasigeodesic, then it is possible to construct an o U 0' -quasigeodesic for any initial data; for T ~ 0 u 0', T -quasigeodesic can be constructed by joining together pieces of 0 and 0' -quasigeodesics and 0 U0' -quasigeodesic can be constructed as a limit of Tn -quasigeodesics as Tn --* OUO'. Let us denote by 0 the maximal open set such that for any initial data one can construct an O-quasigeodesic. We have to show then that 0 = A. Let Q: = A\O, and let p E Q: be the point with almost maximal VOlm -1 ~p. We will arrive to a contradiction by constructing a Be(P) U O-quasigeodesic for any initial data. Choose a finite set of points qi as in A.4.6. Given E > 0, it is enough to construct an O-quasigeodesic 'Ye : [0, T) -t A, for some fixed T > with the given initial data x E Be (p), ~ E ~x, such that the entropies J.Ll'e ((0, T)) --* asc:--*O. The O-quasigeodesic 'Ye which we are going to construct will have the following property: one can present [0, T) as a countable union of disjoint half-open intervals [ai, ad such that

°

°

if and if

Existence of 'Ye is being proved the same way as in the 8 -strained case, with the use of one additional observation: if

then any O-quasigeodesic in this direction has zero entropy for a short time. Then, just as in the 8-strained case, applying inequality A.4.6 we get that J.Ll'e (0, T) -t as E -t 0. Therefore, passing to a partial limit 'Ye -t 'Y gives a Be(P) U O-quasigeodesic 'Y: [0, T) --* A for any initial data in Be(P). 0

°

SEMICONCAVE FUNCTIONS IN ALEXANDROV'S GEOMETRY

199

A.5. Quasigeodesics in extremal subsets. The second part of theorem A.l.4 follows from the above construction, but we have to modify Milka's lemma A.4.2: EXTREMAL MILKA'S LEMMA A.5.l. Let E c Tp be an extremal subset of a tangent cone then for any vector vEE there is a polar vector v* E E such that Ivl = Iv* I· PROOF. Set X = En Ep. If E~X =1= 0 then the proof is the same as for the standard Milka's lemma; it is enough to choose a direction in E~X and shoot a quasigedesic 'Y of length 7r in this direction such that 'Y C X ('Y exists from the induction hypothesis). If X = {O then from the extremality of E we have B1r/2(~) = Ep. Therefore ~ is polar to itself. Otherwise, if E~X = 0 and X contains at least two points, choose C to be closest point in X\~ from ~. Since Xc Ep is extremal we have that for any 1] E Ep LEp1]C~:::;; 7r/2 and since E~X = 0 we have LEp1]~C :::;; 7r/2. Therefore, from triangle comparison we have

o References [AGS] Ambrosio, Luigi; Gigli, Nicola; Savan~, Giuseppe Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zrich. Birkhuser Verlag, Basel, 2005. viii+333 pp. [AKP] S. Alexander, V. Kapovitch, A. Petrunin, An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds. in preparation. [Alexander-Bishop 2003] S.Alexander, R.Bishop, :FK -convex functions on metric spaces. Manuscripta Math. 110, 115133 (2003). [Alexander-Bishop 2004] Alexander, S. B.; Bishop, R. L. Curvature bounds for warped products of metric spaces. Geom. Funct. Anal. 14 (2004), no. 6, 1143-1181. [Alexandrov 1945] A.D. Alexandrov, Curves on convex surfaces, Doklady Acad. Nauk SSSR v.47 (1945), p.319-322. [Alexandrov 1949] A.D. Alexandrov, Quasigeodesics, Doklady Acad. Nauk SSSR v.69 (1949), p.717-720. [Alexandrov 1957] A.D. Alexandrov, tiber eine Verallgemeinerung der Riemannschen Geometrie, Schriftenreihe Inst. Math. 1 (1957), p.33-84. [Alexandrov-Burago] A.D.Alexandrov, Yu.D.Burago, Quasigeodesics, Trudy Mat. Inst. Steklov. 76 (1965), p.49-63. [BBI] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6 [Berestovskii] Berestovskii, V.N. Borsuk's problem on metrization of a polyhedron. (Russian) Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 273-277. [BGP] Burago, Yu.; Gromov, M.; Perelman, G.A.D. Aleksandrov spaces with curvatures bounded below. (Russian) Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222; translation in Russian Math. Surveys 47 (1992), no. 2, 1-58. [Buyalo] Buyalo, S., Shortest paths on convex hypersurface of a Riemannian manifold (Russian), Studies in Topology, Zap. Nauchn. Sem. LOMI 66 (1976) 114-132; translated in J. of Soviet. Math. 12 (1979), 73-85.

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[Fukaya-Yamaguchi] Fukaya, Kenjij Yamaguchi, Takao Almost nonpositively curved manifolds. J. Differential Geom. 33 (1991), no. 1, 67-90. [Gromov] Gromov, Michael Curvature, diameter and Betti numbers. Comment. Math. Helv. 56 (1981), no. 2,179-195. [Grove-Petersen 1988] Grove, K; Petersen, P., Bounding homotopy types by geometry. Ann. of Math. (2) 128 (1988), no. 1, 195-206. [Grove-Petersen 1993] Grove, K; Petersen P., A radius sphere theorem, Invent. Math. 112 (1993), 577-583. [Grove-Wilhelm] Grove, K; Wilhelm, F., Metric constraints on exotic spheres via Alexandrov geometry. J. Reine Angew. Math. 487 (1997), 201-217. [Kapovitch 2002] Kapovitch V., Regularity of limits of noncollapsing sequences of manifolds Geom. Funet. Anal. 12 (2002), no. 1, 121-137. [Kapovitch 2005] Kapovitch, V., Restrictions on collapsing with a lower sectional curvature bound. Math. Z. 249 (2005), no. 3, 519-539. [Kapovitch 2007] Kapovitch, V., Perelman's stability theorem. this volume. [KPT] Kapovitch, V.; Petrunin, A.; Thschmann, W., Nilpotency, Almost Nonnegative Curvature and the Gradient Push, to appear in Annals of Mathematics. [Liberman] Liberman, J., Geodesic lines on convex surfaces. C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, (1941). 310-313. [Lytchak] Lytchak, A. Open map theorem for metric spaces. Algebra i Analiz 17 (2005), no. 3,139-159; translation in St. Petersburg Math. J. 17 (2006), no. 3,477-491. [Lytchak-Nagano] Lytchak, A.; Nagano, K, Local geometry of spaces with an upper curvature bound, in preparation. [Matveev-Shevchishin] Matveev, V. S.; Shevchishin, V. V., Closed polyhedral 3-manifold of nonnegative curvature. in preparation. [Mendonc;a] Mendonc;a, S. The asymptotic behavior of the set of rays, Comment. Math. Helv. 72 (1997) 331-348. [Milka 1968] Milka, A. D., Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5-6 1968 103-114. [Milka 1969] Milka, A. D., Multidimensional spaces with polyhedral metric of nonnegative curvature. II. (Russian) Ukrain. Geometr. Sb. No.7 (1969), 68-77, 185 (1970). [Milka 1971] Milka, A. D., Certain properties of quasigeodesics. Ukrain. Geometr. Sb. No. 11 (1971), 73-77. [Milka 1979] Milka, A. D. Shortest lines on convex surfaces (Russian), Dokl. Akad. Nauk SSSR 248 1979, no. 1, 34-36; translated in Soviet Math. Dokl. 20 1979, 949-952. [Nikolaev] Nikolaev, I. G., Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov, (Russian) Sibirsk. Mat. Zh. 24 (1983), no. 2, 114-132. [Otsu] Otsu, Y., Differential geometric aspects of Alexandrov spaces. Comparison geometry (Berkeley, CA, 1993-94), 135-148, Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997. [Otsu-Shioya] Otsu, Y.; Shioya, T., The Riemannian structure of Alexandrov spaces. J. Differential Geom. 39 (1994), no. 3, 629-658. [Perelman 1991] Perelman, G., Alexandrov paces with curvature bounded from below II. Preprint LOMI, 1991. 35pp. [Perelman 1993] Perelman, G. Ya., Elements of Morse theory on Aleksandrov spaces. (Russian) Algebra i Analiz 5 (1993), no. 1, 232-241; translation in St. Petersburg Math. J. 5 (1994), no. 1, 205-213. [Perelman DC] Perelman, G., DC Structure on Alexandrov Space, http://www.math.psu. edu/petrunin/papers/papers.html

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[Perelman 1997] Perelman, G., Collapsing with no proper extremal subsets. Comparison geometry (Berkeley, CA, 1993-94), 149-155, Math. Sci. Res. Inst. Pub!., 30, Cambridge Univ. Press, Cambridge, 1997. [Perelman-Petrunin 1993] Perelman, G. Ya.; Petrunin, A. M. Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem. (Russian) Algebra i Analiz 5 (1993), no. 1,242-256; translation in St. Petersburg Math. J. 5 (1994), no. 1,215-227. [Perelman-Petrunin QG] Perelman, G.; Petrunin A., Quasigeodesics and Gradient curves in Alexandrov spaces. http://www.math.psu.edu/petrunin/papers/papers.html [Petersen 1996] Petersen, P. Comparison geometry problem list. Riemannian geometry (Waterloo, ON, 1993),87-115, Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI,1996. [Petrunin 1997] Petrunin, A., Applications of quasigeodesics and gradient curves. Comparison geometry (Berkeley, CA, 1993-94), 203-219, Math. Sci. Res. Inst. Pub!., 30, Cambridge Univ. Press, Cambridge, 1997. [Petrunin 2003] Petrunin, A., Polyhedral approximations of Riemannian manifolds. TUrkish J. Math. 27 (2003), no. 1, 173-187. [Petrunin 2007] Petrunin, A., An upper bound for curvature integra!. to appear in Algebra i Analiz. [Petrunin GH] Petrunin, A., Gradient homotopy, in preparation. [Plaut 1996] Plaut, C., Spaces ofWaid curvature bounded below, J. Geom. Ana!., 6,1996, 1, 113-134. [Plaut 2002] Plaut, C., Metric spaces of curvature ~ k. Handbook of geometric topology, 819-898, North-Holland, Amsterdam, 2002. [Pogorelov] Pogorelov, A. V., Qusigeodesic lines on a convex surface, Mat. Sb. v.25/2 (1949), p.275-306. [PWZ] Petersen, P.; Wilhelm, F.; Zhu, S.-H., Spaces on and beyond the boundary of existence. J. Geom. Ana!. 5 (1995), no. 3, 419-426. [Reshetnyak] Reshetnyak Yu. G., Two-dimensional manifolds of bounded curvature. (English. Russian original) [CAl Geometry IV. Nonregular Riemannian geometry. Encyc!. Math. Sci. 70,3-163 (1993); translation from Itogi Nauki Tekh., Ser. Sovrem. Prob!. Mat., Fundam. Napravleniya 70, 7-189 (1989). [Sharafutdinov] Sharafutdinov, V. A., The Pogorelov-Klingenberg theorem for manifolds homeomorphic to IR n , Sib. Math. J. v.18/4 (1977), 915-925. [Shiohama] Shiohama, K., An introduction to the geometry of Alexandrov spaces. Lecture Notes Series, 8. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. ii+78 pp. [Shioya] Shioya, T., Splitting theorems for nonnegatively curved open manifolds with large ideal boundary, Math. Zeit. 212 (1993), 223-238. [Yamaguchi] Yamaguchi, T., Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 113 (1991), 317-357.

Surveys in Differential Geometry XI

Manifolds with a Lower Ricci Curvature Bound Guofang Wei ABSTRACT. This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.

1. Introduction

The purpose of this paper is to give a survey on the structure of manifolds with a lower Ricci curvature bound. A Ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. Ricci curvature is also special in that it occurs in the Einstein equation and in the Ricci flow. The study of manifolds with lower Ricci curvature bound has experienced tremendous progress in the past fifteen years. Our focus in this article is strictly restricted to results with only Ricci curvature bound, and no result with sectional curvature bound is presented unless for straight comparison. The reader is referred to John Lott's article in this volume for the recent important development concerning Ricci curvature for metric measure spaces by Lott-Villani and Sturm. We start by introducing the basic tools for studying manifolds with lower Ricci curvature bound (Sections 2-4), then discuss the structures of these manifolds (Sections 5-9), with examples in Section 10. The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula. From there one can derive powerful comparison tools like the mean curvature comparison, the Laplacian comparison, and the relative volume comparison. For the Laplacian comparison (Section 3) we discuss the global version in three weak senses (barrier, distribution, viscosity) and clarify their relationships (I am very grateful to my colleague Mike Crandall for many helpful discussions and references on this issue). A generalization of the volume comparison theorem to an integral Ricci curvature bound is also presented (Section 4). Important tools such as Cheng-Yau's gradient estimate and Cheeger-Colding's segment inequality are presented in Sections 2 and 4 respectively. Cheeger-Gromoll's splitting The author was partially supported by NSF grant DMS-0505733. ©2007 International Press 203

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theorem and Abresch-Gromoll's excess estimate are presented in Sections 5 and 8 respectively. From comparison theorems, various quantities like the volume, the diameter, the first Betti number, and the first eigenvalue are bounded by the corresponding quantity of the model. When equality occurs one has the rigid case. In Section 5 we discuss many rigidity and stability results for nonnegative and positive Ricci curvature. The Ricci curvature lower bound gives very good control on the fundamental group and the first Betti number of the manifold; this is covered in Section 6 (see also the very recent survey article by Shen-Sormani [97] for more elaborate discussion). In Sections 7, 8, and 9 we discuss rigidity and stability for manifolds with lower Ricci curvature bound under Gromov-Hausdorff convergence, almost rigidity results, and the structure of the limit spaces, mostly due to Cheeger and Colding. Examples of manifolds with positive Ricci curvature are presented in Section 10. Many of the results in this article are covered in the very nice survey articles [118, 23], where complete proofs are presented. We benefit greatly from these two articles. Some materials here are adapted directly from [23] and we are very grateful to Jeff Cheeger for his permission. We also benefit from [49, 24] and the lecture notes [108] of a topics course I taught at UCSB. I would also like to thank Jeff Cheeger, Xianzhe Dai, Karsten Grove, Peter Petersen, Christina Sormani, and William Wylie for reading earlier versions of this article and for their helpful suggestions. 2. Bochner's formula and the mean curvature comparison

For a smooth function f on a complete Riemannian manifold (M n , g), the gradient of f is the vector field \7 f such that (\7 f, X) = X (f) for all vector fields X on M. The Hessian of f is the symmetric bilinear form Hess(f) (X, Y)

=

XY(f) - \7xY(f)

=

(\7x\7f, Y),

and the Laplacian is the trace Ilf = tr(Hessf). For a bilinear form A, we denote IAI2 = tr(AAt). The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula. Here we state the formula for functions. THEOREM 2.1 (Bochner's Formula). For a smooth function f on a complete Riemannian manifold (M n , g), (2.1)

tlll\7 fl2 = IHessfl2

+ (\7 f, \7(Ilf)) + Ric (\7 f, \7 f).

This formula has many applications. In particular, we can apply it to the distance function, harmonic functions, and the eigenfunctions among others. The formula has a more general version (Weitzenb6ck type) for vector fields (I-forms), which also works nicely on Riemannian manifolds with a smooth

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measure [70, 85] where Ricci and all adjoint operators are defined with respect to the measure. Let r( x) = d(p, x) be the distance function from p EM. r( x) is a Lipschitz function and is smooth on M \ {p, Cp }, where Cp is the cut locus of p. At smooth points of r, (2.2)

l'Vrl

== 1, Hessr = II,

~r =

m,

where I I and m are the second fundamental form and mean curvature of the geodesics sphere 8B(p, r). Putting f(x) = r(x) in (2.1), we obtain the Riccati equation along a radial geodesic, (2.3)

0=

IIII2 + m' + Ric('Vr, 'Vr).

By the Schwarz inequality, 2

IIII2 2:~. n-l

Thus, if RicMn 2: (n - I)H, we have the Riccati inequality, (2.4)

m2 n-l

m'::; - - - - (n -1)H.

Let 1\tfj} denote the complete simply connected space of constant curvature Hand mH the mean curvature of its geodesics sphere; then 2

(2.5)

m~ = - m H n-l

-

(n - I)H.

Since limr-+o(m - mH) = 0, using (2.4), (2.5) and the standard Riccati equation comparison, we have THEOREM 2.2 (Mean Curvature Comparison). If RicMn 2: (n - I)H, then along any minimal geodesic segment from p,

(2.6) Moreover, equality holds if and only if all radial sectional curvatures are equal to H.

By applying the Bochner formula to f = log u with an appropriate cutoff function and looking at the maximum point one has Cheng-Yau 's gradient estimate for harmonic functions [34]. THEOREM 2.3 (Gradient Estimate, Cheng-Yau 1975). Let RicMn 2: (n - 1) H on B(p, R2) and u : B(p, R 2) -+ lR. satisfying u > 0, ~u = O. Then for Rl < R2, on B(p, R 1 ),

(2.7)

l'Vul 2 - 2 - ::; U

c(n, H, R 1 , R2)'

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If ~u = K(u), the same proof extends and one has [23]

(2.8)

1\7~12 ::; max{2u- 1 K(u), c(n, H, R 1 , R 2 ) + 2u- 1 K(u) u

2K'(u)}.

3. Laplacian comparison

Recall that m = ~r. From (2.6), we get the local Laplacian comparison for distance functions

(3.1) Note that if x E Cp , then either x is a (first) conjugate point of p or there are two distinct minimal geodesics connecting p and x [29], so x E {conjugate locus of p} u {the set where r is not differentiable}. The conjugate locus of p consists of the critical values of expp. Since expp is smooth, by Sard's theorem, the conjugate locus has measure zero. The set where r is not differentiable has measure zero since r is Lipschitz. Therefore the cut locus Cp has measure zero. One can show Cp has measure zero more directly by observing that the region inside the cut locus is star-shaped [18, Page 112]. The above argument has the advantage that it can be extended easily to show that Perelman's l-cut locus [85] has measure zero since the C-exponential map is smooth and the l-distance function is locally Lipschitz. In fact the Laplacian comparison (3.1) holds globally in various weak senses. First we review the definitions (for simplicity we only do so for the Laplacian) and study the relationship between these different weak senses. For a continuous function f on M, q E M, a function fq defined in a neighborhood U of q is an upper barrier of f at q if fq is C 2 (U) and (3.2)

fq(q) = f(q),

fq(x)

~

f(x) (x E U).

DEFINITION 3.1. For a continuous function f on M, we say ~f(q) ::; c in the barrier sense (f is a barrier subsolution to the equation ~f = cat q), if for all E > 0, there exists an upper barrier fq,E such that ~fq,E(q) ::; c + E.

This notion was defined by Calabi [17] back in 1958 (he used the terminology "weak sense" rather than "barrier sense"). A weaker version is in the sense of viscosity, introduced by Crandall and Lions in [38]. 3.2. For a continuous function f on M, we say ~f(q) ::; c in the viscosity sense (f is a viscosity subsolution of ~f = cat q), if ~ 0), using a heat kernel estimate Li showed that 71"1 (M) is finite [63]. Anderson also derived this using volume comparison [4]. Using the splitting theorem of Cheeger and Gromoll [30] (Theorem 5.1) on the universal cover Sormani showed that a noncompact manifold with positive Ricci curvature has the loops-to-infinity property [99]. As a consequence she showed that a noncompact manifold with positive Ricci curvature is simply connected if it is simply connected at infinity. See [96, 113] for more applications of the loops-to-infinity property. From the above one naturally wonders if all nilpotent groups occur as the fundamental group of a complete non-compact manifold with nonnegative Ricci curvature. Indeed, extending the warping product constructions

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in [77, 11], Wei showed [105] that any finitely generated torsion free nilpotent group could occur as fundamental group of a manifold with positive Ricci curvature. Wilking [109] extended this to any finitely generated almost nilpotent group. This gives a very good understanding of the fundamental group of a manifold with nonnegative Ricci curvature except the following long standing problem regarding the finiteness of generators [75]. 6.4 (Milnor, 1968). The fundamental group of a manifold with nonnegative Ricci curvature is finitely generated. CONJECTURE

There is some very good progress in this direction. Using short generators and a uniform cut lemma based on the excess estimate of Abresch and Gromoll [1] (see (8.2)) Sormani [101] proved that if RicM 20 and Mn has small linear diameter growth, then 1I"1(M) is finitely generated. More precisely the small linear growth condition is:

diamoB(p,r) 1. 1m sup < r--+oo r

_ Sn -

(

n -

n1)3 (n_1)n-l --2 n

n -

The constant Sn was improved in [114]. Then in [112] Wylie proved that in this case 1I"1(M) = G(r) for r big, where G(r) is the image Of1l"1(B(p,r)) in 1I"1(B(p,2r)). In an earlier paper [100], Sormani proved that all manifolds with nonnegative Ricci curvature and linear volume growth have sublinear diamter growth, so manifolds with linear volume growth are covered by these results. Any open manifold with nonnegative Ricci curvature has at least linear volume growth [116]. In a very different direction Wilking [109], using algebraic methods, showed that if RicM 20 then 1I"1(M) is finitely generated iff any abelian subgroup of 11"1 (M) is finitely generated, effectively reducing the Milnor conjecture to the study of manifolds with abelian fundamental groups. The fundamental group and the first Betti number are very nicely related. So it is natural that Ricci lower bound also controls the first Betti number. For compact manifolds Gromov [52] and Gallot [45] showed that if M n is a compact manifold with (6.1)

RicM 2 (n - l)H, diamM S; D,

then there is a function C(n,HD2) such that bl(M) S;C(n,HD2) and lim C(n,x) = nand C(n,x) = 0 for x > O. In particular, if HD2 is small, x--+o-

b1 (M) S; n. The celebrated Betti number estimate of Gromov [50] shows that all higher Betti numbers can be bounded by sectional curvature and diameter. This is not true for Ricci curvature. Using semi-local surgery Sha-Yang constructed metrics of positive Ricci curvature on the connected sum of k copies of 8 2 x 8 2 for all k 2 1 [95]. Recently, using Seifert bundles over orbifolds with a Kahler Einstein metric, Kollar showed that there are Einstein metrics

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with positive Ricci curvature on the connected sums of arbitrary number of copies of 8 2 x 8 3 [61]. Kapovitch-Wilking [59] recently announced a proof of the compact analog of Milnor's conjecture that the fundamental group of a manifold satisfying (6.1) has a presentation with a universally bounded number of generators (as conjectured by this author), and that a manifold which admits almost nonnegative Ricci curvature has a virtually nilpotent fundamental group. The second result would greatly generalize Fukaya-Yamaguchi's work on almost nonnegative sectional curvature [44]. See [106, 107] for earlier partial results. When the volume is also bounded from below, by using a clever covering argument M. Anderson [5] showed that the number of the short homotopically nontrivial closed geodesics can be controlled and for the class of manifolds M with RicM 2: (n - l)H, VolM 2: V and diamM ~ D there are only finitely many isomorphism types of 7r1(M). Again, if the Ricci curvature is replaced by sectional curvature then much more can be said. Namely there are only finitely many homeomorphism types of the manifolds with sectional curvature and volume bounded from below and diameter bounded from above [53, 81]. By [84] this is not true for Ricci curvature unless the dimension is 3 [117]. Contrary to a Ricci curvature lower bound, a Ricci curvature upper bound does not have any topological constraint [68]. THEOREM 6.5 (Lohkamp, 1994). If n 2: 3, any manifold M n admits a complete metric with RicM < O. An upper Ricci curvature bound does have geometric implications, e.g., the isometry group of a compact manifold with negative Ricci curvature is finite. In the presence of a lower bound, an upper bound on Ricci curvature forces additional regularity of the metric, see Theorem 9.8 in Section 9 by Anderson. It's still unknown whether it will give additional topological control. For example, the following question is still open. QUESTION 6.6. Does the class of manifolds M n with IRicMI ~ D have finite many homotopy types?

~

H, VolM

2: V and diamM

There are infinitely many homotopy types without the Ricci upper bound [84].

7. Gromov-Hausdorff convergence Gromov-Hausdorff convergence is very useful in studying man folds with a lower Ricci bound. The starting point is Gromov's precompactness theorem. Let's first recall the Gromov-Hausdorff distance. See [52, Chapter 3,5], [86, Chapter 10], [15, Chapter 7] for more background material on Gromov-Hausdorff convergence.

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Given a metric space (X, d) and subsets A, B eX, the Hausdorff distance is dH(A, B) = inf{f > 0: B

c

Tf(A) and A C Tf(B)},

where Tf(A) = {x EX: d(x,A) < f}. DEFINITION 7.1 (Gromov, 1981). Given two compact metric spaces X, Y, the Gromov-Hausdorff distance is dCH(X, Y) = inf {dH(X, Y) : all metrics on the disjoint union, X II Y, which extend the metrics of X and Y}. The Gromov-Hausdorff distance defines a metric on the collection of isometry classes of compact metric spaces. Thus, there is the naturally associated notion of Gromov-Hausdorff convergence of compact metric spaces. While the Gromov-Hausdorff distance makes sense for non-compact metric spaces, the following looser definition of convergence is more appropriate. See also [52, Defn 3.14]. These two definitions are equivalent [103, Appendix]. DEFINITION 7.2. We say that non-compact metric spaces (Xi, Xi) converge in the pointed Gromov-Hausdorff sense to (Y, y) if for any r > 0, B(Xi' r) converges to B(y, r) in the pointed Gromov-Hausdorff sense. Applying the relative volume comparison (4.5) to manifolds with lower Ricci curvature bound, we have THEOREM 7.3 (Gromov's precompactness theorem). The class of closed manifolds M n with RicM ~ (n - l)H and diamM ::; D is precompact. The class of pointed complete man'ifolds Mn with RicM ~ (n-1)H is precompact. By the above, for an open manifold Mn with RicM ~ 0 any sequence {(Mn, X, r;2g)} , with ri -+ 00, subconverges in the pointed GromovHausdorff topology to a length space Moo. In general, Moo is not unique [83]. Any such limit is called an asymptotic cone of M n , or a cone of M n at infinity. Gromov-Hausdorff convergence defines a very weak topology. In general one only knows that Gromov-Hausdorff limit of length spaces is a length space and diameter is continuous under the Gromov-Hausdorff convergence. When the limit is a smooth manifold with same dimension Colding showed the remarkable result that for manifolds with lower Ricci curvature bound the volume also converges [37], which was conjectured by Anderson-Cheeger. See also [23] for a proof using mod 2 degree. THEOREM 7.4 (Volume Convergence, Colding, 1997). If (Mi, Xi) has RicMi ~ (n - l)H and converges in the pointed Gromov-Hausdorff sense to smooth Riemannian manifold (Mn,x), then for all r > 0 (7.1 )

.lim Vol(B(Xi' r)) = Vol(B(x, r)). ~-+oo

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217

The volume convergence can be generalized to the non collapsed singular limit space (by replacing the Riemannian volume with the n-dimensional Hausdorff measure tin) [26, Theorem 5.9], and to the collapsing case with smooth limit Mk in terms of the k-dimensional Hausdorff content [27, Theorem 1.39J. As an application of Theorem 7.4, Colding [37J derived the rigidity result that if Mn has RicM 2': 0 and some Moo is isometric to JRn, then M is isometric to JR n . We also have the following wonderful stability result [26J which sharpens an earlier version in [37J. THEOREM 7.5 (Cheeger-Colding, 1997). For a closed Riemannian manifold M n there exists an E(M) > 0 such that if N n is a n-manifold with RicN 2': -(n - 1) and dCH(M, N) < E then M and N are diffeomorphic. Unlike the sectional curvature case, examples show that the result does not hold if one allows M to have singularities even on the fundamental group level [80, Remark (2)J. Also the E here must depend on M [3J. Cheeger-Colding also showed that the eigenvalues and eigenfunctions of the Laplacian are continuous under measured Gromov-Hausdorff convergence [28J. To state the result we need a definition and some structure result on the limit space (see Section 9 for more structures). Let Xi be a sequence of metric spaces converging to Xoo and J.Li, J.L00 be Radon measures on Xi, Xoo. DEFINITION 7.6. We say (Xi, J.Li) converges in the measured GromovHausdorff sense to (Xoo, J.L00) if for all sequences of continuous functions fi : Xi -+ JR converging to foo : Xoo -+ JR, we have (7.2) If (Moo,p) is the pointed Gromov-Hausdorff limit of a sequence of Riemannian manifolds (Mr,Pi) with RicMi 2': -(n - 1), then there is a natural collection of measures, J.L, on Moo obtained by taking limits of the normalized Reimannian measures on for a suitable subsequence [43], [26, Section 1],

AI;

(7.3)

J.L = .lim Volj )-+00

(.)

