Topics in extrinsic geometry of codimension-one foliations

SpringerBriefs in Mathematics For further volumes: http://www.springer.com/series/10030 Vladimir Rovenski • Paweł Wa...

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SpringerBriefs in Mathematics

For further volumes: http://www.springer.com/series/10030

Vladimir Rovenski • Paweł Walczak

Topics in Extrinsic Geometry of Codimension-One Foliations

123

Vladimir Rovenski Department of Mathematics University of Haifa 31905 Haifa, Israel [email protected]

Paweł Walczak Department of Mathematics University of Lodz 90238 Lodz, Poland [email protected]

ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4419-9907-8 e-ISBN 978-1-4419-9908-5 DOI 10.1007/978-1-4419-9908-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011931089 Mathematics Subject Classification (2010): 53C12, 35L45, 26B20 © Vladimir Rovenski and Paweł Walczak 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

V. Rovenski dedicates this book to his parents: Ira Rushanova and Yuzef Rovenski P. Walczak dedicates this book to his family.

Foreword

The authors of this work asked me to read it and write a foreword. I did so with pleasure because differential geometry of foliations was one of my research subjects decades ago. Foliations, i.e., partitions into submanifolds of a constant, lower dimension, are beautiful structures on manifolds that encode a lot of geometric information. The topological study of foliations was initiated by Ch. Ehresmann and G. Reeb in the 1940s and soon became a research subject of many mathematicians. In particular, the study of the smooth case and of the differential geometric aspects became an important part of foliation theory, developed in the early stages by B. Reinhart, R. Bott, F. Kamber, Ph. Tondeur, P. Molino, and many others. The present work is a research monograph and is addressed to readers who have enough knowledge of differential and Riemannian geometry. Its first two chapters are devoted to the development of a computational machinery that provides integral and variational formulas for the most general, extrinsic invariants of the leaves of a foliation of a Riemannian manifold. The third chapter defines a very general notion of extrinsic geometric flow and studies the evolution of the leaf-wise Riemannian metric along the trajectories of this flow. The authors give existence theorems and estimations of the maximal evolution time and make a study of soliton solutions. The authors of the present monograph are well known specialists in the field, with previously published books and papers on the differential geometry of foliations of Riemannian manifolds. Here, they succeed a technical tour de force, which will lead to important geometric results in the future and I recommend this work to all those who have an interest in the differential geometry of submanifolds and foliations of Riemannian manifolds. They will find methods and results that bring profit to their research. University of Haifa, Israel

Izu Vaisman

vii

Preface

The subject and the history. Foliation theory is about 60 years old. The notion of a foliation appeared in the 1940s in a series of papers of G. Reeb and Ch. Ehresmann, culminating in the book [40]. Since then, the subject has enjoyed a rapid development. Foliations relate with such topics as vector fields, integrable distributions, almost-product structures, submersions, fiber bundles, pseudogroups, Lie groups actions, and explicit constructions (Hopf and Reeb foliations). Reeb also published a paper [41] on extrinsic geometry of foliation in which he proved that the integral of the mean curvature of the leaves of any codimension-one foliation on any closed Riemannian manifold equals zero. By extrinsic geometry we mean properties of foliations on Riemannian manifolds which can be expressed in terms of the second fundamental form of the leaves and its invariants (principal curvatures, scalar mean curvature, higher mean curvatures, and so on). More precisely, if F is a smooth foliation of a Riemannian manifold (M, g) then the second fundamental forms BL of all the leaves {L} of F provide a vector-valued symmetric tensor B on M defined by: B(X ,Y ) = (∇X Y )⊥ , where ∇ is the Levi-Civita connection on (M, g), X and Y are tangent to F , and ( · )⊥ denotes the projection of the tangent bundle T M onto the orthogonal complement T ⊥ F of the bundle T F consisting of all the vectors tangent to (the leaves of) F . The tensor B can be extended to the whole tangent bundle of M by B(N, ·) = 0 whenever N is orthogonal to F . If F is of codimension 1 and transversely oriented, B induces a symmetric scalar (0, 2)-tensor field b (the second fundamental form) given by b(X ,Y ) = g(B(X ,Y ), N) for all X and Y . All the properties of F which can be expressed in terms of B (respectively, b) belong to extrinsic geometry. For example, a foliation F is called