M;

= Vol(·)/Vol(B(pj, 1)).

In particular, for all z E Moo and 0 < r1 ::; r2, we have the renormalized limit measure J.L satisfy the following comparison (7.4)

J.L(B(z, r1)) > Voln ,-1 (B(rd) J.L(B(z, r2)) - Voln ,-1 (B(r2))·

With this, the extension of the segment inequality (4.6) to the limit, the gradient estimate (2.8), and Bochner's formula, one can define a canonical

218

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self-adjoint Laplacian d oo on the limit space Moo by means of limits of the eigenfunctions and eigenvalues for the sequence of the manifolds. In [19, 28] an intrinsic construction of this operator is also given on more general metric measure spaces. Let {.Xl,i··· ,}, {.Xl,oo,··· ,} denote the eigenvalues for di, d oo on Mi, Moo, and (hi,1>j,oo the eigenfunctions of the jth eigenvalues Aj,i, Aj,oo. In [28] Cheeger-Colding in particular proved the following theorem, establishing Fukaya's conjecture [43]. THEOREM 7.7 (Spectral Convergence, Cheeger-Colding, 2000). Let (Mr,pi, Voli ) with RicMi 2:: -(n-1) converge to (Moo,p, /1) under measured Gromov-HausdorfJ sense and Moo be compact. Then for each j, Aj,i --+ Aj,oo and 1>j,i --+ 1>j,oo uniformly as i --+ 00. As a natural extension, in [42] Ding proved that the heat kernel and Green's function also behave nicely under the measured Gromov-Hausdorff convergence. The natural extension to the p-form Laplacian does not hold; however, there is still very nice work in this direction by John Lott, see [69, 71]. 8. Almost rigidity and applications

Although the analogous stability results for maximal diameter in the case of positive/nonnegative Ricci curvature do not hold, Cheeger-Colding's significant work [25] provides quantitative generalizations of Cheng's maximal diameter theorem and Cheeger-Gromoll's splitting theorem (Theorem 5.1), and the volume annulus implies the metric annulus theorem in terms of Gromov-Hausdroff distance. These results have important applications in extending rigidity results to the limit space. An important ingredient for these results is Abresch-Gromoll's excess estimate [1]. For Yl,Y2 E Mn, the excess function E with respect to Yl,Y2 is

(8.1) Clearly E is Lipschitz with Lipschitz constant ::;2. Let 'Y be a minimal geodesic from Yl to Y2, s(x) = min(d(Yl' x), d(Y2, x)) and h(x) = mint d(x, 'Y(t)), the height from x to a minimal geodesic 'Y(t) connecting Yl and Y2. By the triangle inequality 0::; E(x) ::; 2h(x). Applying the Laplacian comparison (Theorem 3.6) to E(x) and with an elaborate (quantitative) use of the maximum principle (Theorem 3.5) Abresch-Gromoll showed that if RicM 2:: 0 and h(x) ::; s~), then ([1], see also [22])

(8.2) This is the first distance estimate in terms of a lower Ricci curvature bound. The following version (not assuming E(p) = 0, but without the sharp estimate) is from [23, Theorem 9.1].

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THEOREM 8.1 (Excess Estimate, Abresch-Gromoll, 1990). If M n has RicM ~ -(n -1)8, and for p E M, s(p) ~ Land E(p) ~ €, then on B(p, R), E ~ W= W(8, L -1, €I n, R), where W is a nonnegative constant such that for fixed nand R W goes to zero as 8, € -t 0 and L -t 00. This can be interpreted as a weak almost splitting theorem. CheegerColding generalized this result tremendously by proving the following almost splitting theorem [25], see also [23]. THEOREM 8.2 (Almost Splitting, Cheeger-Colding, 1996). With the same assumptions as Theorem 8.1, there is a length space X such that for some ball, B((O,x), ~R) c IR x X, with the product metric, we have dGH

(B (p, ~R) ,B

((O,x),

~R)) ~ W.

Note that X here may not be smooth, and the Hausdorff dimension could be smaller than n - 1. Examples also show that the ball B(p, ~ R) may not have the topology of a product, no matter how small 8, €, and L -1 are

[6,73]. The proof is quite involved. Using the Laplacian comparison, the maximum principle, and Theorem 8.1 one shows that the distance function bi = d(x, yd - d(p, Yi) associated to p and Yi is uniformly close to bi, the harmonic function with same values on {)B(p, R). From this, together with the lower bound for the smallest eigenvalue of the Dirichlet problem on B(p, R) (see Theorem 4.2) one shows that \lbi , \lb i are close in the L2 sense. In particular \lbi is close to 1 in the L2 sense. Then applying the Bochner formula to b i multiplied with a cut-off function with bounded Laplacian one shows that IHessbi I is small in the L2 sense in a smaller ball. Finally, in the most significant step, by using the segment inequality (4.6), the gradient estimate (2.7) and the information established above one derives a quantitative version of the Pythagorean theorem, showing that the ball is close in the Gromov-Hausdorff sense to a ball in some product space; see [25, 23]. An immediate application of the almost splitting theorem is the extension of the splitting theorem to the limit space. THEOREM 8.3 (Cheeger-Colding, 1996). If MJ: has RicMi ~ -(n - 1)8i with 8i -t 0 as i -t 00, converges to Y in the pointed Gromov-HausdorJJ sense, and Y contains a line, then Y is isometric to IR x X for some length space X. Similarly, one has almost rigidity in the presence of finite diameter (with a simpler a proof) [25, Theorem 5.12]. As a special consequence, we have that if MJ: has RicM; ~ (n -1), diamMi -t 7r as i -t 00, and converges to Y in the Gromov-Hausdorff sense, then Y is isometric to the spherical metric suspension of some length space X with diam(X) ~ 7r. This is a kind of stability for diameter.

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Along the same lines (with more complicated technical details) Cheeger and Colding [25] have an almost rigidity version for the volume annulus implies metric annulus theorem (see Section 5). As a very nice application to the asymptotic cone, they showed that if M n has RicM 2: 0 and has Euclidean volume growth, then every asymptotic cone of M is a metric cone.

9. The structure of limit spaces As we have seen, understanding the structure of the limit space of manifolds with lower Ricci curvature bound often helps in understanding the structure of the sequence. Cheeger-Colding made significant progress in understanding the regularity and geometric structure of the limit spaces [26, 27, 28]. On the other hand, Menguy constructed examples showing that the limit space could have infinite topology in an arbitrarily small neighborhood [73]. In [102, 103] Sormani-Wei showed that the limit space has a universal cover. Let (ym, y) (Hausdorff dimension m) be the pointed Gromov-Hausdorff limit of a sequence of Riemannian manifolds (Min, Pi) with RicMi 2: - (n-1). Then m :::; nand ym is locally compact. Moreover Cheeger-Colding [26] showed that if m = dim Y < n, then m:::; n - 1. The basic notion for studying the infinitesimal structure of the limit space Y is that of a tangent cone. DEFINITION 9.1. A tangent cone, yy, at y E (ym, d) is the pointed GromovHausdorff limit of a sequence of the rescaled spaces (ym, rid, y), where ri -+ 00 as i -+ 00. By Gromov's precompactness theorem (Theorem 7.3), every such sequence has a converging subsequence. So tangent cones exist for all y E ym, but might depend on the choice of convergent sequence. Clearly if Mn is a Riemannian manifold, then the tangent cone at any point is isometric to ]R.n. Motivated by this one defines [26] DEFINITION 9.2. A point, y E y, is called k-regular if for some k, every tangent cone at y is isometric to ]R.k. Let Rk denote the set of k-regular points and R = UkRk, the regular set. The singular set, Y \ R, is denoted S. Let /-t be a renormalized limit measure on Y as in (7.3). Cheeger-Colding showed that the regular points have full measure [26]. THEOREM 9.3 (Cheeger-Colding, 1997). For any renormalized limit measure /-t, /-t(S) = 0, in particular, the regular points are dense. Furthermore, up to a set of measure zero, Y is a countable union of sets, each of which is bi-Lipschitz equivalent to a subset of Euclidean space [28].

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DEFINITION 9.4. A metric measure space, (X, J-L), is called J-L-rectifiable if 0< J-L(X) < 00, and there exist N < 00 and a countable collection of subsets, A j , with J-L(X \ UjAj ) = 0, such that each Aj is bi-Lipschitz equivalent to a subset of jRl(j) , for some 1 S l(j) S N. In addtion, on the sets A j , the measures J-L and and the Hausdorff measure 1{1(j) are mutually absolutely continous. THEOREM 9.5 (Cheeger-Colding, 2000). Bounded subsets of Y are J-Lrectifiable with respect to any renormalized limit measure J-L. At the singular points, the structure could be very complicated. Following a related earlier construction of Perelman [84], Menguy constructed 4-dimensional examples of (noncollapsed) limit spaces with RicM:n, > 1, for which there exists a point so that any neighborhood of the point has infinite second Betti number [73]. See [26, 72, 74] for examples of collapsed limit space with interesting properties. Although we have very good regularity results, not much topological structure is known for the limit spaces in general. E.g., is Y locally simply connected? Although this is unknown, using the renormalized limit measure and the existence of regular points, together with 8-covers, Sormani-Wei [102, 103] showed that the universal cover of Y exists. Moreover when Y is compact, the fundamental group of Mi has a surjective homomorphism onto the group of deck transforms of Y for all i sufficiently large. When the sequence has the additional assumption that Vol(B(Pi, 1)) 2: v > 0,

(9.1)

the limit space Y is called noncollapsed. This is equivalent to m = n. In this case, more structure is known. DEFINITION 9.6. Given € > 0, the €-regular set, 'R€, consists of those points y such that for all sufficiently small r,

dCH(B(y,r),B(O,r)) S €r, where 0 E

jRn. o

Clearly 'R

= n€ 'R€. Let 'R€ denote the interior of 'R€.

THEOREM 9.7 (Cheeger-Colding 1997, 2000). There exists €(n) > 0 such that if Y is a noncollapsed limit space of the sequence Mr with RicMi 2: o

-(n - 1), then for 0 < € < €(n), the set 'R€ is a(€)-bi-Holder equivalent to a smooth connected Riemannian manifold, where a (€) ~ 1 as € ~ O. Moreover, o

(9.2)

dim(Y\ 'R€) S n - 2.

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In addition, for all y E Y, every tangent cone Yy at y is a metric cone and the isometry group of Y is a Lie group. This is proved in [26, 27J. If, in addition, Ricci curvature is bounded from two sides, we have stronger regularity [2J. THEOREM 9.8 (Anderson, 1990). There exists E(n) > 0 such that if Y is a noncollapsed limit space of the sequence Mt with IRicMi I ::; n - 1, then for 0 < E < E(n), R€ = R. In particular the singular set is closed. Moreover, R is a C1,0! Riemannian manifold, for all Q < 1. If the metrics on Mt are Einstein, RicMnt = (n - I)Hgi , then the metric on R is actually Coo. Many more regularity results are obtained when the sequence is Einstein, Kahler, has special holonomy, or has bounded V-norm of the full curvature tensor; see [7, 20, 21, 31], especially [24J which gives an excellent survey in this direction. See the recent work [32J for Einstein 4-manifolds with possible collapsing. 10. Examples of manifolds with nonnegative Ricci curvature Many examples of manifolds with nonnegative Ricci curvature have been constructed, which contribute greatly to the study of manifolds with lower Ricci curvature bound. We only discuss the examples related to the basic methods here, therefore many specific examples are unfortunately omitted (some are mentioned in the previous sections). There are mainly three methods: fiber bundle construction, special surgery, and group quotient, all combined with warped products. These method are also very useful in constructing Einstein manifolds. A large class of Einstein manifolds is also provided by Yau's solution of the Calabi conjecture. Note that if two compact Riemannian manifolds M m , Nn(n, m ~ 2) have positive Ricci curvature, then their product has positive Ricci curvature, which is not true for sectional curvature but only needs one factor positive for scalar curvature. Therefore it is natural to look at the fiber bundle case. Using Riemannian submersions with totally geodesic fibers J. C. Nash [78], W. A. Poor [91], and Berard-Bergery [10J showed that the compact total space of a fiber bundle admits a metric of positive Ricci curvature if the base and the fiber admit metrics with positive Ricci curvature and if the structure group acts by isometries. Furthermore, any vector bundle of rank ~ 2 over a compact manifold with Ric> 0 carries a complete metric with positive Ricci curvature. In [48J Gilkey-Park-Thschmann showed that a principal bundle P over a compact manifold with Ric > 0 and compact connected structure group G admits a G invariant metric with positive Ricci curvature if and only if 7rl (P) is finite. Unlike the product case, the corresponding statements for Ric ~ 0 are not true in all these cases, e.g., the nilmanifold 8 1 -+ N3 -+ T2 does not admit a metric with Ric ~ O. On the other hand Belegradek-Wei

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[9J showed that it is true in the stable sense. Namely, if E is the total space of a bundle over a compact base with Ric 2: 0, and has either a compact Ric 2: 0 fiber or vector space as fibers, with compact structure group acting by isometry, then E x ]R.P admits a complete metric with positive Ricci curvature for all sufficiently large p. See [110J for an estimate of p. Surgery constructions are very successful in constructing manifolds with positive scalar curvature, see Rothenberg's article in this volume. Sha-Yang [94, 95J showed that this is also a useful method for constructing manifolds with positive Ricci curvature in special cases. In particular they showed that if M m +1 has a complete metric with Ric > 0, and n, m 2: 2, then sn-l X

( Mm+1 \

117=o D;n+l) Uld Dn

x

117=o sf,

which is diffeomorphic to

(sn-l x Mm+l) # (#f=lsn X sm), carries a complete metric with Ric>O for all k, showing that the total Betti number of a compact Riemannian n-manifold (n 2: 4) with positive Ricci curvature could be arbitrarily large. See also [6J, and [111J when the gluing map is not the identity. Note that a compact homogeneous space admits an invariant metric with positive Ricci curvature if and only if the fundamental group is finite [78, Proposition 3.4J. This is extended greatly by Grove-Ziller [55J showing that any cohomogeneity one manifold M admits a complete invariant metric with nonnegative Ricci curvature and if M is compact then it has positive Ricci curvature if and only if its fundamental group is finite (see also [93]). Therefore, the fundamental group is the only obstruction to a compact manifold admitting a positive Ricci curvature metric when there is enough symmetry. It remains open what the obstructions are to positive Ricci curvature besides the restriction on the fundamental group and those coming from positive scalar curvature (such as the A-genus). Of course, another interesting class of examples are given by Einstein manifolds. For these, besides the "bible" on Einstein manifolds [12], one can refer to the survey book [62J for the development after [12], and the recent articles [14, 13J for Sasakian Einstein metrics and compact homogenous Einstein manifolds. References [1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc., 3(2) (1990), 355-374. [2] M.T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102(2) (1990), 429-445. [3] M.T. Anderson, Metrics of positive Ricci curvat7~re with large diameter, Manuscripta Math., 68(4) (1990), 405-415. [4] M.T. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology, 29(1) (1990),41-55. [5] M.T. Anderson, Short geodesics and gmvitational instantons, J. Differential Geom., 31(1) (1990), 265-275. [6] M.T. Anderson, Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem, Duke Math. J., 68(1) (1992), 67-82.

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[99] C. Sormani, On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature, Indiana Univ. Math. J., 50(4) (2001), 1867-1883. [100] C. Sormani, The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth, Comm. Ana!. Geom., 8(1) (2000), 159-212. [101] C. Sormani, Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups, J. Differential Geom., 54(3) (2000), 547-559. [102] C. Sormani and G. Wei, Hausdorff convergence and universal covers, Trans. Amer. Math. Soc., 353(9) (2001), 3585-3602 (electronic). [103] C. Sormani and G. Wei, Universal covers for Hausdorff limits of noncompact spaces, Trans. Amer. Math. Soc., 356(3) (2004), 1233-1270 (electronic). [104] C. Sprouse, Integral curvature bounds and bounded diameter, Comm. Ana!. Geom., 8(3) (2000), 531-543. [105] G. Wei, Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups, Bull. Amer. Math. Soci., 19(1) (1988), 311-313. [106] G. Wei, On the fundamental groups of manifolds with almost-nonnegative Ricci curvature, Proc. Amer. Math. Soc., 110(1) (1990), 197-199. [107] G. Wei, Ricci curvature and betti numbers, J. Geom. Ana!., 7 (1997), 493-509. [108] G. Wei, Math. Lecture Notes, 241, http://www.math.ucsb.edu/wei/241notes.htm!. [109] B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. App!., 13(2) (2000), 129-165. [110] D. Wraith, Stable bundles with positive Ricci curvature, preprint. [111] D. Wraith, Surgery on Ricci positive manifolds, J. Reine Angew. Math., 501 (1998), 99-113. [112] W. Wylie, Noncompact manifolds with nonnegative Ricci curvature, J. Geom. Ana!., 16(3) (2006), 535-550. [113] W. Wylie, On the fundamental group of noncompact manifolds with nonnegative Ricci curvature, Ph.D. thesis at UC Santa Barbara, 2006. [114] S. Xu, Z. Wang, and F. Yang, On the fundamental group of open manifolds with nonnegative Ricci curvature, Chinese Ann. Math. Ser. B, 24(4) (2003), 469-474. [115] D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature, I, Ann. Sci. Ecole Norm. Sup. (4),25(1) (1992),77-105. [116] S.- T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 25(7) (1976), 659-670. [117] S.-H. Zhu, A finiteness theorem for Ricci curvature in dimension three, J. Different.ial Geom., 37(3) (1993), 711-727. [118] S.-H. Zhu, The comparison geometry of Ricci curvature, in 'Comparison geometry' (Berkeley, CA, 1993-94), volume 30 of Math. Sci. Res. Inst. Publ, 221-262, Cambridge Univ. Press, Cambridge, 1997. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SANTA BARBARA, CA 93106 E-mail address: wei~ath. ucsb. edu

Surveys in Differential Geometry XI

Optimal Transport and Ricci Curvature for Metric-Measure Spaces John Lott ABSTRACT. We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.

An intriguing question is whether one can extend notions of smooth Riemannian geometry to general metric spaces. Besides the inherent interest, such extensions sometimes allow one to prove results about smooth Riemannian manifolds, using compactness theorems. There is a good notion of a metric space having "sectional curvature bounded below by K" or "sectional curvature bounded above by K", due to Alexandrov. We refer to the articles of Petrunin and Buyalo-Schroeder in this volume for further information on these two topics. In this article we address the issue of whether there is a good notion of a metric space having "Ricci curvature bounded below by K" . A motivation for this question comes from Gromov's precompactness theorem [14, Theorem 5.3]. Let M denote the set of compact metric spaces (modulo isometry) with the Gromov-Hausdorff topology. The precompactness theorem says that given N E Z+, D < 00 and K E JR, the subset of M consisting of closed Riemannian manifolds (M, g) with dim(M) = N, Ric ~ Kg and diam ::; D, is precompact. The limit points in M of this subset will be metric spaces of Hausdorff dimension at most N, but generally are not manifolds. However, one would like to say that in some generalized sense they do have Ricci curvature bounded below by K. Deep results about the structure of such limit points, which we call Ricci limits, were obtained by Cheeger and Colding [8, 9, 10]. We refer to the article of Guofang Wei in this volume for further information. In the work of Cheeger and Colding, and in earlier work of Fukaya [13], it turned out to be useful to consider not just metric spaces, but rather metric spaces equipped with measures. Given a compact metric space (X, d), let The author was partially supported by NSF grant DMS-0604829 during the writing of this article. @2007 International Press 229

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P(X) denote the set of Borel probability measures on X. That is, v E P(X) means that v is a nonnegative Borel measure on X with Jx dv = 1. We put the weak-* topology on P(X), so limHoo Vi = v if and only if for all f E C(X), we have limHoo Jx f dVi = Jx f dv. Then P(X) is compact. DEFINITION 0.1. A compact metric measure space is a triple (X, d, v) where (X, d) is a compact metric space and v E P(X). DEFINITION 0.2. Given two compact metric spaces (Xl, d l ) and (X2' d2), an E-Gromov-Hausdorff approximation from Xl to X2 is a (not necessarily continuous) map f : X I --+ X 2 so that (i) For all xI,xi E Xl, Id 2(f(XI),f(xi)) -dl(XI,xi)1 ~ E. (ii) For all X2 E X2, there is an Xl E Xl so that d2(f(xd, X2) ~ E. A sequence {(Xi, di, vd }~l of compact metric-measure spaces converges to (X, d, v) in the measured Gromov-Hausdorff topology if there are Borel Ei-approximations Ii : Xi --+ X, with limi~oo Ei = 0, so that limi~oo(fd*Vi = v in P(X). REMARK 0.3. There are other interesting topologies on the set of metricmeasure spaces, discussed in [14, Chapter 3~l. If M is a compact manifold with Riemannian metric 9 then we also let (M, g) denote the underlying metric space. There is a canonical probability measure on M given by the normalized volume form ~~(}e/). One can easily extend Gromov's precompactness theorem to say that given N E Z+, D < 00 and K E lR, the triples (M, g, ~~(~)) with dim(M) = N, Ric ;::: Kg and diam ~ D form a precompact subset in the measured Gromov-Hausdorff (MGH) topology. The limit points of this subset are now metric-measure spaces (X, d, v). One would like to say that they have "Ricci curvature bounded below by K" in some generalized sense. The metric space (X, d) of a Ricci limit is necessarily a length space. Hereafter we mostly restrict our attention to length spaces. So the question that we address is whether there is a good notion of a compact measured length space (X, d, v) having "Ricci curvature bounded below by K". The word "good" is a bit ambiguous here, but we would like our definition to have the following properties.

WISHLIST 0.4. 1. If {(Xi, di, Vi)}~l is a sequence of compact measured length spaces with "Ricci curvature bounded below by K" and limHoo(Xi , di , Vi) = (X, d, v) in the measured Gromov-Hausdorff topology then (X, d, v) has "Ricci curvature bounded below by K". 2. If (M, g) is a compact Riemannian manifold then the triple

( M, g, ~~(lb) has "Ricci curvature bounded below by K" if and only if Ric ;::: Kg in the usual sense.

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231

3. One can prove some nontrivial results about measured length spaces having "Ricci curvature bounded below by K". It is not so easy to come up with a definition that satisfies all of these properties. One possibility would be to say that (X, d, II) has "Ricci curvature bounded below by K" if and only if it is an MGH limit of Riemannian manifolds with Ric ~ Kg, but this is a bit tautological. We want instead a definition that depends in an intrinsic way on (X, d, II). We refer to [8, Appendix 2] for further discussion of the problem. In fact, it will turn out that we will want to specify an effective dimension N, possibly infinite, of the measured length space. That is, we want to define a notion of (X, d, II) having "N-Ricci curvature bounded below by K", where N is a parameter that is part of the definition. The need to input the parameter N can be seen from the Bishop-Gromov inequality for complete n-dimensional Riemannian manifolds with nonnegative Ricci curvature. It says that r- n vol(Br(m)) is nonincreasing in r, where Br(m) is the r-ball centered at m. We will want a Bishop-Gromov-type inequality to hold in the length space setting, but when we go from manifolds to length spaces there is no a priori value for the parameter n. Hence for each N E [1,00], there will be a notion of (X, d, II) having "N-Ricci curvature bounded below

byK". The goal now is to find some property which we know holds for Ndimensional Riemannian manifolds with Ricci curvature bounded below, and turn it into a definition for measured length spaces. A geometer's first inclination may be to just use the Bishop-Gromov inequality, at least if N < 00, for example to say that (X, d, II) has "nonnegative N-Ricci curvature" if and only if for each x E SUpp(II), r- N II(Br(x)) is nonincreasing in r. Although this is the simplest possibility, it turns out that it is not satisfactory; see Remark 4.9. Instead, we will derive a Bishop-Gromov inequality as part of a more subtle definition. The definition that we give in this paper may seem to come from left field, at least from the viewpoint of standard geometry. It comes from a branch of applied mathematics called optimal transport, which can be informally considered to be the study of moving dirt around. The problem originated with Monge in the paper [27], whose title translates into English as "On the theory of excavations and fillings" . (In that paper Monge also introduced the idea of a line of curvature of a surface.) The problem that Monge considered was how to transport a "before" dirt pile to an "after" dirt pile with minimal total "cost", where he took the cost of transporting a unit mass of dirt between points x and y to be d(x, y). Such a transport F : X ---+ X is called a Monge transport. An account of Monge's life, and his unfortunate political choices, is in [4]. Since Monge's time, there has been considerable work on optimal transport. Of course, the original case of interest was optimal transport on Euclidean space. Kantorovich introduced a important relaxation of Monge's

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original problem, in which not all of the dirt from a given point x has to go to a single point y. That is, the dirt from x is allowed to be spread out over the space. Kantorovich showed that there is always an optimal transport scheme in his sense. (Kantorovich won a 1975 Nobel Prize in economics.) We refer to the book [38] for a lively and detailed account of optimal transport. In Section 1 we summarize some optimal transport results from a modern perspective. We take the cost function of transporting a unit mass of dirt to be d(x, y)2 instead of Monge's cost function d(x, y). The relation to Ricci curvature comes from work of Otto-Villani [30] and Cordero-ErausquinMcCann-Schmuckenschlager [11]. They showed that optimal transport on a Riemannian manifold is affected by the Ricci tensor. To be a bit more precise, the Ricci curvature affects the convexity of certain entropy functionals along an optimal transport path. Details are in Section 2. The idea now, implemented independently by Lott-Villani and Sturm, is to define the property "N-Ricci curvature bounded below by K", for a measured length space (X, d, II), in terms of the convexity of certain entropy functionals along optimal transport paths in the auxiliary space P(X). We present the definition and its initial properties in Section 3. We restrict in that section to the case K = 0, where the discussion becomes a bit simpler. We show that Condition 1. from Wishlist 0.4 is satisfied. In Section 4 we show that Condition 2. from Wishlist 0.4 is satisfied. In Section 5 we give the definition of (X, d, II) having N -Ricci curvature bounded below by K, for K E R Concerning Condition 3. of the Wishlist, in Sections 3, 4 and 5 we give some geometric results that one can prove about measured length spaces with Ricci curvature bounded below. In particular, there are applications to Ricci limit spaces. In Section 6 we give some analytic results. In Section 7 we discuss some further issues. We mostly focus on results from [23] and [24], mainly because of the author's familiarity with those papers. However, we emphasize that many parallel results were obtained independently by Karl-Theodor Sturm in [36, 37]. Background information on optimal transport is in [38] and [39]. The latter book also contains a more detailed exposition of some of the topics of this survey. I thank Cedric Villani for an enjoyable collaboration.

1. Optimal transport Let us state the Kantorovich transport problem. We take (X, d) to be a compact metric space. Our "before" and "after" dirtpiles are measures /-lo, /-ll E P(X). They both have mass one. We want to move the total amount of dirt from /-lo to /-ll most efficiently. A moving scheme, maybe not optimal, will be called a transference plan. Intuitively, it amounts to specifying how much dirt is moved from a point Xo to a point Xl. That is, we have a probability measure 7r E P(X x X), which we informally write as 7r(xo, Xl)' The

OPTIMAL TRANSPORT AND RICCI CURVATURE ...

statement that that

7r

233

does indeed transport J.Lo to J.LI translates to the condition

(1.1) where PO,PI : X X X -+ X are projections onto the first and second factors, respectively. We will use optimal transport with quadratic cost function (square of the distance). The total cost of the transference plan 7r is given by adding the contributions of d(xo, XI)2 with respect to 7r. Taking the infimum of this with respect to 7r gives the square of the Wasserstein distance W2(J.LO, J.Ld between J.Lo and J.LI, i.e., (1.2) where 7r ranges over the set of all transference plans between J.Lo and J.LI. Any minimizer 7r for this variational problem is called an optimal transference plan. In (1.2), one can replace the infimum by the minimum [38, Proposition 2.1], i.e., there always exists (at least) one optimal transference plan. It turns out that W 2 is a metric on P(X). The topology that it induces on P(X) is the weak-* topology [38, Theorems 7.3 and 7.12]. When equipped with the metric W2, P(X) is a compact metric space. In this way, to each compact metric space X we have assigned another compact metric space P(X). The Wasserstein space (P(X), W2) seems to be a very natural object in mathematics. It generally has infinite topological or Hausdorff dimension. (If X is a finite set then P(X) is a simplex, with a certain metric.) It is always contractible, as can be seen by fixing a measure J.Lo E P(X) and linearly contracting other measures J.L E P(X) to J.Lo by t -+ tJ.Lo + (1 - t)J.L. PROPOSITION 1.3 ([23, Corollary 4.3]). If limi-too(Xi,di) = (X,d) in the Gromov-Hausdorff topology then limi-+oo(P(Xi), W 2) = (P(X), W2) in the Gromov-Hausdorff topology. A Monge transport is a transference plan coming from a map F : X -+ X with F*J.Lo = J.LI, given by 7r = (Id, F)*J.Lo. In general an optimal transference plan does not have to be a Monge transport, although this may be true under some assumptions. What does optimal transport look like in Euclidean space ~n? Suppose that J.Lo and J.LI are compactly supported and absolutely continuous with respect to Lebesgue measure. Brenier [5] and Rachev-Riischendorf [33] showed that there is a unique optimal transference plan between J.Lo and J.LI, which is a Monge transport. Furthermore, there is a convex function V on ~n so that for almost all X, the Monge transport is given by F(x) = VxV. So to find the optimal transport, one finds a convex function V such that the pushforward, under the map 'VV : ~n -+ ~n, sends J.Lo to J.LI. This solves

J.

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LOTT

the Monge problem for such measures, under our assumption of quadratic cost function. The solution to the original problem of Monge, with linear cost function, is more difficult; see [12J. The statement of the Brenier-Rachev-Riischendorf theorem may sound like anathema to a geometer. One is identifying the gradient of V (at x), which is a vector, with the image of x under a map, which is a point. Because of this, it is not evident how to extend even the statement of the theorem if one wants to do optimal transport on a Riemannian manifold. The extension was done by McCann [26J. The key point is that on ]Rn, we can write VxV = X - Vx¢, where ¢( x) = 1~2 - V (x). To understand the relation between V and ¢, we note that if the convex function V were smooth then ¢ would have Hessian bounded above by the identity. On a Riemannian manifold (M,g), McCann's theorem says that an optimal transference plan between two compactly supported absolutely continuous measures is a Monge transport F that satisfies F (m) = eXPm ( - V'm ¢) for almost all m, where ¢ is a function on M with Hessian bounded above by g in a generalized sense. More precisely, ¢ is ~ -concave in the sense that it can be written in the form

(1.4)

¢(m) = inf (d(m, m'? _ ¢(m')) m'EM

2

for some function ¢ : M -+ [-00, (0). Returning to the metric space setting, if (X, d) is a compact length space and one has an optimal transference plan 7r then one would physically perform the transport by picking up pieces of dirt in X and moving them along minimal geodesics to other points in X, in a way consistent with the transference plan 7r. The transference plan 7r tells us how much dirt has to go from Xo to Xl, but does not say anything about which minimal geodesics from Xo to Xl we should actually use. After making such a choice of minimizing geodesics, we obtain a I-parameter family of measures {JLthE[O,lj by stopping the physical transport procedure at time t and looking at where the dirt is. This suggests looking at (P(X), W2) as a length space. PROPOSITION 1.5 ([23, Corollary 2.7], [36, Proposition 2.1O(iii)]). If

(X, d) is a compact length space then (P(X), W2) is a compact length space. Hereafter we assume that (X, d) is a compact length space. By definition, a Wasserstein geodesic is a minimizing geodesic in the length space (P(X), W 2 ). (We will always parametrize minimizing geodesics in length spaces to have constant speed.) The length space (P(X), W2) has some interesting features; even for simple X, there may be an uncountable number of Wasserstein geodesics between two measures JLo, JLI E P(X) [23, Example 2.9J. As mentioned above, there is a relation between minimizing geodesics in (P(X), W2) and minimizing geodesics in X. Let r be the set of minimizing

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235

geodesics 'Y : [0,1] -+ X. It is compact in the uniform topology. For any t E [0, 1], the evaluation map et : r -+ X defined by (1.6) is continuous. Let E : r -+ X x X be the "endpoints" map given by E("() = (eo ('Y), el ('Y)). A dynamical transference plan consists of a transference plan 7r and a Borel measure IT on r such that E*IT = 7r; it is said to be optimal if 7r itself is. In words, the transference plan 7r tells us how much mass goes from a point Xo to another point Xl, but does not tell us about the actual path that the mass has to follow. Intuitively, mass should flow along geodesics, but there may be several possible choices of geodesics between two given points and the transport may be divided among these geodesics; this is the information provided by IT. If IT is an optimal dynamical transference plan then for t E [0, 1], we put (1.7) The one-parameter family of measures {!.Lt}tE[O,I] is called a displacement interpolation. In words, /.Lt is what has become of the mass of /.Lo after it has travelled from time to time t according to the dynamical transference plan IT.

°

PROPOSITION 1.8 ([23, Lemma 2.4 and Proposition 2.10]). Any displacement interpolation is a Wasserstein geodesic. Conversely, any Wasserstein geodesic arises as a displacement interpolation from some optimal dynamical transference plan. In the Riemannian case, if /.Lo, /.LI are absolutely continuous with respect to dvolM, and F( m) = eXPm ( - V' m 0 then .\1(:6.) ~ N~1 K [2]. Geometric properties of RicN were studied in [22] and [32]. Finally, if N < n, or if N = nand W is not locally constant, then we take the effective Ricci tensor RicN to be -00. To summarize, DEFINITION 4.4. For N E [1,00]' define the N-Ricci tensor RicN of (M,g,1/) by

(4.5)

. R ~N=

I

Ric + Hess (w) Ric + Hess (w) - N~ dw ® dw n

~~+ Hess (w) - 00 (dw ® dW)

if N = 00, if n < N < 00, if N = n, if N < n,

where by convention 00 . 0 = O. We can now state what the abstract notion of nonnegative N-Ricci curvature boils down to in the smooth case. THEOREM 4.6 ([23, Theorems 7.3 and 7.42], [36, Theorem 4.9], [37, Theorem 1. 7]) . Given N E [1, 00], the measured length space (M, g, 1/) has nonnegative N -Ricci curvature in the sense of Definition 3.13 if and only if RicN ~ O.

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The proof of Theorem 4.6 uses the explicit description of optimal transport on Riemannian manifolds. In the special case when W is constant, and so v = ~~(}(:b, Theorem 4.6 shows that we recover the usual notion of nonnegative Ricci curvature from our length space definition as soon as N ~ n. 4.1. Ricci limit spaces. We give an application of Theorems 3.20 and 4.6 to Ricci limit spaces. From Gromov precompactness, given N E Z+ and D > 0, the Riemannian manifolds with nonnegative Ricci curvature, dimension at most N and diameter at most D form a precompact subset of the set of measured length spaces, with respect to the MGH topology. The problem is to characterize the limit points. In general the limit points can be very singular, so this is a hard problem. However, let us ask a simpler question: what are the limit points that happen to be smooth measured length spaces? That is, we are trying to characterize the smooth limit points. COROLLARY 4.7 ([23, Corollary 7.45]). If (B,9B,e-'iJ! dvolB) is a measured Gromov-HausdorJJ limit of Riemannian manifolds with nonnegative Ricci curvature and dimension at most N then RicN(B) ~ O. (Here B has dimension n, which is less than or equal to N.) PROOF. Suppose that {(Mi,gi)}~l is a sequence of Riemannian manifolds with nonnegative Ricci curvature and dimension at most N, with limi--+oo (Mi' 9i, :~(~j)) = (B, gB, e-lJ! dvolB)' From Theorem 4.6, the mea-

sured length space (Mi' gi, :~(~j)) has nonnegative N-Ricci curvature. From Theorem 3.20, (B, 9B, e-'iJ! dvolB) has nonnegative N-Ricci curvature. From Theorem 4.6 again, RicN(B) ~ O. D There is a partial converse to Corollary 4.7. 4.8 ([23, Corollary 7.45]). (i) Suppose that N is an integer. If (B,9B,e-'iJ! dvolB) has RicN(B) ~ o with N ~ dim(B) + 2 then (B, gB, e-lJ! dvolB) is a measured Gromov-HausdorJJ limit of Riemannian manifolds with nonnegative Ricci curvature and dimension N. (ii) Suppose that N = 00. If (B, gB, e-'iJ! dvolB) has Ricoo(B) ~ 0 then (B,9B,e-'iJ!dvoIB) is a measured Gromov-HausdorJJ limit of Riemannian manifolds Mi with Ric(Mi) ~ 9Mi'

PROPOSITION

-i

Let us consider part (i). The proof uses the warped product construction of [22]. Let gsN-dim(B) be the standard metric on the sphere sN-dim(B). Let Mi be B X sN-dim(B) with the warped product metric PROOF.

Iji

9i = 9B + i- 2 e - N-dim(B) 9SN-dim(B). The metric 9i is constructed so that if p : B x SN -dim(B) -+ B is projection onto the first factor then p* dvolMi

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is a constant times e- w dvolB. In terms of the fibration p, the Ricci tensor of Mi splits into horizontal and vertical components, with the horizontal component being exactly RicN. As i increases, the fibers shrink and the vertical Ricci curvature of Mi becomes dominated by the Ricci curvature of the small fiber sN-dim(B) , which is positive as we are assuming that N - dim(B) 2: 2. Then for large i, (Mi' gd has nonnegative Ricci curvature. Taking Ii = p, we see that limHoo (Mi' gi, :~~~\) = (B, gB, e- w dvoIB)' The proof of (ii) is similar, except that we also allow the dimensions of the fibers to go to infinity. D Examples of singular spaces with nonnegative N-Ricci curvature come from group actions. Suppose that a compact Lie group G acts isometrically on a N-dimensional Riemannian manifold M that has nonnegative Ricci curvature. Put X = MIG, let p : M --+ X be the quotient map, let d be the quotient metric and put 1/ = p* (~a';(17)). Then (X, d, 1/) has nonnegative N-Ricci curvature [23, Corollary 7.51]. Finally, we recall the theorem of 0 'Neill that sectional curvature is nondecreasing under pushforward by a Riemannian submersion. There is a Ricci analog of the O'Neill theorem, expressed in terms of the modified Ricci tensor RicN [22]. The proof of this in [22] was by explicit tensor calculations. Using optimal transport, one can give a "synthetic" proof of this Ricci O'Neill theorem [23, Corollary 7.52]. (This is what first convinced the author that optimal transport is the right approach.) REMARK 4.9. We return to the question of whether one can give a good definition of "nonnegative N-Ricci curvature" by just taking the conclusion of the Bishop-Gromov theorem and turning it into a definition. To be a bit more reasonable, we consider taking an angular Bishop-Gromov inequality as the definition. Such an inequality, with parameter n, does indeed characterize when an n-dimensional Riemannian manifold has nonnegative Ricci curvature. Namely, from comparison geometry, nonnegative Ricci curvature implies an angular Bishop-Gromov inequality. To go the other way, suppose that the angular Bishop-Gromov inequality holds. We use polar coordinates around a point m E M and recall that the volume of a infinitesimally small angular sector centered in the direction of a unit vector vETmM, and going up to radius r, has the Taylor expansion (4.10)

V(v, r)

= const.

rn

(1 - 6(n:

2) Ric(v, v) r2

+ ... ) .

If r-nV(v,r) is to be nonincreasing in r then we must have Ric(v,v) 2: O. As m and v were arbitrary, we conclude that Ric 2: O. There is a version of the angular Bishop-Gromov inequality for measured length spaces, called the "measure contracting property" (MCP) [28, 37]. It satisfies Condition 1 of Wishlist 0.4.

OPTIMAL TRANSPORT AND RICCI CURVATURE ...

247

The reason that the MCP notion is not entirely satisfactory can be seen by asking what it takes for a smooth measured length space (M, g, e- wdvolM) to satisfy the N-dimensional angular Bishop-Gromov inequality. (Here dim(M) = n.) There is a Riccati-type inequality

(4.11)

. oW ) 2 -o ( TrII- -oW ) < -RICN(8 8) - -1- ( TrII-or or r, r N - 1 or'

which looks good. Again there is an expansion for the measure of the infinitesimally small angular sector considered above, of the form V(r) = rn (ao + al r+ a2r 2 + ... ), where the coefficents ai can be expressed in terms of curvature derivatives and the derivatives of W. However, if N > n then saying that r-NV(r) is nonincreasing in r does not imply anything about the coefficients. Thus having the N-dimensional angular Bishop-Gromov inequality does not imply that RicN ~ O. In particular, it does not seem that one can prove Corollary 4.7 using MCP. Having nonnegative N-Ricci curvature does imply MCP [37].

5. N-Ricci curvature bounded below by K

In Section 3 we gave the definition of nonnegative N-Ricci curvature. In this section we discuss how to extend this to a notion of a measured length space having N-Ricci curvature bounded below by some real number K. We start with the case N = 00. As mentioned in Section 2, formal computations indicate that in the case of a smooth measured length space (M, g, e- w dvolM), having Ric oo ~ Kg should imply that Hoo has Hessian bounded below by KgH-l on P(M). In particular, if {J..LtltE[O,lj is a geodesic in P(M) then we would expect that Hoo(J-tt) - ~ W2(J-tO, J-tI)2 t 2 is convex in t. This motivates an adaption of Definition 3.13. In order to handle all U E DCoo , we first make the following definition. Given a continuous convex function U : [0, 00) -+ JR, we define its "pressure" by

(5.1)

p(r) = rU~(r) - U(r),

where U~(r) is the right-derivative. Then given K E JR, we define>. : DCoo -+ JRU {-oo} by

(5.2)

. p(r) {Klimr-to+ ~ >'(U) = mf K = 0 r>O r l1!:2. K limr-too r

if K > 0, if K = 0, if K < O.

Note that if U = Uoo (recall that Uoo(r) = r logr) then p(r) = r and so >,(Uoo ) = K.

248

J.

LOTT

DEFINITION 5.3 ([23, Definition 5.13]). Given (X, d, II) has oo-Ricci curvature bounded below by K with supp(/-La) C SUpp(lI) and supp(/-Lr) C SUpp(lI), stein geodesic {/-LdtE[a,lj from /-La to /-LI so that for t E

K E lR, we say that if for all /-La, /-LI E P(X) there is some Wasserall U E DCoo and all

[0,1]'

(5.4) REMARK 5.5. A similar definition, but in terms of U

= Uoo instead of

U E DCco , was used in [36, Definition 4.5].

°

Clearly if K = then we recover the notion of nonnegative oo-Ricci curvature in the sense of Definition 3.13. The N = 00 results of Sections 3 and 4 can be extended to the present case where K may be nonzero. A good notion of (X, d, II) having N-Ricci curvature bounded below by K E lR, where N can be finite, is less clear and is essentially due to Sturm [37]. The following definition is a variation of Sturm's definition and appears in [24]. Given K E lR and N E (1,00], define

(5.6) if N = 00, 00

(

sin.(ta)) N-I tsma

and a > 1[',

if N < 00, K

and a E [0,1['],

if N < 00 and K

1

(

° >°

if N < 00, K >

sinh(ta)) N-I tsinha

if N

< 00 and

K

= 0, < 0,

where

(5.7) When N = 1, define

(5.8)

oo

/3t(xa, Xl) = { 1

if K > 0, if K ~ 0,

Although we may not write it explicitly, a and /3 depend on K and N. We can disintegrate a transference plan 1[' with respect to its first marginal /-La or its second marginal J.Ll. We write this in a slightly informal way:

DEFINITION 5.10. [24] We say that (X, d, II) has N-Ricci curvature bounded below by K if the following condition is satisfied. Given /-La, /-LI E

OPTIMAL TRANSPORT AND RICCI CURVATURE ...

249

P(X) with support in supp(v), write their Lebesgue decompositions with respect to v as J.Lo = Po v + J.LO,s and J.LI = PI V + J.LI,s, respectively. Then there is some optimal dynamical transference plan IT from J.Lo to J.LI, with corresponding Wasserstein geodesic J.Lt = (et}*IT, so that for all U E VCN and all t E [0,1]' we have

(5.11) U",(J.Lt)

~(1 -

t) [

JXxX

{31-t(XO, Xl) U ({3 po/xo) )) d-rr(xllxo) dv(xo) l-t Xo, Xl

{3t(XO,XI)U ({3~I(XI) )) d-rr(xoIXI)dv(XI) Jxxx tXO,XI + U'(oo) [(1 - t)J.LO,s[X] + tJ.LI,s[Xl].

+t [

Here if {3t(xo, xt}

=

00 then we interpret {3t(xo, Xl) U

(.B:(~:'~1

) as

U'(O) PI (Xl), and similarly for {31-t(XO, Xl) U (.Bl~~(:~:Xl))· REMARK 5.12. If J.Lo and J.LI are absolutely continuous with respect to v then the inequality can be rewritten in the more symmetric form (5.13)

U",(J.Lt) ~(1 - t) [

JXxX

+t

[

Jxxx

{31-t(XO, Xl) U (

pO(Xo)

pO(Xo) ) d7r(xo, Xl) {31-t(XO, xt}

{3t(XO,XI) U ( PI(XI) ) d7r(XO,XI). PI(XI) {3t(XO,XI)

REMARK 5.14. Given K ~ K' and N ~ N', if (X, d, v) has N-Ricci curvature bounded below by K then it also has N'-Ricci curvature bounded below by K'. REMARK 5.15. The case N = 00 of Definition 5.10 is not quite the same as what we gave in Definition 5.3! However, it is true that having oo-Ricci curvature bounded below by K in the sense of Definition 5.10 implies that one has oo-Ricci curvature bounded below by K in the sense of Definition 5.3 [24]. Hence any N = 00 consequences of Definition 5.3 are also consequences of Definition 5.10. We include the N = 00 case in Definition 5.10 in order to present a unified treatment, but this example shows that there may be some flexibility in the precise definitions. The results of Sections 3 and 4 now have extensions to the case K =1= O. However, the proofs of some of the extensions, such as that of Theorem 3.20, may become much more involved [37, Theorem 3.1], [39]. Using the extension of Proposition 3.27, one obtains a generalized Bonnet-Myers theorem. PROPOSITION 5.16 ([37, Corollary 2.6]). If (X, d, v) has N-Ricci curvature bounded below by K > 0 then supp(v) has diameter bounded above by JNi( I 7r.

250

J. LOTT

6. Analytic consequences

Lower Ricci curvature bounds on Riemannian manifolds have various analytic implications, such as eigenvalue inequalities, Sobolev inequalities and local Poincare inequalities. It turns out that these inequalities pass to our generalized setting. 6.1. Log Sobolev and Poincare inequalities. Let us first discuss the so-called log Sobolev inequality. If a smooth measured length space (AI, g, e- w dvolM) has Ric oo 2: Kg, with K > 0, then for all f E COO(M) with P e- w dvolM = 1, it was shown in [3] that

IM

The standard log Sobolev inequality on

dv

=

~n

comes from taking

(47r)-~ e- 1x12 dnx,

giving

P

whenever (47r)-~ IlR n e- 1x12 dnx = l. The log Sobolev inequality for (M, g, e- w dvolM) was given both heuristic and rigorous optimal transport proofs by Otto and Villani [30]. We describe the heuristic proof here. From Section 2, having Ric oo 2: Kg formally implies that Hess (Hoo) 2: KgH-l on P(M). Take J.Lo = e- w dvolM and J.Ll = e- w dvolM. Let {J.LtltE[O,lj be a Wasserstein geodesic from J.Lo to J.Ll along which

P

(6.3) is convex in t. As F(O)

= 0, we have F(1) :S F'(1), or

(6.4)

2K W2(J.LO, J.Ld 2

Hoo(J.Ll) -

:S \

d~t It=l' (grad Hoo)(J.Ld )

- KW2(J.LO,J.Ll)2 gH-l

Here grad Hoo is the formal gradient of Hoo on P(M) and the last norms denote lengths with respect to gH-l. As {J.LthE[O,lj is a minimizing geodesic

OPTIMAL TRANSPORT AND RICCI CURVATURE ...

251

from J-Lo to J-L1, we should have

Id~t=ll = W 2(J-LO , J-L1)'

(6.5)

A formal computation gives

(6.6) Then

(6.7)

1M f2 log(f2)

e-\lI

dvolM

r---------

:::; 2 W2(J-LO, J-L1)

:::; sup wElR

=

1M IV fl2

(2 1Mr IV w

e-\lI

dvolM -

fl2 e-\lI dvolM _ K

2

~ W2(J-LO, J-Ld 2 W 2)

~ 1M IV fl2 e-\lI dvolM

which is the log Sobolev inequality. The rigorous optimal transport proof in [30] extends to measured length spaces. To give the statement, we first must say what we mean by IV fl. We define the local gradient norm of a Lipschitz function f E Lip(X) by the formula

(6.8)

IV fl(x) = lim sup If(y) y--+x

f(x)l. d(x, y)

We don't claim to know the meaning of the gradient V f on X in this generality, but we can talk about its norm anyway! Then we have the following log Sobolev inequality for measured length spaces. THEOREM 6.9 ([23, Corollary 6.12]). Suppose that a compact measured length space (X, d, v) has oo-Ricci curvature bounded below by K > 0, in the sense of Definition 5.3. Suppose that f E Lip(X) satisfies f2 dv = 1. Then

Ix

(6.10)

In the case of Riemannian manifolds, one recovers from (6.10) the log Sobolev inequality (6.1) of Bakry and Emery. REMARK 6.11. The proof of Theorem 6.9, along with the other inequalities in this section, uses the K > 0 analog of Proposition 3.29. In turn, the proof of Proposition 3.29 uses the fact that (3.14) holds for all U E 'DeN, as opposed to j ust UN.

J.

252

LOTT

As is well-known, one can obtain a Poincare inequality from (6.10). Take h E Lip(X) with hdv = 0 and put P = 1 + Eh. Taking E small and expanding the two sides of (6.10) in E gives the following result.

Ix

COROLLARY 6.12 ([23, Theorem 6.18]). Suppose that a compact measured length space (X, d, v) has oo-Ricci curvature bounded below by K > O. Then for all h E Lip(X) with hdv = 0, we have

L

Ix

h 2 dv

(6.13)

~~

L

l'Vhl 2 dv.

In case of a smooth measured length space (lVI, g, e- w dvoIM), the inequality (6.13) coincides with the Bakry-Emery extension of the Lichnerowicz inequality, namely >'1 (Z) ~ K. For a general measured length space as in the hypotheses of Corollary 6.12, we do not know if there is a well-defined Laplacian. The Poincare inequality of Corollary 6.12 can be seen as a generalized eigenvalue inequality that avoids this issue. To say a bit more about when one does have a Laplacian, if Q(h) = l'Vhl 2 dv defines a quadratic form on Lip(X), which in addition is closable in L2(X, v), then there is a self-adjoint Laplacian 61/ associated to Q. In this case, Corollary 6.12 implies that 61/ ~ K on the orthogonal complement of the constant functions. In the case of a Ricci limit space, Cheeger and Colding used additional structure in order to show the Laplacian does exist [10].

Ix

6.2. Sobolev inequality. The log Sobolev inequality can be viewed as an infinite-dimensional version of an ordinary Sobolev inequality. As such, it is interesting because it is a dimension-independent result. However, if one has N-Ricci curvature bounded below by K > 0 with N finite then one gets an ordinary Sobolev inequality, which is a sharper result. PROPOSITION 6.14 ([24]). Given N E (1,00) and K > 0, suppose that (X,d,v) has N-Ricci curvature bounded below by K. Then for any nonnegPo dv = 1, one has ative Lipschitz function Po E Lip(X) with

Ix

-1-~

2

(6.15)

N - N

Ixp:-* dv';; 2~ (N;. 1) Ix ~ £p:* IVpol'dv. 1 3

3 0

To put Proposition 6.14 into a more conventional form, we give a slightly weaker inequality. PROPOSITION 6.16 ([24]). Given N E (2,00) and K > 0, suppose that (X,d,v) has N-Ricci curvature bounded below by K. Then for any nonnegative Lipschitz function f E Lip(X) with

(6.17)

1-

,;

Ix f

2N N-2

dv = 1, one has

(Ix I dvr~' K~ (~ =D' DvII' dv.

OPTIMAL TRANSPORT AND RICCI CURVATURE ...

253

Putting (6.17) into a homogeneous form, the content of Proposition 6.16 is that there is a bound of the form I / I 2N ~ F (II/ 111, I 'V/ 112) for N-2 some appropriate function F. This is an example of Sobolev embedding. The inequality (6.17) is not sharp, due to the many approximations made in its derivation. One can use Proposition 6.14 to prove a sharp Poincare inequality. PROPOSITION 6.18 ([24]). Given N E (1, (0) and K > 0, suppose that (X, d, 1/) has N -Ricci curvature bounded below by K. Suppose that h E Lip(X) has hdl/ = O. Then

Ix

(6.19)

Ix

h 2 dl/

~ :-;/

Ix l'Vhl

2

dl/.

In the case of an N-dimensional Riemannian manifold with Ric 2: Kg, one recovers the Lichnerowicz inequality for the lowest positive eigenvalue of the Laplacian [20]. It is sharp on round spheres. 6.3. Local Poincare inequality. When doing analysis on metricmeasure spaces, a useful analytic property is a "local" Poincare inequality. A metric-measure space (X, d, 1/) admits a local Poincare inequality if, roughly speaking, for each function / and each ball B in X, the mean deviation (on B) of / from its average value on B is quantitatively controlled by the gradient of / on a larger ball. To make this precise, if B = Br(x) is a ball in X then we write >"B for B >'1' (x). The measure 1/ is said to be doubling if there is some D > 0 so that for all balls B, 1/(2B) ~ D I/(B). An upper gradient for a function u E C(X) is a Borel function 9 : X -+ [0,00] such that for each curve 'Y : [0,1] -+ X with finite length L( 'Y) and constant speed, (6.20)

IU(!(l)) - u(!(O))1

~ L(!)

11

g(!(t)) dt.

If u is Lipschitz then l'Vul is an example of an upper gradient. There are many forms of local Poincare inequalities. The strongest one, in a certain sense, is as follows: DEFINITION 6.21. A metric-measure space (X, d, 1/) admits a local Poincare inequality if there are constants >.. 2: 1 and P < 00 such that for all u E C(X) and B = Br(x) with I/(B) > 0, each upper gradient 9 of u satisfies

(6.22)

flu - (u) BI dl/ B

~ Pr f

gdl/.

>'B

Here the barred integral is the average (with respect to 1/), e.g., f>'B 9 dl/ =

Jtf>.~~v,

and (u) B is the average of u over the ball B. In the case of a length

254

J. LOTT

space, the local Poincare inequality as formulated in Definition 6.21 actually implies stronger inequalities, for which we refer to [15, Chapters 4 and 9]. It is known that the property of admitting a local Poincare inequality is preserved under measured Gromov-Hausdorff limits [18, 19]. (This was also shown by Cheeger in unpublished work.) Cheeger showed that if a metric-measure space has a doubling measure and admits a local Poincare inequality then it has remarkable extra local structure [6]. Cheeger and Colding showed that local Poincare inequalities exist for Ricci limit spaces [10]. The method of proof was to show that Riemannian manifolds with lower Ricci curvature bounds satisfy a certain "segment inequality" [7, Theorem 2.11] and then to show that the property of satisfying the segment inequality is preserved under measured Gromov-Hausdorff limits [10, Theorem 2.6]. The segment inequality then implies the local Poincare inequality. It turns out that the argument using the segment inequality can be abstracted and applied to certain measured length spaces. For simplicity, we restrict to the case of nonnegative N-Ricci curvature. We say that (X, d, v) has almost-everywhere unique geodesics if for v 0 v-almost all (xo, Xl) E X x X, there is a unique minimizing geodesic "( E r with "((0) = Xo and "((1) = Xl. 6.23 ([24, 34, 37]). If a compact measured length space (X, d, v) has nonnegative N -Ricci curvature and almost-everywhere unique geodesics then it satisfies the local Poincare inequality of Definition 6.21 with >. = 2 and P = 22N+I. THEOREM

As is well-known, a Riemannian manifold has almost-everywhere unique geodesics. A sufficient condition for (X, d, v) to have almost-everywhere unique geodesics is that almost every X E X is nonbranching in a certain sense [34, 37]. The result of Theorem 6.23 holds in greater generality. What one needs is a way of joining up points by geodesics, called a "democratic coupling" in [24], and a doubling condition on the measure. We do not know whether the condition of nonnegative N-Ricci curvature is sufficient in itself to imply a local Poincare inequality. Having nonnegative N-Ricci curvature does not imply almost-everywhere unique geodesics. For a noncompact example, the finite-dimensional Banach space ~n with the h norm and the Lebesgue measure has nonnegative n-Ricci curvature, but certainly does not have almost-everywhere unique geodesics.

7. Final remarks In this survey we have concentrated on compact spaces. There is also a notion of Ricci curvature bounded below for noncom pact measured length spaces (X, d, v) [23, Appendix E]. Here we want X to be a complete pointed locally compact length space and v to be a nonnegative nonzero Radon

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255

measure on X. We do not require v to be a probability measure. There is a Wasserstein space P2(X) of probability measures on X with finite second moment, i.e.,

where * is the basepoint in X. Many of the results described in this survey extend from compact spaces to such noncompact spaces, although interesting technical points arise. In particular, if (X, d, v) is a compact or noncom pact space with nonnegative N - Ricci curvature and supp(v) = X, and if x is a point in X, then a tangent cone at x has nonnegative N-Ricci curvature [23, Corollary E.44]. There are many directions for future research. Any specific problems that we write here may become obsolete, but let us just mention two general directions. One direction is to see whether known results about Riemannian manifolds with lower Ricci curvature bounds extend to measured length spaces with lower Ricci curvature bounds. As a caution, this is not always the case. For example, the Cheeger-Gromoll splitting theorem says that if there is a line in a complete Riemannian manifold M with nonnegative Ricci curvature then there is an isometric splitting M = lR x Y. This is not true for measured length spaces with nonnegative N-Ricci curvature. Counterexamples are given by nonEuclidean n-dimensional normed linear spaces, equipped with Lebesgue measure, which all have nonnegative nRicci curvature [39]. However, it is possible that there is some vestige of the splitting theorem left. The splitting theorem does hold for a pointed Gromov-Hausdorff limit of a sequence {(Mi' gd }~1 of complete Riemannian manifolds with Ricci curvature bounded below by [7], so not every finite-dimensional (X, d, v) with nonnegative N-Ricci curvature arises as a limit in this way. (The analogous statement is not known for finite-dimensional Alexandrov spaces, but there are candidate Alexandrov spaces that may not be Gromov-Hausdorff limits of Riemannian manifolds with sectional curvature uniformly bounded below [17].) One's attitude towards this fact may depend on whether one intuitively feels that finite-dimensional normed linear spaces should or should not have nonnegative Ricci curvature. Another direction of research is to find classes of measured length spaces (X, d, v) which do or do not have lower Ricci curvature bounds. This usually amounts to understanding optimal transport on such spaces.

-t

References [1] C. Am"), S. Blachere, D. Chafai, P. Fougeres, 1. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Sur les inegalites de Sobolev logarithmiques, Panoramas et Syntheses, 10, Societe MatMmatique de France, 2000.

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[2] D. Bakry, L 'hypercontractivite et son utilisation en theorie des semigroupes, in 'Lectures on probability theory' (Saint-Flour, 1992), Lecture Notes in Math., 1581, Springer, Berlin, 1994, 1-114. [3] D. Bakry and M. Emery, Diffusions hypercontractives, in 'Seminaire de probabilites XIX', Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177-206. [4] E. Bell, Men of Mathematics, Simon and Schuster, 1937. [5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. [6] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funet. Anal., 9 (1999), 428-517. [7] J. Cheeger and T. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math., 144 (1996), 189-237. [8] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below I, J. Differential Geom., 46 (1997), 37-74. [9] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below II, J. Differential Geom., 54 (2000), 13-35. [10] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below III, J. Differential Geom., 54 (2000), 37-74. [11] D. Cordero-Erausquin, R. McCann, and M. Schmuckenschliiger, A Riemannian interpolation inequality d la Borell, Brascamp and Lieb, Inv. Math., 146 (2001), 219-257. [12] 1. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137(653) (1999). [13] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 87 (1987), 517-547. [14] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhiiuser, Boston, 1999. [15] J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 200l. [16] R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the FokkerPlanck equation, SIAM J. Math. Anal., 29 (1998), 1-17. [17] V. Kapovitch, Restrictions on collapsing with a lower sectional curvature bound, Math. Zeit., 249 (2005), 519-539. [18] S. Keith, Modulus and the Poincare inequality on metric measure spaces, Math. Z., 245 (2003), 255-292. [19] P. Koskela, Upper gradients and Poincare inequalities, in 'Lecture notes on analysis in metric spaces', Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, 2000, 55-69. [20] A. Lichnerowicz, Geometrie des groupes de transformations, Travaux et Recherches Mathematiques III, Dunod, Paris, 1958. [21] F. Liese and I. Vajda, Convex statistical distances, Teubner-Texte zur Mathematik, 95, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987. [22] J. Lott, Some geometric properties of the Bakry-Emery-Ricci tensor, Comment. Math. Helv., 78 (2003), 865-883. [23] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, to appear, Annals of Math. [24] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, http://www.arxiv.org/abs/math.DG/0506481, 2005. [25] R.J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. [26] R.J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funet. Anal., 11 (2001), 589-608. [27] G. Monge, Memoire sur la theorie des deblais et des remblais, Histoire de l'Academie Royale des Sciences de Paris, 1781,666-704.

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[28] S.-I. Ohta, On measure contraction property of metric measure spaces, preprint, http://www.math.kyoto-u.ac.jp;-sohta/. 2005. [29] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. [30] F. Otto and C. Villani, Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. [31] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://www.arxiv.org/abs/math.DG/0211159, 2002. [32] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford, 48 (1997), 235-242. [33] S. Rachev and L. Riischendorf, A characterization of random variables with minimum L 2 -distance, J. Multivariate Anal., 32 (1990), 48-54. [34] M. von Renesse, On local Poincare via transportation, preprint, http://www. arxiv.org/abs/math.MG/0505588, 2005. [35] M.-K von Renesse and K-T. Sturm, Transport inequalities, gradient estimates and Ricci curvature, Comm. Pure Appl. Math., 68 (2005), 923-940. [36] K-T. Sturm, On the geometry of metric measure spaces, Acta Math., 196 (2006), 65-13l. [37] K-T. Sturm, On the geometry of metric measure spaces II, Acta Math., 196 (2006), 133-177. [38] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, 2003. [39] C. Villani, Optimal transport, old and new, Springer-Verlag, to appear DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR,

MI 48109-1109, USA E-mail address: lottCDumich. edu

Surveys in Differential Geometry XI

Manifolds of Positive Scalar Curvature: a Progress Report Jonathan Rosenberg

The scalar curvature /'i, is the weakest curvature invariant one can attach (pointwise) to a Riemannian n-manifold Mn. Its value at any point can be described in several different ways: (1) as the trace of the Ricci tensor, evaluated at that point. (2) as twice the sum of the sectional curvatures over all 2-planes ei 1\ ej , i < j, in the tangent space to the point, where el, ... , en is an orthonormal basis. (3) up to a positive constant depending only on n, as the leading coefficient in an expansion [22, Theorem 3.1]

VM(r) = VE(r)

(1 _

/'i,

6(n + 2)

r2

+ ... )

telling how the volume VM(r) of a small geodesic ball in M of radius r differs from volume VE(r) = Cnrn of a corresponding ball in Euclidean space. Positive scalar curvature means balls of radius r for small r have a smaller volume than balls of the same radius in Euclidean space; negative scalar curvature means they have larger volume. In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a given manifold (always assumed COO) admits a Riemannian metric with positive or non-negative scalar curvature. (If the manifold is non-compact, we require the metric to be complete; otherwise this is no restriction at all.) We will not go over the historical development of this subject or everything that is known about it; instead, our focus here will be on updating the existing surveys [20], [68], [69] and [58]. This work was partially supported by NSF grant number DMS-0504212. @2007 International Press 259

260

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We should explain why we care so much about positivity of the scalar curvature. Why not ask about metrics of negative scalar curvature, or of vanishing scalar curvature, or of non-negative scalar curvature? More generally, we could ask which smooth functions on a closed manifold M are realized as the scalar curvature function of some metric on M. It is a remarkable result of Kazdan and Warner that (in dimensions >2) the answer to this question only depends on which of the following classes the manifold M belongs to: (1) Closed manifolds admitting a Riemannian metric whose scalar curvature function is non-negative and not identically O. (2) Closed manifolds admitting a Riemannian metric with vanishing scalar curvature, and not in class (1). (3) Closed manifolds not in classes (1) or (2). All these three classes are non-empty if n 2:: 2. By a simple application of the Gauss-Bonnet Theorem, if n = 2, class (1) consists of 8 2 and JR.JlD2; class (2) consists of T2 and the Klein bottle; and class (3) consists of surfaces with negative Euler characteristic. THEOREM 0.1 ("Trichotomy Theorem" [34]' [33], [32]). Let M n be a closed connected manifold of dimension n 2:: 3. (1) If M belongs to class (1), every smooth function is realized as the scalar curvature function of some Riemannian metric on M. (2) If M belongs to class (2), then a function f is the scalar curvature of some metric if and only if either f(x) < 0 for some point x E M, or else f == O. If the scalar curvature of some metric g vanishes identically, then g is Ricci flat (i. e., not only does the scalar curvature vanish identically, but so does the Ricci tensor). (3) If M belongs to class (3), then f E COO(M) is the scalar curvature of some metric if and only if f(x) < 0 for some point x E M. This Theorem thus shows that deciding whether a manifold M belong to class (1) is equivalent to determining whether M admits a metric of strictly positive scalar curvature. Furthermore, in this case, there are no restrictions at all on possibilities for the scalar curvature. We will include some more results about class (2) in the last section of this paper. REMARK 0.2. Note that Theorem 0.1 partially justifies the comment above, that existence of a metric of positive scalar curvature on a noncompact (connected) manifold M is no restriction at all if the metric is not required to be complete. Indeed, suppose M is diffeomorphic to an open subset of a compact manifold with boundary M. (If M is homotopically finite with "tame" ends, this is not much of a restriction.) Take the double of M along 8M; this is now a closed manifold X in which M is embedded as an open subset with complement having non-empty interior. By Theorem 0.1, there is a metric on X whose scalar curvature function is positive on M but negative somewhere in the complement of M. The general case (where

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261

M cannot be embedded in a closed manifold) can be deduced from this case with somewhat more work.

Most of the results presented in this paper are due to other authors, but the organization here may be a bit different than in the original sources. In Section 1, we will discuss necessary and sufficient conditions for a closed manifold M to admit a metric of positive scalar curvature. Part of this discussion (e.g., Conjecture 1.19, Theorem 1.20, and Remark 1.25) has been known for a while to the experts but may not be in the literature in its present form. In Section 2, we will discuss the topology of the space of metrics of positive scalar curvature in cases where this space is non-empty. Some of the proofs in this section are new. Then Section 3 will discuss the question of what non-compact manifolds admit a complete metric of positive scalar curvature. Here, Theorem 3.4, Corollary 3.5, and part of Theorem 3.9 are new results. Section 4 will discuss a few other miscellaneous topics. 1. The obstruction problem: which closed manifolds admit

a metric of positive scalar curvature? If Mn is a closed n-manifold, when can M be given a Riemannian metric for which the scalar curvature function is everywhere strictly positive? (For simplicity, such a metric will henceforth be called a metric of positive scalar curvature. ) Answering this basic question involves two disjoint sets of techniques: obstruction results, showing that some manifolds do not admit metrics of positive scalar curvature, and positive results, showing that many manifolds do admit such metrics. 1.1. Obstruction results. All known obstruction results follow from one of three basic principles:

(1) The result of Lichnerowicz [40], that if I/J is the Dirac operator on a spin manifold M (a self-adjoint elliptic first-order differential operator, acting on sections of the spinor bundle), then (1.1) Here 'V is the covariant derivative on the spinor bundle induced by the Levi-Civita connection, and 'V* is the adjoint of 'V. Since the operator 'V*'V is obviously self-adjoint and non-negative, it follows from equation (1.1) that the square of the Dirac operator for a metric of positive scalar curvature is bounded away from 0, and thus that the Dirac operator cannot have any kernel. It follows that any index-like invariant of M which can be computed in terms of harmonic spinors (Le., the kernel of I/J) has to vanish.

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(2) The Schoen-Yau minimal surface technique [61J, which implies that if M n is an oriented manifold of positive scalar curvature, and if N n - 1 is a closed stable minimal hypersurface in M dual to a nonzero class in H1(M, Z), then N also admits a metric of positive scalar curvature. (3) The Seiberg-Witten technique [73], which implies that if M4 is a closed 4-manifold with a non-zero Seiberg-Witten invariant, then M does not admit a metric of positive scalar curvature. Each of these three techniques has its own advantages and disadvantages. Technique (1) applies to manifolds of all dimensions, and is usually the most powerful, but it only applies to spin manifolds, or at least to manifolds with a spin cover (Le., to manifolds M such that w2(M) = 0, where M is the universal cover of M and W2 is the second Stiefel-Whitney class). Technique (2) applies whether or not M and N are spin, but it requires H 1 (M, Z) to be non-zero, which is quite a restriction on 1l'1 (M). In addition, since solutions to the minimal hypersurface equations in general have singularities, this technique only works without modification up to dimension 7 or 8. There have been hopes for a long time (see for example [64]) that one could "excise the singularities" to make this technique work in high dimensions, and now Lohkamp [43J has announced a precise result of this sort, based in part on joint work with Ulrich Christ, though as of the time of writing this paper, the details have not yet appeared. Finally, technique (3) again does not require a spin condition, but works only in the special dimension 4. (Sometimes one can reduce problems about manifolds in dimensions 5 through 8 to this case using technique (2).) Let's now go into the three techniques in somewhat more detail. 1.1.1. The Dirac obstruction. We start with (1) of §1.1, the Dirac operator method. If M is a spin manifold of dimension n, there is a version of the Dirac operator which commutes with the action of the Clifford algebra Gin (see [37, §II.7]). In particular, its kernel is a (graded) Gin-module, which represents an element a(M) in the real K-theory group KO n = Ko-n(pt) (see [37, Def. II.7.4]). THEOREM 1.1 (Lichnerowicz [40J; Hitchin [29]). If M n is a closed spin manifold for which a(M) i= 0 in KO n1 then M does not admit a metric of positive scalar curvature. We recall that KO n ~ Z for n == 0 mod 4, that KO n ~ Zj2 for n == 1,2 mod 8, and KO n = 0 for all other values of n. Furthermore, for n == 0 mod 4, the invariant a(M) is essentially equal to Hirzebruch's A-genus A(M), namely a(M) = A(M) for n == 0 mod 8, and a(M) = A(M)j2 for n == 4 mod 8. So this result immediately shows that there are many manifolds, even simply connected ones, which do not lie in class (1) of the KazdanWarner trichotomy (see Theorem 0.1). E.g., the Kummer surface K 4 , the hyperplane in the complex projective space cJID3 given by the equation

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zg + zt + z~ + zf

263

= 0, is spin and has A(K) = 2, and hence does not

admit a metric of positive scalar curvature. We observe that a(M) depends only on the spin bordism class [M] E n:;rin. In fact, we can interpret a(M) as the image of [M] under a natural transformation of generalized homology theories as follows. Let KO*(X) and ko*(X) denote the periodic and connective real K-homology of a space X, respectively (so KO*(X) satisfies Bott periodicity, and the spectrum defining ko. is obtained from the periodic KO-spectrum by killing all homotopy groups in negative degrees). Then there are natural transformations

(1.2)

n~pin(x) ~ ko.(X) per) KO.(X),

the first of which sends the bordism class [M, f] to f.([M]ko), where [M]ko E ko.(M) denotes the ko-fundamental class of M determined by the spin structure, and the second of which builds in Bott periodicity by inverting the Bott generator of k08 ~ Z. With this notation, a(M) = per 0 D([M]) (in the case X = pt). A stronger result than Theorem 1.1 can be obtained by taking the fundamental group into account and coupling the Dirac operator with flat or almost flat vector bundles. To get good results, we need to use infinitedimensional bundles, or at least sequences of bundles whose dimensions go to infinity. Here we will use the index theory of Mishchenko and Fomenko [46] and bundles of Hilbert C· -modules over the real C· -algebra of the fundamental group. This algebra, denoted CiJ7r), is the completion of the group ring lR[7r] for the largest C· -norm, or in other words the largest operator norm on a Hilbert space, when one lets lR[7r] act on Hilbert spaces via representations of 7r by invertible isometries. Ordinary flat vector bundles can't give very much, since the rational characteristic classes of any finitedimensional flat vector bundle are trivial by Chern-Weil theory. We will also need a topological construction that will play a big role later. For any (discrete) group 7r, there is a classifying space B7r, which we can choose to be a CW complex, having 7r as fundamental group and with contractible universal cover E7r. This space is unique up to homotopy equivalence. If M has fundamental group 7r, then there is a classifying map f: M -7 B7r which induces an isomorphism on fundamental groups. This map is determined up to homotopy by an identification of 7rl(M) with 7r. Thus we can replace X by B7r in (1.2) above and define, if Mn is a spin manifold, an invariant aB7r(M) E KOn{B7r). The best result one can obtain on the obstruction problem using the index theory of the Dirac operator can be stated in the case of spin manifolds as follows: THEOREM 1.2 (Rosenberg [54]). For any discrete group 7r, there is a natural assembly map A: KO.(B7r) -7 KO.(CJi(7r)) from the KO-homology of the classifying space to the topological K -theory of the real group C*algebra. (The Baum-Connes Conjecture implies, in particular, that this map

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is injective if 7r is torsion-free.) If Mn is a closed spin manifold for which A(UB1l'(M)) i=. 0 in KO n(Ci(7r)), then M does not admit a metric of positive scalar curvature. SKETCH. Form the bundle VB1l' = E7r X1l'Ci(7r) over B7r whose fibers are rank-one free (right) modules over Ci(7r). As a "Ci(7r)-vector bundle" over B7r, this has a stable class [VB1l'J in a K-group KOO(B7r; Ci(7r)), and A is basically the "slant product" with [VB1l'J. This relies on an index theory, due to Mishchenko and Fomenko, for elliptic operators with coefficients in a Ci(7r)-vector bundle. If M is as in the theorem, then the (Clifford algebra linear) Dirac operator on M, with coefficients in the bundle VB1l" has an index uB1l'(M, 1) E KO n (Ci(7r)), which one can show by the Kasparov calculus is just AoperoD([M, fl). Since VB1l' is by construction a flat bundle, there are no correction terms due to curvature of the bundle, and formula (1.1) applies without change. Hence if M has positive scalar curvature, the square of this Dirac operator is bounded away from 0, and the index vanishes.

o REMARK 1.3. For purposes of the construction above, one could just as well use the reduced real group C*-algebra Ci,red(7r), which is the completion of lR[7rJ for its action on L2(7r) by left convolution. When 7r is amenable, this algebra coincides with Ci(7r); otherwise, it is a proper quotient. The assembly map into KO*(Ci(7r)) potentially has a smaller kernel than the one into KO*(Ci red(7r)), but on the other hand, the latter has a better chance of being an i~omorphism. The Baum-Connes Conjecture would imply that if 7r is torsion-free, the assembly map into KO*(Cired(7r)) is an isomorphism, whereas one cannot generally expect this for the assembly map into KO*(Ci(7r))· It was conjectured in [55], admittedly on the basis of rather flimsy evidence, that when 7r is finite and Mn is a spin manifold, the vanishing of A(UB1l'(M)) is not only necessary, but also sufficient, for M to admit a metric of positive scalar curvature. This conjecture is usually called the GromovLawson-Rosenberg Conjecture. (See Conjecture 1.22 below.) There are no known counterexamples to this conjecture in dimensions n 2 5, though as we will see, there are reasons to be skeptical about it. There is an analogue of Theorem 1.2 that holds when M does not admit a spin structure, but w2(M) = O. We call this the "twisted" case. The statement appears a bit technical, and those not so interested in having the most general possible result can ignore it and concentrate on just two cases: the case where M is spin, which we have already discussed, and the case where M is oriented and w2(M) i=. 0, in which case the Dirac operator method gives no information at all. The following definitions are due to Stolz.

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DEFINITION 1.4. Let'"'( be a triple Crr, w, 1?), where w: 7r -+ 7l/2 is a group homomorphism (this will correspond to WI of our manifold) and 1? -* 7r is an extension of 7r such that ker(1? -+ 7r) is either 7l/2 or the trivial group. Let u: Spin(n) -+ SO(n) be the non-trivial double covering of the special orthogonal group SO(n). We note that the conjugation action of O(n) on SO(n) lifts to an action on Spin(n). Let 1? ~ Spin(n) be the semi direct product, where 9 E 1? acts on the normal subgroup Spin(n) by conjugation by rw(g). Here r E O(n) is the reflection in the hyperplane perpendicular to el = (1,0, ... ,0) E IRn. Abusing notation, we also use the notation W for the composition 1? -+ 7r -+ 7l/2. We define Gh, n) to be the quotient of 1? ~ Spin(n) by the central subgroup generated by (k, -1), where k E 1? is the (possibly trivial) generator of ker(1? -+ 7r). Sending [a, bj E Gh, n) to rw(a)u(b) defines a homomorphism ph, n): Gh, n) -+ O(n). A '"'(-structure on an n-dimensional Riemannian manifold M is a principal Gh, n)-bundle P -+ M together with a Gh, n)-equivariant map p: P -+ O(M). Here O(M) is the orthogonal frame bundle of M, a principal bundle for the orthogonal group O(n), and Gh, n) acts on O(M) via the homomorphism ph, n). REMARK 1.5. Let M be a connected manifold with fundamental group 7r and with wI(M) = w: 7r -+ 7l/2. Then M always admits a ,",(-structure for som~'"'( = (7r, W, 1?). We can arrange to have ker(1? -+ 7r) # 0 exactly when w2(M) = O. (1) If 7r is the trivial group, then Gh, n) = SO(n) (resp. Spin(n)) if ker(1? -+ 7r) is trivial (resp. non-trivial). In this case a '"'(-structure on M amounts to an orientation (resp. spin structure) on M (cf. [37, Def. II.1.3]). (2) More generally, if W = 0 and 1? = 7r (resp. 1? = 7r x 7l/2), then Gh, n) = 7r X SO(n) (resp. Gh, n) = 7r x Spin(n)); in this case, a ,",(-structure amounts to an orientati~ (resp. spin structure) on M, together with a principal 7r-bundle M -+ M. (3) If M is not orientable, so W # 0, but w2(M) # 0, then Ch, n) = 7r X O(n) and the '"'(-structure on M is determined by the classifying map M -+ B7r x BO(n), where the first component of the map is the classifying map for the universal covering, and the second component is the classifying map for the tangent bundle. (4) A '"'(-structure determines a principal 7r-bundle M ~f P/G 1 -+ M, where Gl is the identity component of Gh, n). We note that Gl = SO(n) if ker(1? -+ 7r) is trivial~nd G 1 = Spin(n) otherwise. Hence the principal G1-bundle P -+ M can be identified with the oriented frame bundle of M or a double cover thereof.

The substitute for Theorem 1.2 in the twisted case involves a substitute C* ('"'() for C~ (7r). This is defined as follows.

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DEFINITION 1.6. Let "( = (11', w, 1i') be a "(-structure as in Definition 1.4. We define a Z/2-graded C*-algebra C*"( to be the -I-eigenspace for the involution on the group C* -algebra C*1i' defined by multiplication by the central generator k of ker(1i' -+ 7r). The Z/2-grading is given by the {±I}eigenspaces of the involution C*"( -+ C*"( which is the restriction of the involution C*1i' -+ C*1i' given by 9 t-+ (-1 )w(g) for 9 E 1i' c C*1i', where 'Ii; is the composition of the projection map 1i' -+ 7r and w: 7r -+ Z/2. In particular, C*"( = 0 if 1i' = 7r and C*"( = C*7r (with the trivial grading) if w = 0 and 1i' = 7r x Z/2. The analogue of Theorem 1.2 in the twisted case is then: THEOREM 1.7 (Stolz; see [58, §5]). If M n is a closed manifold with w2(M) = 0 and with ,,(-structure "( = (7r, w, 1i'), then there is a "twisted Dirac obstruction" in KOn(C*,,(), whose vanishing is necessary for M to admit a metric of positive scalar curvature. Roughly speaking, this theorem, like Theorem 1.2, is proved by taking the "index of the Dirac operator" in a suitable sense. Since our understanding of positive scalar curvature is incomplete enough even for spin manifolds, or for oriented manifolds with non-spin universal cover, we will concentrate hereafter on these simpler cases and not mention the twisted case any further. 1.1.2. The minimal hypersurface method. Now let's discuss (2) of §1.I, the minimal hypersurface method. This relies on the following inequality, found in [61]: LEMMA 1.8 (Schoen-Yau [61]). Let M n be a closed oriented n-manifold with Hi (M, Z) f= 0 and with positive scalar curvature, and let Hn-l be a stable minimal hypersurface, minimizing (n - I)-dimensional volume in its homology class. Then (1.3)

i (K~2

+ 1V'12) dvol > 0,

with K the scalar curvature of H in the induced metric from M and d vol the measure on H defined by the induced metric, for all functions E COO(H) not vanishing identically. (Here V' is to be computed with respect to the induced metric on H.) From this one can deduce that H also has a metric of positive scalar curvature, and in some cases, this leads to a contradiction, with the result that M could not have had a metric of positive scalar curvature in the first place. For example, if n = 3, then taking == 1 in (1.3), we deduce that the integral of Kover H is positive, which by Gauss-Bonnet implies that H must be a sphere. In particular, since the homology class of H in H2(M, Z) is represented by a sphere, it lies in the image of the Hurewicz map 7r2 (M) -+

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H2(M,Z). This is impossible if Mis aspherical with bl(M) > 0 (so that we could construct H in the first place), so we see that an aspherical oriented closed 3-manifold M with bi (M) > 0 cannot have a metric of positive scalar curvature. This sort of reasoning was refined in [62] to show that if M is a compact oriented closed 3-manifold M with 7l"1 (M) containing a product of two cyclic groups or a subgroup isomorphic to the fundamental group of a compact Riemann surface of genus >1, then M cannot admit a metric of positive scalar curvature. If n > 3, the reasoning to get from (1.3) to the fact that H admits a metric of positive scalar curvature is a bit more complicated. Basically, (1.3) implies that the conformal Laplacian of H (for the metric induced from M) is strictly positive, which in turn implies (by an argument of Kazdan and Warner) that one can make a conformal change in the metric of H to achieve positive scalar curvature. Iterating use of this technique, one can show that many manifolds do not admit metrics of positive scalar curvature, as long as one can produce chains of stable minimal hypersurfaces going down in dimension from n to 2. For example, one can formalize this as follows: THEOREM 1.9 (Schick, [60, Corollary 1.5]). Let X be any space, let 3 ~ k ~ 8, and let 0 E HI(X, Z). Let Hk(X, Z)+ denote the subset of Hk(X, Z) consisting of classes f*([N]), where f: Nk ---t X and N is an oriented closed manifold of positive scalar curvature. Then cap product with o maps Hk(X, Z)+ to Hk-I (X, Z)+.

Here the restriction to k ~ 8 is simply for the purpose of knowing that if f: Nk ---t X, then there is a smooth, nonsingular minimal hypersurface of N dual to Schick's original paper had k ~ 7; this can be improved, as remarked in [31], using better regularity results for minimal hypersurfaces in [65]. Presumably the dimensional restriction can be removed altogether using the results of [43]. The minimal hypersurface technique is especially powerful in low dimensions. For example, it was used to prove:

ro.

THEOREM 1.10 (Schoen and Yau [64, Theorem 6]). No closed aspherical 4-manifold can admit a metric of positive scalar curvature.

1.1.3. The Seiberg- Witten method. Finally we get to (3) of §1.1, the Seiberg-Witten method. This applies only to oriented closed 4-manifolds. Any such manifold M always admits a spinc structure ~. It is not unique (in fact, the set of spinc structures compatible with the orientation is a principal homogeneous space for H2 (M, Z)), but for each choice, provided bt (M) > 1, there is an integer invariant, called the Seiberg- Witten invariant SW(~), which counts the number of solutions to a certain non-linear elliptic system of partial differential equations. (The equations concern a spinor field 1/J, i.e.,

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a section of the positive half-spinor bundle

st.

st associated

to~, plus a connec-

tion A on the line bundle /\ 2 We require 'l/J to satisfy the Dirac equation defined by the connection, I!Jf.,A('l/J) = 0, and in the "unperturbed" version of the equations, l also require the self-dual part of the curvature of A to be given by the pairing of 'l/J with itself under the nontrivial bundle map

st®st -+ nt, which of course is quadratic, not linear, in 'l/J.) When bt(M) = 1, it is still possible to define SW(~), but in general it also depends on the Riemannian metric (or the perturbation made to the equation). The basic connection between Seiberg-Witten invariants and scalar curvature is the following: THEOREM 1.11 (Witten [73], [47, Corollary, 5.1.8]). Let M be an oriented closed 4-manifold with bt(M) > 1. If SW(~) 0 for some spinc structure ~, then M does not admit a metric of positive scalar curvature.

t=

On the other hand, for some special classes of 4-manifolds, one knows that the Seiberg-Witten invariant can be non-zero. For example, one has: THEOREM 1.12 (Taubes [70]). Let M n be a closed, connected oriented 4manifold with bt (M) > 1. If M admits a symplectic structure (in particular, if M admits the structure of a Kahler manifold of complex dimension 2), then SW(~) 0 for some spinc structure~, and thus M does not admit a positive scalar curvature metric (even one not well-behaved with respect to the symplectic structure).

t=

This dramatic result implies that many smooth 4-manifolds do not admit metrics of positive scalar curvature, even if they are non-spin and simply connected. In other words: COUNTEREXAMPLE 1.13. In dimension 4, there exist: (1) a simply connected spin manifold M4 with A(M) = 0 but with no positive scalar curvature metric. (2) simply connected non-spin manifolds with no positive scalar curvature metric. Still more subtle things go wrong in dimension 4, such as: COUNTEREXAMPLE 1.14 (Hanke, Kotschick, and Wehrheim [25]). For any odd prime p, there exists a smooth spin 4-manifold M4 with fundamental group cyclic of order p, such that M does not admit a metric of positive scalar curvature, but its universal cover does. ISometimes, for technical reasons, one needs to make a small perturbation. See [47, Chapter 6] for a detailed explanation. We'll ignore this for the moment.

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When bt (111) = 1, then Seiberg-Witten invariants still exist, but they are not necessarily independent of the metric g. However, it is still true that if SW(~, g) =F 0, then the metric g cannot have positive scalar curvature. This is useful in some cases, since for example, one can show that if M is homeomorphic to ((::Jp2 # nClP'2 with n :s 9, then the Seiberg-Witten invariants are actually independent of the choice of metric. This was used in [48], [66], [15], and [49] to construct exotic 4-manifolds, homeomorphic to classical simply connected manifolds of positive scalar curvature, without metrics of positive scalar curvature. We will see that all of this is very different from what happens in dimensions 2:5.

1.2. Positive results. The known positive results about existence of metrics of positive scalar curvature come from a combination of (1) specific constructions for certain special classes of manifolds, such as fiber bundles with fibers of positive scalar curvature and the structure group consisting of isometries (see, e.g., [67, Observation, p. 512]), manifolds with a non-trivial action of SU(2) or SO(3) [38], or Toda brackets [7]; (2) surgery techniques for "propagating" positive scalar curvature from one manifold to another. Here (1) is self-explanatory, but only covers a rather small number of examples, built out of standard building blocks such as spheres, projective spaces, and lens spaces, using fairly standard constructions. Such techniques only work on "highly symmetric" manifolds, and so one needs a way to get from these to more general manifolds. That is what is provided by the surgery method (item (2) above). The basic result on which everything is based is the following: THEOREM 1.15 (Surgery Theorem of Gromov-Lawson [23], Schoen-Yau

[61]). Let M' be a closed manifold of positive scalar curvature, not necessarily connected, and suppose M is a manifold that can be obtained from M' by surgery in codimension 2:3. Then M also admits a metric of positive scalar curvature.

While this may not seem like much, this Surgery Theorem, together with the method of proof of the s-cobordism theorem, implies a reduction of the question of what manifolds M n admit metrics of positive scalar curvature, provided that n 2: 5, to bordism theory. Further application of techniques developed in [67], together with additional ideas of Jung based on the Baas theory of "bordism with singularities," ultimately reduce one down to the following statement. For simplicity, we have ignored the "twisted case," which is much more complicated to describe, though the ideas are roughly the same. THEOREM 1.16 (Jung and Stolz). Let M n be a closed connected oriented manifold with fundamental group 7r and dimension n 2: 5. Let B7r be the

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classifying space of 7r, or in other words, a K(7r,I) space, which is well defined up to homotopy equivalence, and let u: M -+ B7r be a classifying map for the universal cover of 7r (so that the universal cover of l'v! is the pull-back under u of the universal principal 7r-bundle). (1) If M is spin, let [M]ko E ko*(M) denote the ko-fundamental class of M determined by the spin structure. Suppose U*([M]ko) E kon (B7r)+, the subset of kon (B7r) consisting of classes f*([N]ko) with N n a spin manifold of positive scalar curvature and with f: N -+ B7r. (2) If w2(M) =1= 0, i.e., the universal cover of M is not spin, suppose u*([M]) E Hn(B7r)+, the subset of Hn(B7r) consisting of classes f*([N]) with Nn an oriented manifold of positive scalar curvature and with f: N -+ B7r. Then (in either case) M admits a metric of positive scalar curvature.

REMARK 1.17. Note there is a certain asymmetry between M and N in the theorem. While M has to be connected, and while u has to be a classifying map for the universal cover of M, N need not be connected, and f can be arbitrary. In addition, in case (2), while we require 'W2(M) =1= 0, there is no such condition on N. Another curious fact is that while, a priori, kon (B7r)+ and Hn(B7r)+ are just sets, they are in fact subgroups of kon (B7r) and Hn(B7r). The reason is that addition is represented in bordism theories by disjoint union of manifolds, and the disjoint union of manifolds of positive scalar curvature clearly has a metric of positive scalar curvature. Similarly, multiplication by -1 is represented by reversal of orientation or spin structure, which has no effect on the positive scalar curvature condition. Still another case of interest is the one where M is not orientable, but still w2(M) =1= 0. In this case, there is an analogue of case (2) of Theorem 1.16, but one needs to replace usual homology by homology with local coefficients. (See for example [7, Theorems 2.5 and 2.7(3)].)

1.3. Classification conjectures. In this subsection, we will discuss what answers might be expected to the question of what closed manifolds should admit metrics of positive scalar curvature, and what is known about the status of these conjectures. First of all, for simply connected manifolds of dimension 2: 5, the problem is fully understood. THEOREM 1.18 (Gromov-Lawson [23], Stolz [67]). Let M n be a connected, simply connected closed manifold, with n 2: 5. Then if w2(M) =1= 0, M admits a metric of positive scalar curvature. If w2(M) = 0, so that M admits a spin structure, then M admits a metric of positive scalar curvature if and only if a(M) = in KOn(pt).

°

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Dimension 2 is of course also fully understood, and we would understand dimension 3 if the Thurston Geometrization Conjecture is true. (Of course, the Poincare Conjecture alone would settle the simply connected case.) But because of Counterexample 1.13, the situation has to be more complicated in dimension 4. The best we might hope for would be: CONJECTURE 1.19. Let M4 be a closed simply connected 4-manifold. Then M admits a metric of positive scalar curvature unless either M is spin with A(M) =1= 0 or bt(M) 2:: 1 and some Seiberg-Witten invariant of M is non-zero. At the moment, we have no methods at all for attacking Conjecture 1.19. The most mysterious case of all may be the one where bt(M) = 1 and b2(M) is large, in which case the Seiberg-Witten invariants are not independent of the choice of metric. In dimension 4, one possibility is to simplify the problem by allowing connected sums with 8 2 x 8 2 . (By a famous argument of Wall [71], this is known to make simply connected surgery theory work in the smooth category, whereas without stabilization, smooth surgery theory fails miserably [36].) Then one obtains a rather simple result. THEOREM 1.20. Let M n be a connected and simply connected smooth 4-manifold. Then M#k(8 2 x 8 2 ) admits a metric of positive scalar curvature for some k if and only if either w2(M) =1= 0, or else w2(M) = 0 and A(M) = O. PROOF. If M is spin and A(M) =1= 0, then these conditions are preserved under taking connected sums with 8 2 x 8 2 . Hence, by Lichnerowicz's Theorem (Theorem 1.1), M#k(8 2 x 8 2 ) does not admit a metric of positive scalar curvature, for any value of k. If M is spin with A(M) = 0, then the signature of M vanishes (since in dimension 4, the signature and the Agenus are proportional to one another), and by Wall [71], M#k(8 2 x 8 2 ) is diffeomorphic to a connected sum of copies of 8 2 x 8 2 , for sufficiently large k, and thus for such k, M #k(8 2 X 8 2 ) admits a metric of positive scalar curvature by the Surgery Theorem, Theorem 1.15. Finally, if M is non-spin, then again by Wall [71], M#k(8 2 x 8 2 ) is diffeomorphic to a connected sum of copies of C]p>2 and C]P>2 once k is sufficiently large, and so once again, M#k(8 2 x 8 2 ) admits a metric of positive scalar curvature by the Surgery Theorem. 0

In dimensions 2::5, various attempts have been made to extrapolate from Theorem 1.18 and Theorem 1.2 to a reasonable guess about necessary and sufficient conditions for positive scalar curvature. The best known conjectures are the following:

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CONJECTURE 1.21 ("Gromov-Lawson Conjecture"). Let 1\IIn be a closed, connected n-manifold with n 2: 5~nd with fundamental group 11" and classifying map u: M -+ B11". If w2(M) f:. 0, then M admits a metric of positive scalar curvature. If W2 (M) = 0 (so we can choose a spin structure on M), then M admits a metric of positive scalar curvature if and only if O!B7r(M) = 0 in KOn(B11"). CONJECTURE 1.22 ("Gromov-Lawson-Rosenberg Conjecture"). Let M n be a closed, connected n-manifold with n 2: 5 and with fundamental group 11" and classifying map u: M -+ B11". If w2(M) f:. 0, then M admits a metric of positive scalar curvature. If w2(M) = 0 (so we can choose a spin structure on M), then M admits a metric of positive scalar curvature if and only if A 0 O!B7r(M) = in KOn(CR(11")).

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CONJECTURE 1.23 ("Stable Gromov-Lawson-Rosenberg Conjecture"). Let J8 be a simply connected spin 8-manifold with A(J) = 1. (This implies that O!( J) is a "geometric representative" for Bott periodicity. For example, we can take J to be a "Joyce manifold" with exceptional holonomy Spin(7).) Say that a closed n-manifold M n stably admits a metric of positive scalar curvature if M x J x ... x J admits a metric of positive scalar curvature for a sufficiently large number of J-factors. Then if M is oriented with w2(M) f:. 0, M always stably admits a metric of positive scalar curvature, and if M is spin, then M stably admits a metric of positive scalar curvature if and only if A 0 O!B7r(M) = 0 in KO n(CR(11")). A few words about the history are in order here. Conjecture 1.21 was hinted at in [24], but in the same paper, it was observed that the conjecture cannot always be right in the non-spin case, because of the minimal hypersurface method. (For example, apply Theorem 1.9 to the case X = Tn, n :s; 8. One sees that Hn(X, /l)+ = 0, since otherwise, Theorem 1.9 shows Hn-l (X, /l)+ f:. O. Iterating the construction, one eventually comes down to the case n :s; 2, where we know this is false. It follows that Conjecture 1.21 fails for (ClP2 x S2)#T6 or for ClP4 #T8, since these are oriented manifolds with non-spin universal cover mapping to non-trivial homology classes in T 6 , resp., T 8 .) Conjecture 1.22 was proposed in [55], but only when the fundamental group 11" is finite. Counterexample 1.14 shows that it fails in dimension 4, but the conjecture was only intended to apply in dimensions 5 and up. It was shown in [60] that the conjecture fails when 11" is a product of a free abelian group and a finite group, because one can use Theorem 1.9 to reduce to a low-dimensional case. A more subtle counterexample was found in [13]; here it is shown that there is a torsion-free group 11" for which the assembly map A is injective, but yet one can construct a manifold M n with fundamental group 11" for which O!B7r(M) = 0 in KO n(B11") and yet M does not admit a metric of positive scalar curvature. What goes wrong is related to the fact

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that the periodization map per: ko*(Brr) --t KO*(Brr) has a big kernel. On the other hand, if rr is such that A and per are injective for rr, which is the case for a large number of "nice" groups, then Conjectures 1.21 and 1.22 both hold for rr in the spin case, by a combination of Theorem 1.2 and Theorem 1.16. Conjecture 1.23 was proposed in [57], where it was observed that the case w2(M) t- 0 is trivial since J is oriented bordant to a non-spin manifold of positive scalar curvature. In this same paper, Conjecture 1.23 was proved (or at least the proof was sketched) when M is spin and rr is finite, and also when rr is torsion-free and the assembly map A is injective. A much more general result was sketched in [68] and [69, §3]. Namely, if the Baum-Connes Conjecture holds for rr, or even if the Baum-Connes assembly map KO:(Err) --t KO*(Ci(rr)) is injective, then Conjecture 1.23 holds for rr. (Here Err is the universal proper rr-space; it coincides with Err, and KO:(Err) coincides with KO*(Brr), provided that rr is torsion-free.) The current status of the positive scalar curvature problem is thus very complicated. There are good reasons to believe that the Stable Conjecture, Conjecture 1.23, holds in general, but this still begs the question of what is true unstably. For finite groups, there is a bit more one can say. First of all, it was shown in [4] that Conjecture 1.22 does hold for finite groups rr with periodic cohomology. (These are exactly the finite groups whose Sylow subgroups are all either cyclic or generalized quaternion.) By a combination of results of [30] and [8]' Conjecture 1.22 in the case of non-spin universal cover also holds for elementary abelian 2-groups. Furthermore, by a combination of the results of [7] and [8], Conjecture 1.22 also holds (in both the spin and non-spin cases) for elementary abelian p-groups with p odd, once n (the dimension of the manifold) exceeds the rank of rr, or more generally, provided one reformulates the conjecture just for "atoral" classes. But even in the elementary abelian case, there is one tricky case nobody has been able to handle. Namely, suppose p is an odd prime and one looks at the homology class in Hn ((B7l./p)n) represented by Tn in the obvious way (the map on classifying spaces induced by "reduction mod p" 7l. n --t (7l./p)n). Is this homology class represented by an oriented (or spin) manifold with positive scalar curvature? As for general finite groups, nothing we know excludes the possibility that Conjecture 1.22 holds for all finite rr, but in the other hand there is no obvious reason why a conjecture that fails for infinite groups should hold for finite ones. Another "stable" conjecture which has the advantage over Conjecture 1.23 of having a simpler statement is the following: CONJECTURE 1.24 ("SI-Stability"). Let M n be a closed, connected n-manifold. Then M admits a metric of positive scalar curvature if and only if M x SI does.

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REMARK 1.25. One direction of this is trivial; certainly if M has a metric of positive scalar curvature, then the obvious product metric on M x 81 has the same property. The converse would follow from most "reasonable" criteria (with good functoriality in the fundamental group) proposed for positive scalar curvature, such as Conjecture 1.21 or Conjecture 1.22 (in dimensions 2:5). This conjecture is also compatible with Theorem 1.9. Unfortunately, Conjecture 1.24 fails in dimension 4. To see this, choose any smooth complex hypersurface V of odd degree 2: 5 in CJIll3. Then (since the degree of V is odd) V is a non-spin smooth simply connected 4-manifold with a Kahler structure and with bt > 1, hence by Theorem 1.12, V does not admit a metric of positive scalar curvature. On the other hand, V x 81 is a closed oriented 5-manifold with fundamental group 7r = Z, representing in H 5(BZ, Z) = H5(81, Z) = 0, so V X 81 has a metric of positive scalar curvature by Theorem 1.16.

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The author does not know of any counterexamples to Conjecture 1.24 with n 2: 5. Of course the big problem with this conjecture is that it doesn't settle the positive scalar curvature problem for any manifold; it simply states the equivalence of the problem on one manifold with the problem on another. 2. The moduli space problem: what does the space of positive scalar curvature metrics look like? In this section, we consider the following problem. If M n is a closed manifold which admits at least one Riemannian metric of positive scalar curvature, what is the topology of the space 91+(M) of all such metrics on M? In particular, is this space connected? In general, the answer to this problem is not known, but a methodology exists for approaching it, and there are lots of partial results. The one case that is totally understood is the one where n = 2. By Gauss-Bonnet, the only closed 2-manifolds admitting metrics of positive scalar curvature are 8 2 and RJIll 2. And we have: THEOREM 2.1 ([58, Theorem 3.4]). The spaces 91+(8 2) and 91+(RJIll2) are contractible. To deal with higher-dimensional manifolds, we first need some definitions. DEFINITION 2.2. Suppose M is a closed manifold, and let 91+(M) denote the space of all Riemannian metrics of positive scalar curvature on M, with the Coo topology. We assume this space is non-empty. Then two metrics 90 and 91 in 91+ (M) are called concordant if there is a smooth metric 9 of positive scalar curvature on M x [0, a], for some a > 0, which restricts in a neighborhood of to the product metric 90 x 0', and in a neighborhood of a to the product metric 91 x 0', where 0' is the standard metric on R

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(corresponding to dt 2 ). The metrics go and gl are called isotopic if they lie in the same connected component (or path component, it doesn't matter) of !Jt+(M). It is easy to show that isotopic metrics of positive scalar curvature are concordant. (The original argument is in [23, Lemma 3]; see also [58, Proposition 3.3].) The converse is not at all obvious and is now known to be false (see Theorem 2.10 below), since a metric of positive scalar curvature on M x [0, a] may not necessarily be a product metric, and there is no obvious way to "straighten it." Thus 7ro(!Jt+(M)), the set of path components of !Jt+(1\,f), surjects onto the set 7ro(!Jt+(M)) of concordance classes, and if the latter has more than one element (respectively, is infinite), then so is the set of path components of !Jt+ (M). The major result in high dimensions (specialized to the case of closed manifolds, as there is also a version for manifolds with boundary) is:

THEOREM 2.3 (Stolz [68, Theorem 3.9], [58]). Let Afn be a connected closed spin n-manifold with fundamental group 7r admitting a metric of positive scalar curvature, and suppose n 2': 5. Then there is a group Rn+1(7r) acting simply transitively on 7ro(!Jt+ (M)). (Thus in some sense the latter only depends on 7r and on n.) Furthermore, there is an "index homomorphism" (): R n+l(7r) -t KOn +1(Ci(7r)). Suppose furthermore that N n +1 is a spin manifold with boundary aN = M. Then a given metric of positive scalar curvature g on M extends to a metric of positive scalar curvature on N which is a product metric in a collar neighborhood of the boundary if and only if an obstruction defined by (N, g) vanishes in Rn+ 1 (7rl (N)). CONJECTURE 2.4 (Stolz [58]). The index map (): Rn+1 (7r) -t K On+1 (Ci (7r)) is "stably" an isomorphism. The notion of stability here is similar to that in Conjecture 1.23; we replace Rn+1(7r) by ~Rn+1+8j(7r), where the maps in the inductive limit come from products with the manifold J8. There is also a version of Theorem 2.3 and of Conjecture 2.4 for non-spin manifolds. In this case, Rn+l(7r) should be replaced by Rn+l(-Y) with "f as in Definition 1.4, and the real group C*-algebra should be replaced by C*(-y). The index map () of Theorem 2.3 can sometimes be used to distinguish different connected components in !Jt+(1\,f). This use of the index map is quite similar to, and presumably generalizes2 , earlier uses of the "relative index" or the relative 'T]-invariant to distinguish different connected components in !Jt+ (M). For example, Hitchin [29, Theorem 4.7] proved that if M n is a closed spin manifold admitting a metric of positive scalar curvature, then7ro(!Jt+(M)) i2The author is not sure if all the details of proving that the two constructions coincide have been verified, but there is reason to believe this shouldn't be so hard.

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oprovided that n == 0 or 1 mod 8, and 71"1 (!)t+ (M)) t= 0 provided that n == -1 or 0 mod 8. Examination of his proof shows that in terms of the language of Theorem 2.3, he was really showing that the composite

is surjective when n + 1 == 1 or 2 mod 8, basically because we know there are exotic spheres in dimensions 1 or 2 mod 8 for which the a-invariant is non-zero. Since there are still no high-dimensional manifolds for which the topology of !)t+(M) is fully understood, the rest of this section will consist largely of a catalog of examples. For example, one of the earliest results on the topology of !)t+(M) is the following: THEOREM 2.5 (Gromov-Lawson [24, Theorem 4.47]). The space !)t+(87 ) has infinitely many connected components; in fact, 7ro(!)t+ (8 7 )) is infinite. PROOF FOLLOWING [24]. The idea is to construct "exotic metrics" on 8 7 by using the fact that there are many ways to write 8 7 as the unit sphere bundle of an oriented JR4-bundle E over 8 4. Such bundles are classified by two integer invariants: the first Pontrjagin class PI (always an even number) and the Euler class e. The unit sphere bundle 8(E) = E7 (with respect to some choice of smooth metric on the vector bundle) is (oriented) homotopy equivalent to 8 7 if e = 1, and as shown by Milnor [44], pi == 4 (mod 7) is necessary for E to be diffeomorphic to 8 7 . The value PI = 2 corresponds to the usual presentation of 8 7 as the unit sphere bundle of a quaternionic line bundle over JH[JP>1 ~ 8 4. But in [35], it is shown that the h-cobordism classes of smooth homotopy 7-spheres constitute a cyclic group 87 of order 28, and thus there are other values of PI (such as PI = 2 + 28 = 30) for which 8(E) is diffeomorphic to 8 7 . Now it is easy to construct a metric of positive scalar curvature on the unit disk bundle D(E) of E which is a product metric in a collar neighborhood of the boundary 8(E) ~ 8 7 . Suppose the metric obtained this way on 8 7 were "standard," i.e., concordant to the standard spherical metric 90. Then we could take a metric on D8 of positive scalar curvature which is a product metric 90 x a (see Definition 2.2 for the notation) in a collar neighborhood of the boundary 8 7 , and glue the two metrics together (after first inserting a "fitting," a copy of 8 7 x I with a concordance metric) to get a metric of positive scalar curvature on M8 = D(E) US(E) (8 7 x I) US7 D8. This is a contradiction, since M is a spin manifold with A(.iVf) t= o. In fact, a slight variant of this calculation shows that if one takes two different values for PI (E) (but for both of which we have 8(E) ~ 8 7 ), then the metrics obtained on 8 7 cannot be concordant. Thus since there are infinitely many values of PI for which 8(E) ~ 8 7 , there are infinitely many concordance classes of positive scalar curvature metrics on 8 7 . 0

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Now let's give a different construction for exotic positive scalar curvature metrics on homotopy 7-spheres, which works in any dimension 4k -1, k ~ 2. Afterwards, we will say a bit about the special case k = 1. We actually do not know if 9{+(S3) has infinitely many connected components or not. THEOREM 2.6. The space 9{+(S4k-l) has infinitely many connected components for k ~ 2; in fact, the index invariant R4k -+ K0 4k ~ Z is non-trivial. PROOF. Let M4k be the parallelizable manifold with boundary obtained from the Es plumbing as in [45]. In more specific terms, M is obtained by starting with the 4k-disk D4k and adding on 8 2k-handles D2k x D 2k , plumbed together to intersect according to the Cartan matrix of Es. Note that M may be viewed as the result of doing surgery on 8 copies of S2k-l embedded into the boundary S4k-l of D4k. Since we are assuming that k ~ 2, the co dimension condition of the Surgery Theorem is satisfied, and so 1M admits a metric of positive scalar curvature extending a standard "round" metric on the original 4k-disk D 4k , and a product metric on the boundary ~4k-l = 8M. (Strictly speaking, we need the version of the Surgery Theorem that applies to manifolds with boundary. This is a variant on Theorem 1.15 due to Gajer. See [17] and [68, Theorem 3.3] for details.) As argued by Kervaire and Milnor, M is parallelizable and ~ is a nonstandard homotopy sphere. To quickly sum up the argument, the main points are these: (1) M and ~ are simply connected, since they are the result of highly connected surgeries. (N.B.: The assumption k ~ 2 is used here; if k = 1, ~ turns out to be the Poincare homology sphere, the quotient of S3 by the binary icosahedral group of order 120, and so is not simply connected. We will come back to this point later.) (2) A1 is parallelizable, since it is built by framed surgery from the tangent bundles of spheres. (3) ~ is a homology sphere, because of the fact that the Cartan matrix of Es is unimodular. (This part is still valid even when k = 1.) (4) Consider N4k = M4kUI:cone(~). This is a topologicaI4k-manifold. However, it cannot be smooth, and so ~ is not diffeomorphic to S4k-l. The reason is the following. Suppose we had ~ ~ S4k-l, cone(~) ~ D4k. Then N would be smooth and almost parallelizable (parallelizable off a disk). Thus all its Pontrjagin classes would vanish except for Pk in degree 4k. (Any lower Pontrjagin class would be detectable by its restriction to a proper skeleton, and thus by its restriction to M4k. But flIf4k is parallelizable.) However, the signature of N is 8, since by construction, its intersection form on middle homology is given by Es, which is unimodular of rank 8. Thus, by the Hirzebruch signature formula, the term in the L-class of !vI involving Pk must evaluate to 8. This is a contradiction, since (Pk, [M]) is integral and we know the coefficient 11k of Pk in L; it's

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a complicated rational number related to the Bernoulli numbers:

J-Lk

=

22k(22k-1 - 1) (2k)! Bk

[28, §1.5]. In particular, its numerator is such that (J-LkPk, [Ml) can't be 8. Now let's go back to the issue of positive scalar curvature metrics. Recall we've constructed using surgery a metric of positive scalar curvature on M4k which restricts to a product metric in a collar neighborhood of the boundary homotopy sphere ~4k-1. By [35], there is a finite number m such m

that the m-fold connected sum 'L,#~# ... #i is diffeomorphic to a standard sphere S4k-1. Thus there is a (spin) cobordism p4k, the trace of a surgery ~4k-1 to #f!! ~4k-1 ~ S4k-1 . Again by on a union on So's , from 11~ 3=1 3=1 the Surgery Theorem, there is a metric of positive scalar curvature on p4k which is a product metric on a neighborhood of each boundary component: the metric constructed above (coming from M4k) on each copy of ~4k-l, and some metric of positive scalar curvature 9 on S4k-l. We claim that 9 is not in the same concordance class as the standard round metric 90 on S4k-1, and in fact that the index obstruction to extending 9 to a metric on D 4k , restricting to a product metric near the boundary, is non-zero. This will prove the theorem. Indeed, if our claim is false, there is a metric of positive scalar curvature on m

Q4k =

(II M4k)

UUj=l E4k-l

P

U S 4k-l

D4k

j=l

extending the metrics we've constructed on each copy of M and on P. (See Figure 1.) Now Q4k is a closed spin manifold, and by an argument similar to the one used above with N 4k, it is almost parallelizable. So all its Pontrjagin classes vanish except for Pk in top degree. Furthermore, the construction shows that the intersection form of Q on middle homology is a direct sum of m copies of E8, so the signature of Q4k is 8m. That tells us, as above, that J-Lk(Pk, [Nfl) = 8m. But the coefficient of Pk in the ..4 polynomial, -(Bn/2(2n)!), is also nonzero. SO ..4(Q) =1= 0, contradicting Theorem 1.1. This completes the proof, since we've shown that the index obstruction to extending 9 over a disk D4k is non-zero. 0 In [24] and [58], the question of whether 9\+(S3) is disconnected was left open. On the one hand, might expect 7I"0(9\+(S3)) to be infinite via a certain index calculation, but on the other hand, Hitchin showed in [29, §3] that the space of left invariant positive scalar curvature metrics on SU(2) ~ S3 is contractible. He also computed the 1]-invariants for these metrics, and showed that it varies continuously. The method of proof used above seems

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FIGURE

279

1. The construction of the manifold Q4k.

at first to be promising, in terms of showing that the Es manifold M4 has a metric of positive scalar curvature restricting to a product metric on a neighborhood of the boundary, the Poincare homology sphere ~3. If this were the case, there would be some hope of lifting the metric on ~3 to the covering space S3, and showing that the metric constructed this way on S3 is "exotic," say by means of an 7J-invariant calculation. But unfortunately, the argument breaks down right at the first step, because the codimension condition in the Surgery Theorem 1.15 isn't satisfied. Next, we discuss various methods for detecting non-triviality of the topology of ryt+(M), M odd-dimensional, via eta invariants. Unfortunately, the ordinary (untwisted) eta invariant of the Dirac operator doesn't help much, since it vanishes identically for metrics of positive scalar curvature on spin manifolds in dimension == 1 mod 4 (see for example [20, Lemma 1.7.10]), while in dimension == 3 mod 4, it only helps if one can control the .A term in the Atiyah-Patodi-Singer Theorem [IJ. However, there is also a reasonably large literature using the twisted eta invariant (for the Dirac operator twisted by a flat bundle, and especially for the formal difference of two such twists of the same dimension) to prove facts about ryt+(M) when M is not simply connected. We will just give a few representative examples, and leave it to the reader to consult [5J, [39], [19J, [18], [29J, and [6J for more results and details. THEOREM 2.7. Let M be a closed connected spin manifold of odd dimension, and let 90 and 91 be metrics of positive scalar curvature on M. Let p be

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a virtual unitary representation of 7r = 7r1 (M) of virtual dimension 0, i.e., a formal difference of two finite dimensional unitary representations p+ and p- of7r with dimp+ = dimp-. Let TJo{p) = TJU/Jo,p,a), TJ1{p) = TJ{l/h,p,a), where f/Jo and f/J1 are the Dirac operators for the metrics go and gl, and TJ{f/Jj, p) is defined by

TJ{f/Jj, p)

=

TJ{f/Jj ® Ivp +) - TJ{f/Jj ® Ivp +)'

V p± denoting the fiat vector bundle defined by p±. Then if TJo (p) =1= TJ1 (p), go and gl and not concordant, and in particular, do not lie in the same connected component of ry:{+ (M). PROOF. Suppose go and gl are concordant. Then there is a metric g of positive scalar curvature on M x [0, 1], restricting to a product of go with a metric on the line in a collar neighborhood of M x {a} (which we may identify with M) and to a product of gl with a metric on the line in a collar neighborhood of M x {I} (which we may identify with -M, M with the orientation reversed). The representations p± of 7r give vector bundles Vp± on M x [0,1]' which we equip with fiat connections. Now consider the index problem for f/J Mx [0,1] ® 1vp ± with Atiyah-Patodi-Singer boundary conditions on M x {a} and M x {I}. Since M x [0,1] has positive scalar curvature and the vector bundle is fiat, the Lichnerowicz identity (1.1) shows the kernel of f/JMx[O,l] vanishes, and thus the index is a. Furthermore, since Vp ± is fiat, its Chern character reduces by Chern-Weil theory simply to dim p±. So by the Atiyah-Patodi-Singer Theorem [1], we have (2.1)

a=~(TJ{f/J1'P±)-TJ{f/Jo,p±))+

(

A·dimp±.

JMX[O,l]

Subtract equation (2.1) for p- from equation (2.1) for p+, and the A terms cancel. So if TJo (p) =1= TJ1 (p), we get a contradiction, and the result follows. 0 COROLLARY 2.8 (Botvinnik and Gilkey [5, Theorem 0.2]). Let M be a closed connected spin (4k + 1) -manifold with finite fundamental group 7r, admitting a metric of positive scalar curvature, and assume that 7r has a non-zero virtual unitary representation p of virtual dimension 0, satisfying the parity condition Tr p{ h) = - Tr p{ h -1) for all h E 7r. Then 7ro (ry:{+ (M)) is infinite. SKETCH OF PROOF. An induction argument reduces everything to the case where 7r ~ Zip is cyclic. We will give the proof in the case p is odd, which results in a slight simplification since, in this case, any non-trivial irreducible representation (J': 7r ~ U (I) gives a non-zero p of virtual dimension a satisfying the parity condition (namely p = (J' - 0'), and any lens space with fundamental group 7r is automatically spin. Then because of Theorem 2.3, it suffices to prove the result when M itself is a 5-dimensional

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lens space L5. (To pass to the case k > 1, simply take a product with copies of a Kummer surface and/or a Joyce manifold J8, using the fact that ",(M x N, p) = ",(M, p)A(N) when N has dimension divisible by 4 [20, Lemma 1.7.18].) By the Atiyah-Hirzebruch spectral sequence for bordism, In~pin(Z/p)1 = p2 when p is odd. But the eta invariants of lens spaces for the standard metric of constant curvature, ",(L'n)(p), are computed in [12] (for the signature operator, but the same method also works for the Dirac operator) and in [20, Theorem 1.8.5]. To fix notation, let L5(7) = 8 5/7 be a lens space of dimension n with fundamental group 1[" = Zip, associated to the representation 7 = (A, Aal , Aa2 ) of 1[", where A sends the generator of 1[" to e27ri / p and 1 ~ aI, a2 ~ P - 1.) Note that 7 acts freely on ([:3 " {a}, and thus freely on the unit sphere 8 5 . The formula in [20, Theorem 1.8.5] gives

(2.2) Because of the parity condition on p, the quantity inside the summation sign is invariant under replacing h by h- 1 . (In fact that's why we need the parity condition, for if Tr(p( h)) = Tr(p( h -1)) for all h, then the quantity being summed is odd and the ",-invariant vanishes.) We might as well take p = A1/ 2 - A-1/2. Then if 7 = (A, A, A), (2.2) becomes ",

(L 5 (A A A))(A 1/ 2 _ A- 1/ 2) _ ~ " - p

""

~

hE7r, h#l

1 p-1

=

while if 7

t;

P

1

(A(h)1/2 _ A(h)-1/2)2 1

4sin2 (21["j/p)

= J.L > a

= (A, A, A-I), (2.2) becomes ",(L 5 (A,A,A- 1))(A 1/ 2 _ A- 1/ 2) 1

1

L

= phE7r, h#l (A(h)1/2 - A(h)-1/2) (A(h)-1/2 - A(h)1/2) 1 p-1 =

t;

P

1

-4sin2 (21["j/p)

= -J.L < a.

Let M 5 = L 5 (A, A, A) and call go the standard metric of positive scalar curvature on M. Then ",(go)(p) = J.L > a. But since In~pin(Z/p)1 < 00, there is a spin bordism (over BZ/p) from a disjoint union of finitely many copies of L 5 (A, A, A-I), say r copies, to M. We can use this (as in the proof of Theorem 2.6) to push the standard metric on this disjoint union over to a metric gl on M, which by an argument similar to that in the proof of Theorem 2.7 must satisfy ",(gI)(p) = -rJ.L < a. Since we just showed the

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standard metric go on M satisfies 7](gt}(p) > 0, Theorem 2.7 shows go and g1 are not concordant. Furthermore, we can construct metrics on M with infinitely many values of the 7]-invariant, since we are free to replace r by r + p2j for any j. (Recall In~pin(lE./p)1 = p2.) This completes the proof for this case. The other cases are similar. 0 REMARK 2.9. In the language of Theorem 2.3, Corollary 2.8 says that under these hypotheses, R4k+2(-7r) is infinite. Another way to prove this would be to show directly that the index invariant

has infinite image. The representation-theoretic hypothesis guarantees that 7r has at least one irreducible representation of complex type, i.e., that IR7r has at least one summand of the form Mm(C). This summand contributes a IE. to K04k+2(IR7r), and Corollary 2.8 says () hits this IE. non-trivially. This lends a bit of support to the surjectivity part of Conjecture 2.4. All of the results so far, showing that the topology of 9\+(M) is nontrivial, have been based on index theory or the eta-invariant. In dimension 4, Seiberg-Witten theory can also be used, leading to a remarkable result: THEOREM 2.10 (Ruberman [59]). There is a simply connected 4-manifold with infinitely many concordant but nonisotopic metrics of positive scalar curvature. I am not aware of any results like this in higher dimensions. Finally, there are some other results that imply something interesting about the space 9\+(M). For example, in [10]' it is shown that if 9 is a metric on M and (M,g) admits a spin cover with nonzero parallel spinors, then 9 cannot be deformed to a metric of positive scalar curvature. If M is spin and simply connected, and has nonzero parallel spinors for the metric g, then there can be no metrics of positive scalar curvature in a neighborhood of g.

3. Complete metrics of positive scalar curvature on noncom pact manifolds The study of complete metrics of positive scalar curvature on noncompact manifolds is noticeably more complicated than for closed manifolds, and in this section we will just touch on a few of the issues involved. We have divided the discussion into two subsections: one on global results and one on metrics within a fixed quasi-isometry class. 3.1. Global results. Some noncompact manifolds do not admit any complete metrics of positive scalar curvature at all; others admit such metrics, but not if the scalar curvature is bounded below by a positive constant.

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While we still don't know what manifolds belong in these classes, we do have the following conjecture. CONJECTURE 3.1 ([56, Conjecture 7.1]). Let xn be a closed manifold which does not admit a metric of positive scalar curvature. Then (1) xn x ]R does not admit a complete metric of positive scalar curvature; (2) xn x ]R2 admits no complete metrics of uniformly positive scalar curvature. This conjecture is known to be true [24, §6 and §7] if n :s: 2, i.e., if X is a point, Sl, T2, a Klein bottle, or a compact surface without boundary with a hyperbolic metric. The conjecture cannot be strengthened any further because of: PROPOSITION 3.2 ([56, Proposition 7.2]). Let xn be any closed manifold. Then xn x ]R2 admits complete metrics of (non-uniformly) positive scalar curvature, and xn x IRk admits complete metrics of un'ijormly positive scalar curvature when k 2: 3. Additional positive evidence for something like Conjecture 3.1 comes from noncompact index theory, at least in the case of spin manifolds. For example, one has: THEOREM 3.3 ([24, Corollary B2]-see also [50]). Let xn be a closed spin manifold with A(X) i= O. Then xn x IR does not admit a complete metric of positive scalar curvatuTe. The following is based on some of the same ideas, but carried out in the context of more sophisticated Kasparov theory. THEOREM 3.4. Let xn be a closed spin manifold, and assume that i= 0 in KOn(C~(7l')). Then xn x IR does not admit a complete metric of uniformly positive scalar- curvatur-e.

A(nBrr(X))

PROOF. As some of the details are a bit complicated, we prefer to begin by explaining first how the proof works when 71' is trivial, i.e., when we replace A(X) in Theorem 3.3 by n(X) E KO n (which can be non-zero also in dimensions 1 and 2 mod 8). Then we will explain how to modify the proof to cover the general case. Fix a Riemannian metric on X, and let f/J x be the CRn-linear Dirac operator on X, as used in the proof of Theorem 1.1. This operator defines a class [f/Jx] in the Kasparov group KKO(C'f?(X), CRn), and a(X) is the image of this class in KKO(IR,CR n ) = KKO(C'f?(pt),CRn ) under the map of Kasparov groups induced by the inclusion IR y C'f?(X), or dually, the "collapse map" c: X -+f pt.

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Next, observe that we have a similar class [QJxxlR], defined by the Dirac operator on X x lR for the product metric on this manifold, in the Kasparov group KKO(C~(X x lR),Cfn+I). This class is just the external Kasparov product of [QJx] with the Dirac operator class on the line. We have a commutative diagram of groups:

KKO(CIR(X), Cfn)

C.

:>

KKO(CIR(pt), Cfn)

®y!~ KKO(C~(X x lR), Cfn+!)

= KOn ,

®y!~ (cXl)\

KKO(C~(lR),Cfn+l)

where y E KKO(C~(lR), Cf l ) is QJIR. or the Bott periodicity operator (see [2, §19.2]), and ®y denotes the (external) Kasparov product. This gives rise to the commutative diagram of Kasparov elements: (3.1)

Now suppose that X x lR admits a complete metric of uniformly positive scalar curvature, say g. Then the Dirac operator QJg for this metric g, which is essentially self-adjoint since 9 is complete, and the Dirac operator for the product metric on X x lR, define the same Kasparov class [QJg] = [QJxxlR]. (This point is made in [27]; the essential fact is that we are dealing with a Kasparov class for the algebra of continuous functions which vanish at infinity, in which continuous functions of compact support are dense, and any two complete metrics on a noncom pact manifold, when restricted to a fixed compact set, are homotopic through homotopies of complete metrics supported on a slightly larger compact set.) Next we note that Bott periodicity implies that the Kasparov class y, which lies in KKO(C~(lR),Cfl)' has an inverse x E KKO(Cfl,C~(lR)) ~ KO-1(lR) (see [2, §19.2]). Thus, putting this and the insensitivity of the Dirac class on X x lR to the choice of complete metric together with (3.1), we obtain the equality

The rest of the proof consists of showing that this Kasparov product vanishes, using the fact that 9 has uniformly positive scalar curvature. For this, we need explicit realizations for the classes x and (c x l)*([QJg]), as well as the Kasparov calculus for computing the product. Note incidentally that Cf l = lR + lRi, where i 2 = -1, so we can identify Cf l with C, the grading given by complex conjugation. The class x is represented by the Cfl-C~(lR) bimodule Cf l ® C~(lR) = C~(lR) + iC~(lR), together with the operator F

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given by multiplication by

0 F = ( if (x)

-if(X)) 0

'

where f is a continuous function on the line that tends to 1 at +00 and to -1 at -00. It will be convenient to assume that -1 ~ f ~ 1, that f is smooth, and that f == -Ion (-oo,-a], f == Ion [a,oo), for some a> O. Thus F2 == 1 except on [-a, a]. The class (c x 1)* ([Q)g]) is represented by the graded real Hilbert space 'Ii of L2 sections of the Gfn+l-linear spinor bundle on (X x R, g), the operator D = Q)g(Q);)-l, and the obvious action of G~(R) by multiplication operators. (The fact that 9 has uniformly positive scalar curvature implies that the differential operator Q); is bounded away from 0, hence invertible, so we can use the above formula for D instead of the more usual Q)9 (1 + Q);) -1 . In particular, our choice of D satisfies D2 = 1 precisely, not just "approximately.") We proceed to the compute the Kasparov product in (3.2). It acts on the graded Hilbert space G~(R)®C~(IR)'Ii = Gfl®'Ii = 'lie, with the obvious action of Gf l on the left, and the issue is to compute the relevant Fredholm operator G = F#D. From the recipe for the Kasparov product (see [2, §18.4]), the operator G should be chosen to be of the form

where 0 ~ M, N ~ 1, M + N = 1, so that G2 - 1 is compact and the anticommutator {G, F} is positive modulo compacts. (We are letting Fact on function spaces on xn x R in the obvious way, through strictly speaking we should write F®l, etc.) We have G2

_

1 = Ml/4 F M l / 2 F M l / 4 + N l / 4DN I / 2DN I / 4 - 1

+ {Ml/4FMl/4, Nl/4DNl/4}. In our situation, D2 = 1, and {F, D} is basically [D, if], which is the commutator of two pseudo differential operators of order 0, hence is pseudodifferential of negative order, but may not be compact since we are on a noncompact manifold. This suggests taking N to be a multiplication operator given by a nonnegative function on R of compact support, with F2 = 1 on the support of M, in which case M will commute with F and

+ N l / 4DN I / 2DN I / 4 - 1 = fl,f F2 + N l / 2D2 N l / 2 _ 1 + N l / 4(DN I / 2D _ == MF2 + Nl/2D2Nl/2 -1 = M +N -1 = 0

Ml/4 F M l / 2 F Ml/4

N l / 4D2 N l / 4)N l / 4

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modulo compacts. On the other hand,

{Ml/4 FMl/4, Nl/4DNI/4} =

N l / 4{Ml/2F,D}N l / 4

is compact since {Ml/2 F, D} is pseudodifferential of negative order and N has compact support. So G2 - 1 is compact, and by a similar calculation, {N l / 4DN l / 4,F} is compact, hence {G,F} is positive modulo compacts. Thus G as we've written it down is a representative for the "sharp product" F#D, and so by (3.2), a(X) can be computed from the finite-dimensional kernel of G, which is a graded Gfl-Gfn +! bimodule. Next, observe that we obtain a homotopy of Kasparov Gfl-Gfn +! bimodules by letting the support of N grow and letting the support of M shrink, so that M tends strongly to 1 and in the limit, the operator G becomes simply D. Since D2 = 1, ker D = O. This means the Kasparov module is trivial, i.e., a(X) = O. Now we indicate how to extend the proof to the case of an arbitrary group 7r. As in the proof of Theorem 1.2, let VB7l" be the "universal flat bundle" over B7r with fibers that are rank-one free (right) modules over GR(7r). Pull this bundle back to a bundle Vx over X via f: X --+ B7r, and extend the bundle in the obvious way to a bundle V over X x JR. We now repeat the whole argument, replacing f/Jg by f/Jg ® 1 acting on the Gfn+!-module spinor bundle with coefficients in V. We construct the operator D as before, this time obtaining a class [DxxlR,v] E KKO(G~(X x JR),GR(7r)®Gf n +!) mapping to A(aB7l" (X)) ®y in KKO(G~(JR), GR(7r)®Gf n +!) under (c® 1)*. As before we take the Kasparov product with the class x and obtain the desired conclusion. 0 COROLLARY 3.5. Conjecture 1.22 implies part (1) of Conjecture 3.1, if we weaken positive scalar curvature to uniformly positive scalar curvature, at least in the spin case with n ~ 5. PROOF. Conjecture 1.22 is simply the statement that the hypothesis of Theorem 3.4 is equivalent to xn not having a metric of positive scalar curvature. 0 REMARK 3.6. We should mention that any counterexample to Conjecture 1.24 is also a counterexample to Conjecture 3.1. Indeed, suppose xn is a closed manifold that does not admit a metric of positive scalar curvature, but such that xn X 8 1 does admit such a metric. Then the lift of this metric to the covering space xn x JR has uniformly positive scalar curvature, contradicting both parts of Conjecture 3.1. Thus the example mentioned in Remark 1.25 above (which incidentally was simply connected but not spin) shows that Conjecture 3.1 fails if n = 4, even if X is simply connected. One can construct a similar example (again with n = 4) with X spin by

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using part (1) of Counterexample 1.13. However, we know of no counterexamples to Conjecture 3.1 with n > 4, and Corollary 3.5 suggests that such counterexamples will be difficult to find. In dimension 3, somewhat more is known; for example we have: THEOREM 3.7 (Schoen and Yau [63, Theorem 4]). Let M be a 3-dimensional connected manifold admitting a complete metric of positive scalar curvature. Then 'lrl( M) cannot contain a subgroup isomorphic to the fundamental group of a closed Riemann surface of positive genus. Aside from products of compact manifolds with Euclidean spaces, another very interesting general class of noncom pact manifolds comes from locally symmetric spaces. The following was proved by Block and Weinberger: THEOREM 3.8 (Block and Weinberger [3]). Let M = r\ G / K be an irreducible locally symmetric space of noncompact type and finite volume. Then M can be given a complete metric of uniformly positive scalar curvature if and only if r is an arithmetic lattice of Q-rank 2:3. Note incidentally that r is cocompact, i.e., M is compact, if and only if has Q-rank O. This case is included in the theorem, but if M is compact, it cannot have a metric of positive scalar curvature because of Theorems 1.2 and 1.7, or other similar results. Also, we are not assuming a priori that r is arithmetic, though if it is not, Theorem 3.8 says that M never has a complete metric of uniformly positive scalar curvature.

r

3.2. Metrics in a fixed quasi-isometry class. Many of the interesting results on positive scalar curvature for noncompact manifolds involve specifying the quasi-isometry class of the metric, or what is almost the same, specifying the rate of growth or decay of the metric at infinity. One of the most effective tools for producing results of this sort is the coarse index theory of Roe, as outlined in [51], [52], and [53]. The basic construction involves a C*-algebra C*(M) attached to a "coarse space," a metric space in which closed bounded sets are compact. For present purposes we should really work with the real version of the construction and write CR(M), but we will suppress the lR for notational convenience. The algebra C*(M) is the completion of the locally compact, finite propagation operators on M, acting on an auxiliary separable Hilbert space, and when M is a complete Riemannian manifold, it really only depends on the quasiisometry class of the metric. For example, when M is compact, C*(M) is just the algebra K of compact operators (which is Morita equivalent to the scalars). Then we have the following result, generalizing Theorem 1.1 to the noncompact case. THEOREM 3.9 (See [52, Definition 3.7 and Proposition 3.8].). Let (M n , g) be a complete Riemannian spin manifold. Then the Dirac operator for the

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metric 9 and the given spin structure defines a class ind Q> g E K On (C* (M)), and if this class is non-zero, 9 cannot have uniformly positive scalar curvature. In fact, when the index is non-zero, there can be no complete metric of uniformly positive scalar curvature in the same quasi-isometry class.

PROOF. Roe states and proves this in the complex case, so we will just indicate how to obtain the refinement in real K-theory. As usual, we work with the Cin-linear Dirac operator Q>g. By (1.1), the spectrum of this operator is bounded away from 0 if 9 has uniformly positive scalar curvature. As in the proof of Theorem 3.4, choose a continuous real-valued (odd) function f on lR with -1 ~ f ~ 1 and with f(x) ---+ 1 as x ---+ +00, f(x) ---+ -1 as x ---+ -00, and observe that f(Q>g) is Cin-linear, odd (with respect to the grading of the spinor bundle) and bounded. Furthermore, since the hyperbolic equation Ut = iQ>gu has finite propagation speed, f(Q>g) is a multiplier of C*(M), and defines a class indQ>g E KKO(lR,C*(M)®Cin) ~ KOn(C*(M)), and ind Q> g is evidently 0 if Q> g has a bounded inverse, which is the case if g has uniformly positive scalar curvature. Furthermore, this index class ind Q> g is invariant under homotopies of the metric within the same quasi-isometry class (since such homotopies give homotopies of the Kasparov class), so if the index is '" 0, there can be no complete metric of uniformly positive scalar 0 curvature in the same quasi-isometry class. Most of the known results about non-existence of complete metrics of uniformly positive scalar curvature in quasi-isometry classes of noncompact manifolds come from applying various tricks to detect the index class ind Q> g topologically. Of course, since it is the coarse geometry of M, not its usual topology, that is relevant here, "topologically" means "in terms of coarse invariants," such as coarse KO-homology KOX*(M) in the sense of [52]. There is a coarse assembly map KOX*(M) ---+ KO*(C*(M)) defined in [52, Ch. 8]. When M is uniformly contractible, this is simply the map that takes the class [D] of an elliptic operator D to ind(D) as defined above. More generally, this map is defined by taking indices of the images of D on "coarse approximations" to X. CONJECTURE 3.10 (Coarse Baum-Connes Conjecture [52, Conjecture 8.2]). For any proper metric space M of bounded geometry, the coarse assembly map KOX*(M) ---+ KO*(C*(M)) is an isomorphism. Incidentally, a counterexample to the surjectivity part of Conjecture 3.10 is known [26, §6], but we shall only need the injectivity part. Putting together Theorem 3.9 and Conjecture 3.10, we obtain: PROPOSITION 3.11 (Roe). If M is a uniformly contractible complete Riemannian manifold of bounded geometry, and if (the injectivity part of the)

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Conjecture 3.10 holds for M, then there is no complete metric of uniformly positive scalar curvature in the quasi-isometry class of the given metric on M. PROOF. In this case, the coarse assembly map takes [Q>g] E KOn(M), which is a generator in KOn(M) ~ KOn(lR n ) ~ :il, to indQ>g, which by Theorem 3.9 is an obstruction to uniformly positive scalar curvature in the quasi-isometry class of the given metric on M. 0 In some cases, one can prove Conjecture 3.10 and apply this result. For example, we have the following results: THEOREM 3.12 (Yu [74, Corollary 7.3]). Let M be a uniformly contractible complete Riemannian manifold with finite asymptotic dimension. Then M cannot have uniformly positive scalar curvature. THEOREM 3.13 (Yu [75, Corollary 1.3]). Let M be a complete Riemannian manifold with bounded geometry. If M is uniformly contractible and admits a uniform embedding into Hilbert space, then M cannot have uniformly positive scalar curvature. THEOREM 3.14 (Gong and Yu [21, Corollary 4.3]). Let M be a uniformly contractible complete Riemannian manifold with bounded geometry. If M has subexponential volume growth, then M cannot have uniformly positive scalar curvature. Another result related to Theorem 3.8 is the following: THEOREM 3.15 (Chang [9]). Let M = r\G/K be an irreducible locally symmetric space of noncompact type and finite volume, and suppose r is an arithmetic lattice ofQ-rank ~ 3 (so that by Theorem 3.8, M admits a metric of positive scalar curvature). Then no metric of positive scalar curvature on M can be quasi-isometric to the standard locally symmetric metric. Finally, there are results on positive scalar curvature in a quasi-isometry class that involve still other versions of noncompact index theory. A typical example is: THEOREM 3.16 (Whyte [72]). Assume that M n is a complete connected spin manifold with bounded curvature and uniformly positive scalar curvature, and that N n is a closed spin manifold with A(N) > o. Let S be a discrete subset of M. Then the connected sum of M with one copy of N attached at each point of S (see Figure 2) admits a complete metric of uniformly positive curvature (in the canonical quasi-isometry class of metrics) if [S] = 0 in H;f(M), the uniformly finite homology of M, and does not admit any complete metric of positive scalar curvature if [S] f. 0 in H;f(M).

J.

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ROSENBERG

s FIGURE 2. The connected sum along a discrete subset.

4. Miscellaneous topics 4.1. The second Kazdan-Warner class. Recall from Theorem 0.1 that if M n is a closed manifold of dimension n ~ 3, and if M admits a metric with nonnegative scalar curvature but not one with positive scalar curvature, then any such metric must be Ricci-flat. Futaki [16] and Dessai [11] have obtained additional restrictions on such manifolds. For example, [16] shows that simply connected manifolds of dimension >4 in class (2) of Theorem 0.1 must be spin, have non-vanishing A-genus, and have exceptional holonomy SU(2m), Sp(n) or Spin(7). Furthermore, Futaki obtains additional constraints on the A-genus, and Dessai shows that the first Pontrjagin class must be non-trivial. One also has certain results that constrain "almost non-negative scalar curvature." A closed manifold M is said [14] to have "almost non-negative scalar curvature" if, for any c > 0, there is a Riemannian metric 9 with sectional curvature ~ 1 and with scalar curvature /'i, and diameter d satisfying /'i, ~ -c/d2 . The results of [14] say that in some cases, this is impossible unless M lies in the second Kazdan-Warner class. 4.2. Metrics with negative scalar curvature. Lohkamp has also shown that the results of Section 2 are also really very special to positive scalar curvature. On any closed manifold Mn with n ~ 3, Lohkamp showed [41] that the space of metrics of negative scalar curvature is contractible, with a retraction onto the subspace of metrics of constant scalar curvature -1. Furthermore, an arbitrary metric can be perturbed so as to decrease its scalar curvature on a prescribed open set, without changing the overall "shape" of the manifold. More precisely, one has: THEOREM 4.1 (Lohkamp [42]). Let (M n , g) be a Riemannian n-manifold with n > 2 and with scalar curvature function /'i,. Let U be an open subset of M, and let f be a smooth function on M such that f < /'i, on U and f = /'i, on M " U. Then for each positive c, there is a smooth metric g{ on

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M such that gf = 9 outside the c-neighborhood of U and such that the scalar curvature function ""e; of gf satisfies f - c ~ ""e; ~ f on the c-neighborhood of U. Moreover, ge; can be chosen arbitrarily close to 9 in the CO topology. References [1] M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc., 77 (1975), 43-69, MR0397797 (53 #1655a). [2] B. Blackadar, K -theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications, 5, Cambridge University Press, Cambridge, 1998, MR1656031 (99g:46104). [3] J. Block and S. Weinberger, Arithmetic manifolds of positive scalar curvature, J. Differential Geom., 52(2) (1999), 375-406, MR1758300 (200lh:53047). [4] B. Botvinnik, P. Gilkey, and S. Stolz, The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology, J. Differential Geom., 46(3) (1997), 374-405, MR1484887 (98i:58227). [5] B. Botvinnik and P.B. Gilkey, The eta invariant and metrics of positive scalar curvature, Math. Ann., 302(3) (1995),507-517, MR1339924 (96f:58159). [6] B. Botvinnik and P.B. Gilkey, The eta invariant and the Gromov-Lawson conjecture for elementary abelian groups of odd order, Topology Appl., 80{1-2) (1997), 43-53, MR1469465 (99f:58194). [7] B. Botvinnik and J. Rosenberg, The Yamabe invariant for non-simply connected manifolds, J. Differential Geom., 62(2) (2002), 175-208, MR1988502 (2004j:53045). [8] B. Botvinnik and J. Rosenberg, Positive scalar curvature for manifolds with elementary abelian fundamental group, Proc. Amer. Math. Soc., 133(2) (2005), 545-556 (electronic), MR2093079 (2005g:53057). [9] S.S. Chang, Coarse obstructions to positive scalar curvature in noncompact arithmetic manifolds, J. Differential Geom., 57(1) (2001), 1-21, MR1871489 (2002j:53037). [10] X. Dai, X. Wang, and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math., 161(1) (2005), 151-176, MR2178660. [11] A. Dessai, On the topology of scalar-flat manifolds, Bull. London Math. Soc., 33(2) (2001), 203-209, MR1815425 (2002b:53063). [12] H. Donnelly, Eta invariants for G-spaces, Indiana Univ. Math. J., 27(6) (1978), 889-918, MR511246 (80m:58042). [13] W. Dwyer, T. Schick, and S. Stolz, Remarks on a conjecture of Gromov and Lawson, High-dimensional manifold topology, World Sci. Publishing, River Edge, NJ, 2003, 159-176, MR2048721 (2005f:53043). [14) F'uquan Fang, Index of Dirac operator and scalar curvature almost non-negative manifolds, Asian J. Math., 7(1) (2003), 31-38, MR2015240 (2005a:58032). [15] R. Fintushel and R.J. Stern, Double node neighborhoods and families of simply connected 4-manifolds with b+ = 1, J. Amer. Math. Soc., 19(1) (2006), 171-180 (electronic), MR2169045. [16] A. F'utaki, Scalar-flat closed manifolds not admitting positive scalar curvature metrics, Invent. Math., 112(1) (1993), 23-29, MR1207476 (94f:53072). [17] P. Gajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom., 5(3) (1987), 179-191, MR962295 (89m:53061). [18] P.B. Gilkey, The eta invariant, manifolds of positive scalar curvature, and equivariant bordism, Geometry, topology and physics (Campinas, 1996), de Gruyter, Berlin, 1997, 157-171, MR1605220 (99k:58175). [19] P.B. Gilkey, The eta invariant of Pin manifolds with cyclic fundamental groups, Period. Math. Hungar., 36{2-3) (1998), 139-170, MR1694601 (2001a:58034).

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[39] E. Leichtnam and P. Piazza, On higher eta-invariants and metrics of positive scalar cU7'Vature, K-Theory, 24(4) (2001),341-359, MR1885126 (2002k:58051). [40] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris, 257 (1963), 7-9, MR0156292 (27 #6218). [41] J. Lohkamp, The space of negative scalar curvature metrics, Invent. Math., 110(4) (1992), 403-407, MR1185590 (93h:58025). [42] J. Lohkamp, Scalar curvature and hammocks, Math. Ann., 313(3) (1999), 385-407, MR1678604 (2000a:53059). [43] J. Lohkamp, Positive scalar curvature in dim ~8, C.R. Math. Acad. Sci. Paris, 343 (2006), no. 9, 585-588. [44] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2),64 (1956), 399-405, MR0082103 (18,498d). [45] J.W. Milnor and M.A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Internat. Congress Math., 1958, Cambridge Univ. Press, New York, 1960,454-458, MR0121801 (22 #12531). [46] A.S. Miscenko and A.T. Fomenko, The index of elliptic operators over C' -algebras, Izv. Akad. Nauk SSSR Ser. Mat., 43(4) (1979),831-859,967, MR548506 (8li:46075). [47] J.W. Morgan, The Seiberg- Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, 44, Princeton University Press, Princeton, NJ, 1996, MR1367507 (97d:57042). [48] J. Park, Simply connected symplectic 4-manifolds with bt = 1 and ci = 2, Invent. Math., 159(3) (2005), 657-667, MR2125736 (2006c:57024). [49] J. Park, A.I. Stipsicz, and Z. SzabO, Exotic smooth structures on CJP'2#5CJP'2, Math. Res. Lett., 12(5-6) (2005),701-712, MR2189231. [50) J. Roe, Partitioning noncompact manifolds and the dual Toeplitz problem, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge, 1988, 187-228, MR996446 (90i:58186). [51) J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc., 104(497) (1993), MR1147350 (94a:58193). [52) J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, 90, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996, MR1399087 (97h:58155). [53) J. Roe, Lectures on coarse geometry, University Lecture Series, 31, American Mathematical Society, Providence, RI, 2003, MR2007488 (2004g:53050). [54) J. Rosenberg, C' -algebras, positive scalar curvatu7'e, and the Novikov conjecture, III, Topology, 25(3) (1986), 319-336, MR842428 (88f:58141). [55] .J. Rosenberg, The KO-assembly map and positive scalar curvature, Algebraic topology Poznan 1989, Lecture Notes in Math., 1474, Springer, Berlin, 1991, 170-182, MR1133900 (92m:53060). [56) J. Rosenberg and S. Stolz, Manifolds of positive scalar curvature, Algebraic topology and its applications, Math. Sci. Res. Inst. Publ., 27, Springer, New York, 1994, 241-267, MR1268192. [57) J. Rosenberg and S. Stolz, A "stable" version of the Gromov-Lawson conjecture, The Cech centennial (Boston, MA, 1993), Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995, 405-418, MR1321004 (96m:53042). [58] J. Rosenberg and S. Stolz, Metrics of positive scalar curvature and connections with surgery, Surveys on surgery theory, 2, Ann. of Math. Stud., 149, Princeton Univ. Press, Princeton, NJ, 2001, 353-386, MR1818778 (2002f:53054). [59] D. Ruberman, Positive scalar curvature, diffeomorphisms and the Seiberg- Witten invariants, Geom. Topol., 5 (2001), 895-924 (electronic), MR1874146 (2002k:57076). [60) T. Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology, 37(6) (1998), 1165-1168, MR1632971 (99j:53049).

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[61] R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math., 28(1-3) (1979), 159-183, MR535700 (80k:53064). [62] R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2), 110(1) (1979), 127-142, MR541332 (81k:58029). [63] R. Schoen and S.-T. Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, NJ, 1982, 09-228, MR645740 (83k:53060). [64] R. Schoen and S.-T. Yau, The structure of manifolds with positive scalar curvature, Directions in partial differential equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, 1987, 235-242, MR1013841 (90e:53059). [65] N. Smale, Generic regularity of homologically area minimizing hypersurfaces in eightdimensional manifolds, Comm. Anal. Geom., 1(2) (1993), 217-228, MR1243523 (95b:49065) . [66] A.1. Stipsicz and Z. Szabo, An exotic smooth structure on CJP'2#6CJP'2, Geom. Topol., 9 (2005), 813-832 (electronic), MR2140993. [67] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2), 136(3) (1992),511-540, MR1189863 (93i:57033). [68] S. Stolz, Positive scalar curvature metrics-existence and classification questions, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994) (Basel), Birkhauser, 1995, 625-636, MR1403963 (98h:53063). [69] S. Stolz, Manifolds of positive scalar curvature, Topology of high-dimensional manifolds, No.1, 2 (Trieste, 2001), ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, Papers from the School on High-Dimensional Manifold Topology held in Trieste, May 21-June 8, 2001; Available electronically at http://users.ictp.it/-pub_off/lectures/ , 661-709, MR1937026 (2003m:53059). [70] C.H. Taubes, The Seiberg- Witten invariants and symplectic forms, Math. Res. Lett., 1(6) (1994), 809-822, MR1306023 (95j:57039). [71] C.T.C. Wall, On simply-connected 4-manifolds, J. London Math. Soc., 39 (1964), 141-149, MR0163324 (29 #627). [72] K. Whyte, Index theory with bounded geometry, the uniformly finite A class, and infinite connected sums, J. Differential Geom., 59(1) (2001), 1-14, MR1909246 (2003a: 58031). [73] E. Witten, Monopoles and four-manifolds, Math. Res. Lett., 1(6) (1994), 769-796, MR1306021 (96d:57035). [74] G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2), 147(2) (1998), 325-355, MR1626745 (99k:57072). [75] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139(1) (2000), 201-240, MR1728880 (2000j:19005). DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARYLAND, COLLEGE PARK, MD 20742, USA E-mail address: jmr 0 such that for every (1,8)-quasi-isometric f : 8 -+ U with perimeter of 8 bounded by P, every (1,8)-quasi-isometric extension OJ, I, z} -+ U of fl{y, z} with the midpoint I E yz defines a (1, E)-quasi-isometric extension 8 U {t} -+ U of f. This stabilized definition is useful for proving that the CBA-condition behaves well under limiting operations.

7:

A very important class of CBA spaces is the class of Hadamard spaces. DEFINITION 2.5 (Hadamard space). A Hadamard space is a complete CAT(O) space.

2.2. Reshetnyak's majorizing theorem. There is a broad generalization of the defining property of a CAT(K,) space which is an important tool for the study of CBA-spaces [100]. THEOREM 2.6 (RMT). Let 'Y be a closed curve of length O. Moreover, we require the following uniformity condition in the limit above: for each c > 0 there is p > 0 such that !r(t)-y'(t)I ::; ct for all positive t < P and " " whose directions at x are p-close to each other; • each tangent space TxX, x EX, is uniformly convex and smooth; • geodesics vary smoothly in X. For precise definitions we refer to [78]. The following classes of spaces are geometric: CBA and CBB (curvature bounded below) spaces; extremal subsets in CBB spaces [96]; surfaces with bounded total curvature (see sect. 10.5); Holder continuous Riemannian manifolds; sufficiently convex and smooth Finsler manifolds; subsets of positive reach in CBA spaces (see sect. 7). The class of geometric spaces is closed under natural metric operations and the first variation formula holds for the geometric spaces.

3.5. Riemannian manifolds with boundary. The following important result is obtained in [14]. THEOREM 3.7. Let M be a Riemannian manifold with boundary B. Then the following two conditions are equivalent: (1) M is CBA(K). (2) The sectional curvatures of the interior of M and the outward sectional curvatures of the boundary B do not exceed K (where an outward sectional curvature of B is one that corresponds to a tangent section all of whose normal curvature vectors point outward). A characteristic difficulty lies in the possibility of unbounded switching behavior, which may, for example, produce Cantor coincidence sets between a geodesic and the boundary. This theorem is proven by establishing the equivalence of conditions 1 and 2 with a third condition, namely, the K-convexity of normal Jacobi field lengths. The most difficult part of Theorem 3.7 is to prove that condition 2 implies K-convexity of normal Jacobi field lengths, especially at points of a geodesic, C M lying in the boundary where the acceleration exists and vanishes. As an application, the following Hadamard-Cartan theorem for manifolds with boundary is obtained in [14]. COROLLARY 3.8. If for a simply connected, complete Riemannian manifold with boundary, the sectional curvatures of the interior and the outward sectional curvatures of the boundary are nonpositive, then any two points are joined by a unique geodesic, and the distance between any two geodesics is convex.

3.6. Higher order properties. On the unit tangent bundle U M of every Riemannian manifold, there is a natural Riemannian metric, called

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the Sasaki metric, which is defined via the Levi-Civita parallel transport, [101, 102J. An attempt to define an analog of the Sasaki metric or, more precisely, of the notion of angle between two directions, possibly based at different points, is made in [40J for general CBA-spaces. The approach is based on the notions of quadrilateral cosine and sine, where the later is defined via the former. The definition of the quadrilateral sine is rather involved and for general CBA-space it is hard to prove anything useful about that notion, see [40J. In the case of spaces with more regularity, one can achieve more advances and more interesting results involving higher order properties, see [91, 92J.

4. Local properties of CBA-spaces 4.1. Geometric dimension. One defines the geometric dimension of CBA spaces to be the smallest function (taking values in NUoo) on the class of CBA spaces such that (1) GeomDim(X) = 0 if X is discrete; (2) GeomDim(X) ~ 1 + GeomDim(~xX) for every x E X. In other words, to find the geometric dimension of a CBA-space we look for the largest number of times that we can pass to spaces of directions without getting the empty set. This notion and the related results are due to B. Kleiner, [65J. THEOREM

4.1. For every CBA -space X we have

GeomDim(X)

= sup{TopDim(K)

: K

c

X is compact},

where TopDim is the topological dimension. Let X be a CBA space with GeomDim(X) = n < 00. Then sup{k : \Ie> 0 :3(1 + e) - bilipschitz embedding U -t X of an open U C jRk} = n. Let X be a locally compact Hadamard space on which Isom(X) acts cocompactly. Then sup{ k : There is an isometric embedding jRk -t X} = 1 + GeomDim(arX). Here aTX is the boundary at infinity of X equipped with the Tits metric. Actually, a number of other properties related to GeomDim(X) are proved in [65J. Here we have listed only the most important ones. A key ingredient in the proofs is the notion of a barycentric simplex. For Z = (zo, ... , zn) C X (with sufficiently small diameter if /'i, > 0), the barycentric simplex determined by Z is the singular simplex (7 z : ~n -t X which maps each a = (ao, ... , an) E ~n to the unique minimum of the uniformly convex function ¢ex = L ai dist(zi, .)2. Barycentric simplices are Lipschitz and possess the following remarkable property. If x E (7z(~n)\(7z(a~n), then GeomDim(~x) ~ n - l. For the proof, one considers the differential gex = d¢ex = L adi : TxX -t R It turns out that for a E a~n, its restriction to the unit sphere ~xX C

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TxX possesses a unique minimum and thus defines a Lipschitz (J f : 8D.. n -+ 1;xX which is a nondegenerate (n-l )-chain of barycentric simplices. Arguing

by induction, one concludes that GeomDim(1;xX) ~ n - l. This leads to the estimate TopDim(K) ~ GeomDim(X) for compact subsets K c X, and using the nondegenerate part (Jz(D.. n)\(Jz(8D..n), one obtains bilipschitz embeddings of open sets U c Rn into X. Though it is not stated explicitly, the existence of bilipschitz embeddings U c ]Rn -+ X should imply (an extension of) the Rademacher-Stepanoff theorem on the differentiability of Lipschitz functions to CBA-spaces, cf.

[67], [54]. 4.2. Branch and singular points. Let X be a locally compact, geodesic ally complete CBA-space. A point x E X is said to be regular, if the direction space 1;xX is isometric to the unit sphere 5 n- 1 for some n E N, while 1-£n (Br (x)) < 00 for some r > o. A point x E X that is not regular is called singular. Let x EX, 8 > O. Following [94], we say that v E 1;xX is a 8-branch direction, if diam Bv ~ 8, where Bv C 1;xX consists of all directions forming the (maximal possible) angle 7r with v. Furthermore, y E X is called a 8branch point of x, if Vyx E 1;yX is a 8-branch direction, where the direction Vyx is tangent to some geodesic segment yx. We denote by 5 x ,8 the set of all 8-branch points of x and by 58 the set of all8-branch points, 58 = UxEX5x,8. Note that if x E 58 then 1;xX is not isometric to 5 n- 1 for any n E N. In particular, it follows that 58 consists of singular points. The following result [94, 88] shows the abundance of regular points.

THEOREM 4.2. Assume that 1-£n(Br (x)) < 00 for some n E N, x E X and a sufficiently small r > O. Then 1-£n(5x ,8 n Br(x)) = 0 for any 8 > O. Moreover, 1-£n(58 n Br(x)) = 0 and 1-£n(An n Br(x)) = 0, where An C X consists of all x with 1;xX not isometric to 5 n - 1 . We conclude that if 0 < 1-£n(Br (x)) < 00 for all sufficiently small r then 1-£n-almost every point of every such ball is regular.

> 0,

4.3. Manifold points. A manifold point in a metric space X is a point with a neighborhood homeomorphic to an open subset in some ]Rn. The following result is due to B. Kleiner. For simplicity, we give a qualitative version.

THEOREM 4.3. If a metric ball Br(x) in a geodesically complete CBAspace X is sufficiently close in the Hausdorff-Gromov metric (see sect. 5) to the ball Br(O) C ]Rn of the same sufficiently small radius r > 0, then a smaller concentric ball Bp(x) C Br(x) with p « r is bi-Lipschitz homeomorphic to an open subset of]Rn.

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For the proof see [49, §3], a quantitative version can be found in [88]. This theorem is similar to [47, Theorem 5.4] for CBB-spaces, and the argument follows a similar line of reasoning by proving the existence of a distance frame or strainer and studying the associated distance map into ]Rn. The essential distinction to the CBB case is only that a lower estimate for angles is based on the extendability of geodesics. For applications to structure results for CBA-metrics on 2-polyhedra see [49] and sect. 10, for applications to a volume convergence theorem see [88] and the end of sect. 5. If every point of X is a manifold point, then X is called a CBA-manifold. Even locally CBA-manifold can differ significantly from Riemannian manifolds, as the examples of CBA-manifolds whose space of directions ~x is not a manifold (section 3.3) show. For every n ~ 5 M. Davis and T. Januszkiewicz [55] constructed CBA(O)-manifolds M n whose universal covering space X = Mn is not simply connected at infinity, in particular not homeomorphic to ]Rn. P. Thurston [104] however showed, that a 4-dimensional CAT(O)manifold X 4 which possesses a tame point, is homeomorphic to ]R4. Here a point x E X is called tame, if for all r > 0 the distance sphere Br(x) is a closed manifold. 4.4. Propagation from local to global. A remarkable property of the comparison conditions CBA as well as CBB is that they propagate from local to global. However, there is a fundamental distinction between CBA and CBB: in any CBB(~) space every geodesic triangle satisfies the angle comparison with MK (Toponogov's theorem), whereas for CBA spaces this is not the case. This is a major source of problems for proving or checking the CBA-condition. A geodesic , : [0, 1] ~ X has no conjugate points, if for some neighborhood U of, (in the space of maps [0, 1] ~ X with the compact-open topology) the map p: U ~ X x X, p(O") = (0"(0),0"(1)) is a homeomorphism on the neighborhood V = p(U) of h(O),,(l)). We have [5] the following: THEOREM CBA(~)-space

4.4. Every geodesic, of length Lh) < DK in a complete X has no conjugate points. Moreover, every narrow triangle

with two sides sufficiently close to , is

~-thin.

The main ingredient of the proof is the following middle-third construction. For geodesics " 0", ,', 0"' : [0, 1] ~ X such that ,'(0) = ,(0), ,'(1) = 0"(1/2),0"'(0) = ,(1/2), 0"'(1) = 0"(1) we denote h',O"') = Ah,O") and put /-lh,O") = max {b(1/2)0"(0)1, 1,(1)0"(1/2)1}. For every 0 < P < ~DK there exist fJ = fJ(P), ,X = 'x(P), fJ > 0,0 < ,X < 1 such that iffor pairs of geodesics h, 0"), h', 0"') = Ah, 0") in a metric space X the triangles A with sides " ,', ,(1)1'(1) and A' with sides 0", (7', 0"(0)0"'(0) are ~-thin and P(A), P(A') ~ P, /-lh,O") < fJ then

/-lh', 0"') ~ 'x/-l(" 0").

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Applied to the case K, ~ 0 this leads to an extension of the classical Hadamard-Cartan theorem to Hadamard spaces. THEOREM 4.5. For every K, (K,)-space satisfies CAT(K,).

~

0, every complete simply connected CBA

There is a standard trap in the proof: if one has a space in which a geodesic between any two points is unique then it is natural to suppose that the geodesics vary continuously with their ends. This is indeed the case for the locally compact CBA(K,)-spaces. However, in general, this is not true. One can observe this effect by looking at the cartwheel. As an example (due to W. Ballmann) one can take the metric completion X of B = UnBn where BI is the circle of length 3, and Bn+! is obtained from Bn by connecting every x, y E Bn with dist(x, y) > 1 by a segment of length 1. Then X is a complete CBA(K,)-space for each K, E IR such that every two points from the dense subset B are connected by a unique geodesic. However, geodesics in X do not vary continuously with their ends. 4.5. Busemann's G-spaces. A G-space of Busemann is a locally compact, complete, inner metric space in which geodesics are not overlapping and locally extendable (geodesics are not overlapping if whenever two of them, , and ,f, have an open common interval, their union , U,f again supports a geodesics). The following result due to V. Berestovskii [33] describes G-spaces which are CBA.

THEOREM 4.6. Every CBA G-space of Busemann is a Riemannian Co_ manifold. The components of the metric tensor are continuous w. r. t. distance coordinates. Every two distance coordinates maps are CI-compatible. REMARK 4.7. A similar result holds true for CBB G-spaces, moreover, in that case the components of the metric tensor are C I / 2-smooth functions of the distance coordinates, [34], [95]. Distance coordinates were introduced in [35] and used there to obtain the first synthetic characterization of Riemannian manifolds as metric spaces which are both CBA and CBB with locally extendable and non-overlapping geodesics. 5. Different types of convergence

In general, the CBA-condition does not survive the Hausdorff-Gromov convergence. The reason is that the size of CAT(K,)-neighborhoods may become arbitrarily small. The standard example is this: the hyperboloids Xc = {(x, y, z) E 1R3 : x 2 + y2 - z2 = c;2} with the induced intrinsic metrics are CBA(O) and they Hausdorff-Gromov converge to the double cone Xo = {(x, y, z) E 1R3 : x 2 + y2 - z2 = O} as c; -+ 0 which is not CBA. However, we easily have:

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THEOREM 5.1. If xn -t X by Hausdorff-Gromov and Xn are CBA(/'i:) with convexity radii uniformly separated from 0, then X is CBA(/'i:). One can define the Hausdorff-Gromov convergence as follows. The distortion of a map f : X -t Y between metric spaces is dis(J)

=

sup Idy(J(x), f(x')) - dx(x, x') I. x,x'EX

Let A be the class of all maps X -t Y. Putting J(X, Y) = inf lEA dis(J), one defines IXYIHG = max {J(X, Y),J(Y,X)}. The convergence with respect to this metric is equivalent to the HausdorffGromov convergence. Replacing the class A of all maps by the class of all homeomorphisms, we arrive at the uniform metric and the uniform convergence respectively. Now, we formulate a useful sufficient condition for retaining CBA(/'i:) under uniform convergence. Recall that the dilatation of a mapping f between metric spaces X, Y is the (possibly infinite) number dil(J) = sup If(x)f(x')I,

Ixx'i

where the supremum is taken over all distinct x, x' EX. We say that a sequence of metrics {dk} on X has no local blow-ups if for every x E X there are n E Nand c > 0 such that the restriction id~,n+k of the identity map idn,n+k : (X, dn ) -t (X, dn+k) on the ball Btn(x) has dilatation dil(id~,n+k) :s < 00 for all kEN. The following sufficient condition is proven in the lecture notes [53J.

en

THEOREM 5.2. Assume that a sequence {d n } of metrics without local blow-ups on X uniformly converges to a metric d. If dn is complete, CBA(/'i:) and its metric topology is locally compact for every n 2: 1, then d is CBA( /'i:). The condition that d is a metric is essential as the example Xc -t Xo from above shows. Here we have the uniform convergence without blow-ups, however, the limiting d is only a pseudo-metric. Finally, we introduce the homotopy metric and the homotopy convergence which is well adapted to the CBA(O)-condition. Let f : X -t Y be a homotopy equivalence with homotopy inverse g : Y -t X. For compact metric spaces X, Y we put

IXYlh = inf max {dis(J), dis(g)}, I,g where the infimum is taken over all homotopy equivalences (J, g) : X +-7 Y. This homotopy distance defines a metric on the classes of isometric compact metric spaces.

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THEOREM 5.3. Assume that IXnXlh --+ 0 as n --+ CBA(K) with K ~ O. Then X is CBA(K).

00,

where all Xn are

This is proven in [19]. For K > 0, the CBA(K)-condition does in general not survive as the following example due to S. Ivanov [61] shows. EXAMPLE 5.4. For every E > 0 there exists a contractible closed 2-polyhedron X with a CBA(l)-metric having diameter less than E. This polyhedron X consists of a huge number of blocks Xi (depending on E), each of which is obtained as follows. Fix a tiny d > 0 (depending on E) and take a unit sphere 8 2 with an open ball B removed, where the boundary curve u of B has length d. Next, take a tree T C 8 2 \ B with the root vertex on u such that the length of every edge of T is d and whose vertices form a sufficiently dense subset in 8 2 . Finally, identify all vertices of T getting Xi' Different blocks are glued together in a way such that the vertices Vi E Xi are identified with a unique vertex v E X and the boundary curve Ui of every block Xi is identified with a curve of another block Xj which originates from an edge of Tj. This gives the contractibility of X (a similar effect provides the contractibility of the dunce cap). To ensure the CBA(l)-condition one should solve a combinatorial problem to guarantee the 27r-systole condition for the link of the vertex v EX. The question if it is possible to find similar E-small CBA(l)-metrics on a fixed closed contractible 2-polyhedron remains open. A volume convergence theorem for CBA-spaces is proven in [88] under the following assumptions. Let X, Xj, j E N, be compact, geodesically complete CBA(K)-spaces of the same Hausdorff dimension n. Assume that the convexity radius r(Xj) is separated from 0 uniformly in j E Nand IXXjlHG --+ 0 as j --+ 00. Then 1in(Xj) --+ 1in(x).

6. Constructions 6.1. Gluing theorems. A simple but very useful tool is the following gluing theorem by Reshetnyak [98]. THEOREM 6.1. Let Xl, X 2 be complete locally compact CBA(K)-spaces. Suppose that there are convex sets C i C Xi and an isometry f : Cl --+ C2' Attach these spaces together along f. Then the resulting space X is CBA(K). The proof is more or less straightforward and uses the comparison of angles. However, there is a standard trap while checking the condition of the theorem. JR2 with an open disk removed is CBA(O) with convex boundary C = aX. Gluing two copies of X along the boundary, we obtain a CBA(O)space y2. However, taking JRn with n ~ 3 instead of JR2 we obtain a yn which is not CBA(O). The reason is that JRn with an open ball removed is only CBA(K) where K = r- 2 and r is the radius of the ball. Thus yn is

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only CBA(r- 2 ). Similarly, the hyperbolic space Hn with an open horoball removed is only CAT(O) for n 2: 3, but not CAT ( -1). On the other hand, if we glue two copies of ~n along closed isometric balls, then the resulting space is CAT(O) for every n 2: 1. These additional pieces which save the CAT(O)-property are called fins and they were successfully used in the proofs of quite different results (see the end of sect. 6.2). Trying to generalize Reshetnyak's gluing theorem, one can ask for conditions which would guarantee that gluing two smooth Riemannian manifolds M 1 , M2 of the same dimension n 2: 3 (the case n = 2 we discuss in sect. 10) along isometric boundaries gives a CBA(Ii:)-space, if both M 1 , M2 are CBA(Ii:). It is natural to conjecture that such a condition must be Ll + L2 :::; 0, where Li is the second fundamental form of 8Mi. This was proven by N. Kosovskii, [68].

zn

THEOREM 6.2. Assume that L := Ll +L2 :::; 0 at the corresponding points of 8M1 , 8M2. Then M = Ml U M2 is locally a CBA-space. Moreover, if in addition the sectional curvatures of both Ml, M2 are :::; Ii: and the sectional curvatures of their common boundary r are at most Ii: at those 2-directions where both L 1 , L2 are negatively determined, then M is CBA(Ii:). The example of the space yn above shows that no condition can be omitted, and in fact the conditions above are necessary. Let reM be the singular hypersurface obtained from 8MI, 8M2 while gluing M. The approach is to extend the Riemannian metric appropriately, say of Ml c M to a neighborhood of r in M and then using L to perturb the metric on Ml smoothing r and pumping its singularity into the curvature of the perturbed metric. The main issue is to obtain a uniform (in the perturbation parameter 6) curvature estimate from above. This is achieved via tremendous analytic calculations. The perturbed metrics form a sequence converging without local blow-ups to the initial one, which shows that the gluing gives a CBAspace. Now, the sharp Ii:-estimate is obtained in two steps. First, it is obtained under the assumption L < 0 using approaches from [14] and the fact that M is CBA. Second, in the general case L :::; 0, the manifold M is appropriately approximated by those with L < O. This is achieved by C 2-small changes of the metrics on Ml, M2 in a way that the forms Ll, L2 decrease while the induced metric on r is not changed. A sharp gluing CBB-theorem for two Riemannian manifolds is obtained by similar arguments in [69]. The result above is generalized in [70] to the case of an arbitrary finite number of manifolds M a , a E A, of the same dimension n 2: 3 glued together along the common boundary r, M = UaMa. THEOREM 6.3. Assume that the sectional curvatures of the manifolds M a , a E A, are bounded from above by Ii: and that La + Lal :::; 0 for each pair

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of different a, a' E A and for the second fundamental forms La, Lal of r with respect to M a , Mal respectively. Furthermore, assume that the sectional curvatures of r are bounded from above by K, in those 2-directions where the forms La, a E A, are negative definite simultaneously. Then M is CBA(K,).

Again, the issue is to prove that M is locally CBA. Then the sharp K,estimate is obtained using methods of [14], see Proposition 3.6. Note that the condition is weaker than that by [68] yielding the CBA(K,) property for each union Ma U Mal with different a, a' E A. It is only known from [68] that every such union is CBA(K,') for some K,' E IR. To prove that M is locally CBA, one needs a good control over minimizers in M. This is achieved under the assumption that the sums La + Lal are locally uniformly negative by introducing a class of curves called almostgeodesics and careful study their properties (the general case La + Lal ~ is obtained by approximation as in [68]). An almost-geodesic "I in M is a C 1 _ smooth curve concatenated from finitely many minimizers each of which is running in its own leaf Ma. Using a sort of linearization argument, the author shows that locally every almost-geodesic consists of at most three such minimizers, and the angle comparison with MK. holds for triangles formed by almost-geodesics. This suffices to prove that locally every minimizer in M is almost-geodesic and therefore M is locally CBA.

°

I

6.2. Warped spaces, and f : B B x f F is defined curve "I = ("tB,"IF)

products. Suppose that Band F are intrinsic metric -+ JR.~o is continuous. Distance in the warped product by the infimum of path-lengths, where the length of a for rectifiable curves "IB and "IF in Band F is given by: L("t) =

J

JvMt)

+ j2("tB(t))v~(t)dt,

where VB and VF are the speeds of "IB and "IF. Equivalently, L("t) is the supremum of the expressions

For example, taking F = sn-l, we obtain that • for the function f : B = [0, 00) -+ JR., f (t) = t, the warped product space B x f F is isometric to JR.n with the metric ds 2 = dt 2 + t2dw~_I'

where dw~_1 is the standard metric of the unit sphere sn-l j • for the function f : B = [0,00) -+ JR., f(t) = sinh t, the warped product space B x f F is isometric to H n with the metric

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• for the function f : B = [0,7r] ---+ JR., f(t) = sin t, the warped product space B x f F is isometric to sn with the metric

The most general sharp conditions for a warped product of metric spaces to have a given curvature bound for CBA- as well as for CBB-spaces are found by S. Alexander and R. Bishop in [9]. We formulate these conditions for CBA-spaces. THEOREM 6.4. Let Band F be complete CAT(~) and CAT(~F) spaces, respectively. Let f : B ---+ JR.>o be ~-convex, where f is Lipschitz on bounded sets or B is locally compact. Set X = f-l(O). (1) If X = 0, suppose ~F ::; K(inf 1)2. (2) If X #- 0, suppose f'(0+)2 ~ ~F at footpoints of dx-minimizers in B, and ~F ::; ~f(p)2 for points p E B further than 7r/2y'K, from X. Then B xfF is CAT(~).

These conditions are close to be necessary ones. Namely, if a warped product of metric spaces B x f F has an upper curvature bound ~, then the same is true for B because its images in B x f F are totally convex. One can also derive ~-convexity of the warped function f. It remains to show that F has an upper curvature bound ~F satisfying conditions (1) and (2). This is obviously true if f takes a positive minimum f(p), since then {p} x F is totally convex. As applications, the theorem above gives rise to a number of constructions of spaces with upper curvature bounds, among which is Reshetnyak's gluing theorem [98], which we discussed above (this theorem is used in the proof). It also covers a result of Ancel and Guilbault [16] saying that the interiors of compact contractible n-manifolds, n ~ 5, support a geodesic metric of strictly negative curvature. The case of Hadamard spaces, that is, ~ = ~F = 0 and f > 0, was studied earlier in [7]. We briefly sketch the proof of that case underling the basic idea of the general proof. It can be illustrated as follows. Consider two copies of JR.n, n ~ 1, glued together along unit balls. Let Xn be the resulting space, Yn = Xn \ (interior of B), where B is the image of the balls. Then Xn is CBA(O) (and even CAT(O)) for every n ~ 1 by Reshetnyak's gluing theorem, while for n ~ 3 the space Yn is only CBA(l) and not CBA(O). This is because the boundary sphere of B is convex in Yn , and for n ~ 3 its dimension is bigger than 1. The effect of lowering a curvature bound by adding B to Yn is crucial for the proof of the warped product theorem. The proof proceeds by reduction to the case F = JR. and by approximation of B x f JR. by subspaces We of Hadamard spaces W; which are constructed as follows. We decompose the Euclidean product B x JR. into three regions wjO) = {(p, u) : -cf(p)::; u ::; cf(P)}, uJO) = {(p, u) : cf(p) ::; u},

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L~O) = {(p, u) : u ~ -ef (P)}, where the two last are closed convex subset n), L~n) of these, we because f is convex. Taking isometric copies win), construct W; identifying isometric pairs n) with L~n+1) for each n E Z. To summarize, the space W; consists of the mutually isometric strips win) and n) = L~n+1) with the appropriate boundary the mutually isometric fins components identified. Although the space We obtained by gluing together the strips win) typically has positive infinite curvature, we recover nonpos-

ui

ui

ui

itive curvature by gluing on the fins. The same sort of construction which takes into account fins has been used in [50] to prove a uniform estimate on the number of collisions in semi-dispersing billiards. The idea is to develop a billiard trajectory into a geodesic in a CBA-space obtained by gluing together step by step convex walls that are hit by the trajectory. This translates a difficult dynamical problem into a geometric one, which can be solved by geometric methods. 7. Gauss equation The well known Gauss equation in Riemannian geometry allows to express intrinsic sectional curvatures of a submanifold via extrinsic curvatures and sectional curvatures of the ambient space. Surprisingly, the equation can be extended in a sense to arbitrary CBA-spaces. To describe such an extension, we recall some definitions. Let M be a CBA(K)-space, K E R. A subset N c M is said to have positive reach ~ r if every point x in the r-neighborhood of N has a unique foot point in N, that is, pEN with Ixpl = dist(x, N). It is proved in [77] that in the case M is a Riemannian manifold, any subset N c M of positive reach has some intrinsic curvature bound from above. The condition of positive reach can be expressed by comparing lengths of arcs and chords as follows [79]: a complete subset N c M has positive reach if there exists p > 0 such that intrinsic distances dN = 8 and extrinsic distances dM = r satisfy 8 - r ~ Cr 3 for r < p (actually, these two conditions are more or less equivalent for subsets of Riemannian manifolds). This estimate is an important step toward the notion of extrinsic curvature. The constant C in front of r3 on the right hand side may serve as a bound for extrinsic curvature. Namely, we say N is a subspace of the extrinsic curvature ~A in M if there is a length-preserving map N -+ M between intrinsic metric spaces, where N is complete and 8 -

r
.d(x, s))

c

Y,

sES

where B(y, r) is the closed ball of radius r around y. Note that if xES, then A(x) = {f(x)} since I(x) E B(f(s), >'d(x, s)) for all s E S. One can prove the following: (a) If a ~ V2, then A(x) =1= 0 for all x E X. (b) For x, x' E X the Hausdorff distance between A(x) and A(x' ) satisfies Hd(A(x), A(x' )) ~ 2v'2>.d(x, x').

Hence, in order to extend 1 to X, it would suffice to find a Lipschitz map 2. Then f is homotopic to an isometry. So the answer to the (smooth) Rigidity Question (1.3) for constant negative curvature is affirmative. It is natural to ask then, as in Question (1.4), if there is a stronger version of this Rigidity. An obvious choice in (1.4) for the weaker property pI is negative curvature (i.e, not necessarily constant). We get the following question: (1. 7) A Stronger Rigidity for hyperbolic manifolds: Suppose that X and Y are homotopy equivalent closed Riemannian manifolds. Assume that X is hyperbolic and Y is negatively curved. Are X and Y diffeomorphic? Or more generally: (1.8) Smooth Rigidity for negative curvature: Suppose that X and Y are homotopy equivalent closed negatively curved Riemannian manifolds. Are X and Y diffeomorphic? Of course an affirmative answer for (1.8) implies an affirmative answer for (1.7). There are some very successful tools from Geometric Analysis that were very promising to prove that (1.8) had an affirmative answer. The most important was the Harmonic Map technique. A smooth map k : X --+ Y between Riemannian manifolds is harmonic if ~IDkI2. An equivalent it is a critical point of the energy functional £(k) = definition is that the tension field Tk of k vanishes everywhere. {The tension field Tk is a section of the bundle k*TY and can be defined in the following way: for x E X choose an orthonormal basis {vd of TxX and define Tk (x) = L: Wi were Wi is the acceleration vector, at t = 0, of k(-Yi), and Ii is the geodesic with li{O) = x and 1t'i{O) = ud Given a map f : X --+ Y between Riemannian manifolds, we can try to associate to it a harmonic map that is the limit k = limHoo kt , where kt is the unique solution of the heat flow equation, that is, the PDE initial value problem ~ = T{k t ), ko = f. If this limit k exists then it is homotopic to f (the homotopy is t f-t k t ).

Ix

Now, it follows from the classical result of Eells and Sampson [15] that if f : X --+ Y is a smooth homotopy equivalence between closed negatively curved manifolds the heat flow equation beginning at f converges to a well defined harmonic map k = limt--+oo k t . Moreover, from the results of Hartman [36] and Al'ber [1] it follows that f is homotopic to a unique harmonic map. Therefore the homotopy equivalences in Questions (1. 7) and (1.8)

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are homotopic to unique harmonic maps. If both manifolds were hyperbolic then this harmonic map is in fact the isometry of Mostow Rigidity Theorem. Therefore we can use the Harmonic Map Technique to try to prove Mostow's Fundamental result. Indeed the Theory of Harmonic Maps had been very successful in showing rigidity results, see for instance Siu [54], Sampson [53], Hernandez [37], Corlette [13], Gromov and Schoen [33], Jost and Yau [39], and Mok, Sui and Yeung [45]. Because of this evidence it seems reasonable that Lawson and Yau conjectured that the answer to (1.8) was affirmative. But the answer to (1.7) is negative (and thus the answer to (1.8) is also negative). This was proved by counterexamples constructed by Farrell and Jones [16]. They proved the following.

7. Let xn be a closed hyperbolic n-manifold, E an exotic n-sphere, n ~ 5, and € > O. Then there is a finite cover Z of X such that (1) The connected sum Z#E is not diffeomorphic to Z. (2) Z#E admits a Riemannian metric with sectional curvatures in (-1-€,-1+€). THEOREM

Note that Theorem 7 above shows that the answer to a strengthened version of (1.7) is also negative: the property pi (which, in (1.7) is negative curvature) can be replaced by €-pinched negative curvature (€ depending only on the dimension). Since en is trivial in dimensions < 7, Theorem 7 does not give counterexamples in dimensions < 7. Furthermore in these dimensions manifolds are diffeomorphic if and only if they are P L-equivalent (with the P L structure induced by the given smooth structure). Hence for dimensions < 7 the Smooth Rigidity Question (1.8) is equivalent to the following P L version: (1.9) PL Rigidity for negative curvature: Suppose that X and Y are homotopy equivalent closed negatively curved Riemannian manifolds. Are X and Y PL-homeomorphic? For a general dimension n a negative answer to (1.9) implies a negative answer for (1.8), because diffeomorphic manifolds are PL homeomorphic. The converse is not true in general, but, as mentioned before, it is true for dimensions < 7. For example an (smoothly) exotic sphere E is not diffeomorphic to the corresponding sphere (by definition) but it is P Lhomeomorphic to it, provided dim E i= 4. In fact there are no P L-exotic spheres in any dimension i= 4. It follows that Z is PL-homeomorphic to Z#E for any manifold Z, and exotic sphere E, dim E i= 4. Therefore Theorem 7 does not answer Question (1.9). Note that, since diffeomorphism implies P L-homeomorphism and this in turn implies homeomorphism, we have that Question (1.9) lies between the Topological Rigidity for negative curvature (which is true, by Theorem 1) and the Smooth Rigidity (which is false, by Theorem 7).

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The answer to Question (1.9) was proved to be negative by Ontaneda [48] in dimension 6 and later by Farrell, Jones and Ontaneda [22] for every dimension> 5. Here is the result: THEOREM 8. For every n > 5 and E > 0 there are closed Riemannian n-manifolds X and Y such that: (1) X is homeomorphic to Y. (2) X is not P L-homeomorphic to Y. (3) X is hyperbolic. (4) Y has sectional curvatures in (-1 - E, -1 + E).

The counterexamples constructed in the proof of Theorem 8 use the results of Millson and Raghunathan [44], based on previous work of Millson

[43]. Theorems 7 and 8 were the first in a sequence of results that shed some light on the relationship between the analysis, geometry and topology of negatively curved manifolds. These results showed certain limitations of well-known powerful analytic tools in geometry, such as the Harmonic Map technique, the Ricci flow technique, the Elliptic deformation technique as well as Besson-Courtois-Gallot's Natural Map technique [10]. Here we shall just present briefly the main conclusions of this research. A more complete exposition on this area and how it evolved in time can be found in the survey article [26]. The main results are described in the following 5 items: 1. Recall that the Topological Rigidity for negative curvature is true, by Theorem 1. One can ask whether there is a Harmonic Map proof of this fact, that is, whether Lawson-Yau conjecture is true "topologically": • Question: Let f : X --t Y be a homotopy equivalence between closed negatively curved manifolds and let k : X --t Y be the unique harmonic map homotopic to f. Is k a homeomorphism? It follows from Theorem 8 and the Coo - Hauptvermutung of Scharlemann and Siebenmann [55], that this unique harmonic map k is not, in general, a homeomorphism. Hence, even though we do know that f is homotopic to a homeomorphism (by Theorem 1), we cannot, in general, obtain a homeomorphism using the Harmonic Map technique, at least not directly. 2. Consider Problem 111 of the list compiled by S.-T. Yau in [61] (it is also Grand Challenge Problem 3.6. in [59]) Here it is a restatement of this problem: • Problem 111 of [61]. Let f : X --t Y be a diffeomorphism between closed negatively curved manifolds and let k : X --t Y be the unique harmonic map homotopic to f. Is k a homeomorphism? Note that the difference between this and the previous question in 1 is that we begin now with a diffeomorphism f. Hence in this case the heat flow begins already with a diffeomorphism and we want to know is the limit harmonic map k = limt-+oo k t is a homeomorphism. The examples

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mentioned in 1 above (given by Theorem 8) are not useful, at least directly, to answer this question because X and Yare not PL equivalent in Theorem 8, hence there is no diffeomorphism between them. The answer to the problem above was proved to be yes when dim X = 2 by Schoen-Yau [57] and Sampson [52]. But it was proved by Farrell, Ontaneda and Raghunathan [24] that the answer to this question is, in general, negative. In fact because of Scharlemann's generalization [56] of [55] together with the recent positive solution of the Poincare Conjecture the maps k t are also all nonunivalent for t sufficiently large. 3. Since a harmonic map (between closed negatively curved manifolds) homotopic to a diffeomorphism is not necessarily a homeomorphism we can ask a deeper question: suppose now that the harmonic map can be approximated by homeomorphisms (or even diffeomorphisms), that is, the harmonic map is cellular. Does this imply that the harmonic map is a diffeomorphism? Farrell and Ontaneda [25] showed that the answer to this question is also negative. 4. It was pointed out by M. Varisco that most of the results mentioned above for harmonic maps can also be applied to the natural maps defined by G. Besson, G. Courtois and S. Gallot [10]. 5. In items 1-4 above we dealt with processes that produce some special type of map, e.g harmonic maps or natural maps. We can also consider some processes that produce a special type of metric: Einstein metrics, that is, metrics of constant Ricci curvature. The best known method for obtaining Einstein metrics is the Ricci flow method introduced by Hamilton in his seminal paper [34]. Starting with an arbitrary smooth Riemannian metric h on a closed smooth n-dimensional manifold xn, he considered the evolution equation &t h = ~ r h - Ric, where r = R dJ-L/ dJ-L is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. For n = 3 Hamilton proved that if the initial Riemannian metric on X 3 has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric hoc. This implies that X3 equipped with hoo is a spherical space-form; i.e., its universal cover is the round sphere. Following Hamilton's approach G. Huisken [38], C. Margerin [42] and S. Nishikawa [47], proved that Riemannian manifolds whose sectional curvatures are pinched close to +1 (the pinching constant depending only on the dimension) can be deformed, through the Ricci flow, to a spherical-space form. Then it was natural to ask whether the same was true for Riemannian manifolds whose sectional curvatures are pinched close to -1, again the pinching constant depending only on the dimension:

J

• Question: Is there a constant

J

En, depending only on n, such that if g is a Riemannian metric on a closed n-manifold with sectional curvatures in (-1 - E, -1 + E), with E < En, then the Ricci flow beginning at g converges to an Einstein metric of negative sectional curvatures?

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Rugang Ye [64] proved that sufficiently pinched to -1 manifolds can be deformed, through the Ricci flow, to hyperbolic manifolds, but the pinching constant in his Theorem depends on other quantities (e.g. the diameter or the volume). It was shown later by Farrell and Ontaneda [27], using the tools developed in [25], that the pinching constant cannot depend solely on the dimension by giving examples of arbitrarily pinched to -1 Riemannian metrics for which the Ricci flow does not converge smoothly to a negatively curved metric: THEOREM 9. Given n > 10 and € > 0 there is a closed smooth ndimensional manifold X such that (i) X admits a hyperbolic metric. (ii) X admits a Riemannian metric h with sectional curvatures in [-1- €, -1 + €] for which the Ricci flow does not converge smoothly to a negatively curved Einstein metric.

As mentioned earlier, a more complete account of the results mentioned in the previous five items can be found in [26]. We now give two different versions of Theorem 7: one version for noncompact finite volume complete hyperbolic manifolds and the other for negatively curved manifolds not homotopy equivalent to a closed locally symmetric space. Let us begin with the former. First note that taking the connected sum of a noncompact manifold X with an exotic sphere can never change the differential structure of xn, n > 4. Therefore we do not have an exact analogue of Theorem 7 for the finite volume noncompact case. Still, Farrell and Jones [20] considered Dehn surgery along a properly embedded tube §l X ][])n-l in Mn to prove: THEOREM 10. For every integer n such that 8 n - 1 i= 0 and any € > 0 there are non-compact Riemannian n-manifolds X and Y with finite volume such that (i) X is homeomorphic to Y. (ii) X is not diffeomorphic to Y. (iii) X is hyperbolic. (iv) Y has sectional curvatures in (-1 - €, -1 + f).

The technique mentioned above (proving Theorem 10) actually gives new cases also for the compact case, i.e., for Theorem 7. Now, the negatively curved manifolds mentioned up to this point were homeomorphic (hence homotopy equivalent) to hyperbolic manifolds. We call these manifolds of hyperbolic homotopy type. In [7] Ardanza gave a version of Theorem 7 for manifolds that are not homotopy equivalent to a closed locally symmetric space; in particular, they do not have a hyperbolic homotopy type. His constructions use branched covers of hyperbolic manifolds. Recall that Gromov and Thurston [32] proved that large branched covers

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P.

ONTANEDA

of hyperbolic manifolds do not have the homotopy type of a closed locally symmetric space. Here is the statement of Ardanza's result: THEOREM 11. For all n = 4k -1, k 2: 2, there exist closed Riemannian n-dimensional manifolds X and Y with negative sectional curvature such that they do not have the homotopy type of a locally symmetric space and (i) X is homeomorphic to Y. (ii) X is not diffeomorphic to Y. Later Farrell and Ontaneda [28] showed that most of the results mentioned in items 1-5 above are also true for examples of non-hyperbolic homotopy type. Up to now the hyperbolic manifolds considered were real hyperbolic manifolds. We now consider Rigidity Questions for complex, quaternionic and Cayley hyperbolic manifolds. These are Riemannian n-manifolds whose universal covers, with the pulled back metric, are isometric to complex hyperbolic space CHm (n = 2m), quaternionic hyperbolic space lHIHm (n = 4m), or Cayley hyperbolic plane ({])H 2 (n = 16), respectively. Recall that these manifolds have sectional curvatures in the interval [-4, -1] and they also are rigid. In fact they satisfy the following stronger Rigidity results, called superrigidity in the quaternionic and Cayley cases. Assume that xn and yn are homeomorphic closed Riemannian manifolds. Then: (a) If X is complex, quaternionic (n = 4m, m 2: 2) or Cayley hyperbolic (n = 16) and Y has sectional curvatures in [-4, -1] then X and Yare isometric. This follows from results proved independently by Hernandez [37] and Yau and Zheng [63]. (b) If X is quaternionic (n = 4m, m 2: 2) or Cayley hyperbolic (n= 16) and Y has nonpositive curvature operator then X and Y are isometric (up to scaling). This follows from results proved Corlette [13]. (c) If X is quaternionic (n = 4m, m 2: 2) or Cayley hyperbolic (n = 16) and the complexified sectional curvatures of Y are nonpositive then X and Y are isometric (up to scaling). This follows from results proved by Mok, Siu and Yeung [45].

REMARK. The conditions in items (a) or (b) for Y imply the condition in (c) for Y. But for the complex case (see item (a)) Farrell and Jones [21] proved that this Rigidity can not be strengthened to requiring that the curvatures lie in the interval [-4 - E, -1 + E], for some E > 0: THEOREM 12. For every integer m of the form 4k + 1, k> 1 and there are closed Riemannian manifolds X 2m and y2m such that (1) X is homeomorphic but not diffeomorphic to Y. (2) X is complex hyperbolic. (3) Y has sectional curvatures in [-4 - E, -1 + E].

E

>0

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Also, Aravinda and Farrell proved similar results (see item (a) above) for the quaternionic [5] and Cayley [4] cases. Here is the statement for the quaternionic version of Theorem 12:

xn

13. For n = 8, 16 any closed quaternionic hyperbolic manifold has a finite sheeted cover Z such that if I::n is an exotic n-sphere then (1) Z is not diffeomorphic to Z#I::. (2) Z#I:: admits a Riemannian metric with negative sectional curvatures.

THEOREM

REMARKS.

1. In the Theorem above the conclusion remains true if we replace Z by any finite sheeted cover of it. 2. The Theorem above holds also for n = 20 with the extra condition: 6I:: =1= 0 in 820. The abelian groups 88, 816 and 820 have orders 2, 2, 24, respectively.

Here is the statement for the Cayley version of Theorem 12: THEOREM 14. Given t: > 0, any closed hyperbolic Cayley manifold X I6 , and the unique exotic 16- sphere I::I6 there is a finite sheeted cover Z of X such that (1) Z is not diffeomorphic to Z#I::. (2) Z#I:: admits a Riemannian metric with sectional curvatures in [-4-t:,-1+t:]. (3) Conclusions 1 and 2 remain true if Z is replaced by any finite sheeted cover of it.

Finally Theorem 13 and 14 together with Corlette's superrigidity result (see item (b) above) were used by Aravinda and Farrell [6] to answer positively the following question posed by Petersen in his text book [51, pp. 239-240]: • Question: Are there any closed smooth manifolds which support a negatively curved Riemannian metric but do not support a Riemannian metric with nonpositive curvature operator? In fact the manifolds Z#I:: of theorems 13 and 14 provide such examples.

5. Classification The idea of studying the space of all Riemannian metrics that satisfy some property is a very natural one and in this section we consider this, i.e., the Classification Question (1.6) mentioned in the Introduction, for the property negative (sectional) curvature. That is, we want to study the space of all negatively curved Riemannian metrics on a manifold. Let us introduce some notation.

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Let X be a closed smooth manifold. We will denote by MET(X) the space of all Riemannian metrics on X and we will consider MET(X) with the smooth topology. Note that the space MET(X) is contractible. A subspace of metrics whose sectional curvatures lie in some interval (closed, open, semi-open) will be denoted by placing a superscript on MET(X). For example, METsec O. In particular T€(X), 0 < f ~ 00, is in general not contractible. Here is a more detailed statement of this result: THEOREM

17. For every integer ko ~ 1 there is an integer no

= no(ko)

such that the following holds. Given f > 0 and a closed real hyperbolic n-manifold X with n ~ no, there is a finite sheeted cover Z of X such that, for every 1 ~ k ~ ko with n + k == 2 mod4, the map 7rk(T€(Z)) --+ 7rk(T(Z)), induced by the inclusion T€(Z) Y T(Z), is non-zero. Consequently 7rk(T€(Z)) =10. In particular, TO(Z) is not contractible, for every 8 such that f ~ 8 ~ 00 (provided ko ~ 4).

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The homotopy groups 7rk in Theorem 17 are based at the class of the hyperbolic metric on Z. REMARK.

As a Corollary of (proof of the) Theorem 17 we get: COROLLARY. Let X be a closed real hyperbolic manifold of dimension n, n ;::: 10. Assume that en+! =f O. Then for every € > 0 there is a finite sheeted cover Z of X such that 7rl('P{Z)) =f o. Therefore 'P{Z) is not contmctible.

References [I] S.1. Al'ber, Spaces of mappings into manifold of negative curvature, Dokl. Akad. Nauk USSR, 168 (1968), 13-16. [2] D. Anderson and W.-C. Hsiang, The functors Ki and pseudoisotopies of polyhedra, Ann. of Math. (2), 105 (1977), 201-223. [3] C.S. Aravinda and F.T. Farrell, Rank 1 aspherical manifolds which do not support any nonpositively curved metric, Comm. Anal. Geom., 2 (1994), 65-78. [4] C.S. Aravinda and F.T. Farrell, Exotic negatively curved structures on Cayley hyperbolic manifolds, J. Differential Geom., 63 (2003), 41-62. [5] C.S. Aravinda and F.T. Farrell, Exotic structures and quaternionic hyperbolic manifolds, in 'Algebraic Groups and Arithmetic', 507-524, Tata Inst. Fund. Res., Mumbai, 2004. [6] C.S. Aravinda and F.T. Farrell, Nonpositivity: curvature vs. curvature operator, Proc. AMS, 132 (2005), 191-192. [7] S. Ardanza, Ph.D. Thesis, Binghamton University, 2000. [8] Bieberbach, Uber die Bewegungsgruppen der Euklidischen Riiume II, die Gruppen mit einen Pundamentalbereich, Math. Ann., 72 (1912),400-412. [9] A.L. Besse, Einstein Manifolds, Ergebnisse Series, 10, Springer-Verlag, Berlin, 1987. [10] G. Besson, G. Courtois, and S. Gallot, Minimal entropy and Mostow's rigidity Theorems, Ergodic Theory & Dynam. Sys., 16 (1996), 623-649. [11] W. Browder, On the action of e n (811"), Differential and Combinatorial Topology, Princeton University Press, Princeton, NJ, 1965, 23-36. [12] K. Burns and A. Katok, Manifolds with nonpositive curvature, Ergodic Theory & Dynam. Sys., 5 (1985), 307-317. [13] K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math., 135 (1992), 165-182. [14] C.J. Earle and J. Eells, Deformations of Riemannian surfaces, LNM, 102, SpringerVerlag, Berlin, 1969, 122-149. [15] J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. [16] F.T. Farrell and L.E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc., 2 (1989), 899--908. [17] F.T. Farrell and L.E. Jones, Rigidity in geometry and topology, Proc. of the ICM, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991,653-663. [18] F.T. Farrell and L.E. Jones, Compact infrasolvmanifolds are smoothly rigid, in 'Geometry from the Pacific Rim', edited by Berrick, Loo and Wang, de Gruyter & Co., Berlin, 1997, 85-97. [19] F.T. Farrell and L.E. Jones, Exotic smoothings of hyperbolic manifolds which do not support pinched negative curvature, Proc. Amer. Math. Soc., 121 (1994), 627-630. [20] F.T. Farrell and L.E. Jones, Nonuniform hyperbolic lattices and exotic smooth structures, J. Differential Geom., 38 (1993), 235-261.

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