ix

x

Preface

totally geodesic when B ≡ 0, minimal when the mean curvature vector H = 1n Tr g (B) of F vanishes, umbilical when B(X ,Y ) = H · g(X ,Y ) for all X ,Y ∈ T F , and so on. One of the principal problems of extrinsic geometry of foliations reads as follows: Given a foliation F on a manifold M and an extrinsic geometric property (P), does there exist a Riemannian metric g on M such that F enjoys (P) with respect to g? Similarly, one may ask the following, analogous question: Given a manifold M and an extrinsic geometric property (P), does there exist a foliation F and a Riemannian metric g on M such that F enjoys (P) with respect to g? Such problems (first posed by H. Gluck for geodesic foliations) were studied already in the 1970s when Sullivan [50] provided a topological condition (called topological tautness) for a foliation, equivalent to geometrical tautness, that is existence of a Riemannian metric making all the leaves minimal. From classical theorem of Novikov [32] and results of Sullivan, it follows directly that the threedimensional sphere S3 admits no two-dimensional foliations which are minimal with respect to any Riemannian metric. For instance, there is no metric making a Reeb foliation FR on a three-dimensional sphere minimal. Umbilizable foliations on M 3 are transversely holomorphic, hence, see [11]: If a closed orientable M 3 admits an umbilical foliation then it is diffeomorphic to the total space of a Seifert fibration (all one-dimensional leaves are closed) or of a torus bundle over the circle. For example, since S3 is the total space of a Seifert fibration, there exist metrics making a Reeb foliation (S3 , FR ) umbilical. Another example of this type may be found in a recent paper by Langevin and the second author [30]: closed Riemannian spaces of negative Ricci curvature admit no codimension-1 umbilical foliations. In recent decades, several tools providing results of this sort have been developed. Among them, one may find Sullivan’s [49] foliated cycles and several Integral Formulae ([3, 9, 45, 46, 54], etc.), the very first of which is G. Reeb’s vanishing of the integral of the mean curvature mentioned earlier. The authors also have been interested in extrinsic geometry of foliations for a long time (see, for example, [42–44, 54–58]) and this work is, in some sense, a continuation of this interest. The contents. The book includes several topics in Extrinsic Geometry of Foliations. The first topic presented in the book (Chap. 1) is a series of new Integral Formulae, for a codimension-one foliation on a closed Riemannian manifold. The formulae depend on the Weingarten operator, the Riemannian curvature tensor (e.g., Jacobi operator), and their scalar invariants. Integral formulae begin with the classic formula by Reeb, for manifolds of constant curvature they reduce, to known formulae by Brito et al. [9], and Asimov [4]. Integral formulae can be useful for the following problems: prescribing higher mean curvatures (or other symmetric functions of principal curvatures) of foliations; minimizing volume and energy

Preface

xi

defined for vector or plane fields on manifolds; existence of foliations whose leaves enjoy a given geometric property such as being totally geodesic, umbilical, minimal, etc. The central topic of the book is Extrinsic Geometric Flow (EGF, for short, see Chap. 3) on foliated manifolds (M, F ), codim F = 1, which may provide more results on geometry of foliations. EGFs arise as solutions to the partial differential equation (PDE) ∂t gt = h(bt ), where (gt ), t ∈ [0, T ), are Riemannian metrics on M along the leaves and h(bt ) the symmetric (0, 2)-tensors along the leaves expressed in terms of the second fundamental form bt of F on (M, gt ); h(bt ) being identically zero in the direction orthogonal to F . In particular, EGF – for suitable choice of the right-hand side in the EGF equation – may provide families (gt ) of Riemannian structures on a given foliated manifold (M, F ) converging as t → T to a metric gT for which F satisfies a given geometric property (P), say, is umbilical, minimal, or just totally geodesic. A Geometric Flow is an evolution of a given geometric structure under a differential equation associated to a functional on a manifold which has geometric interpretation, usually associated with some (either extrinsic or intrinsic) curvature. Geometric flows play an essential role in many fields of mathematics and physics. They all correspond to dynamical systems in the infinite dimensional space of all possible geometric structures (of given type) on a given manifold. The strong interest of scientists in GF of various types is demonstrated by Annual International Workshops (GF in Mathematics and Physics, 2006 – 2011, BIRS Banff; GF in finite or infinite dimension, 2011, CIRM; Geometric Evolution Equations, 2011, University of Constance; GF and Geometric Operators, 2009, Centro De Giorgi, Pisa, and so on). To some extent, the idea of EGF is analogous to that of the famous Ricci flow. In the Ricci flow equation, the configuration space is a single manifold and the Riemannian structures are deformed by quantities which belong to intrinsic geometry, in the case of EGFs, the configuration space is a foliated manifold while the Riemannian structures are deformed by invariants of extrinsic geometry. In both cases, the (EGF or Ricci flow) equation makes sense because both its sides are symmetric tensors of the same type. Notice that the study of the Ricci flow provided the proof of outstanding conjectures: Poincaré Conjecture and Thurston Geometrization Conjecture. To apply EGF to various problems of extrinsic geometry, one needs variational formulae (see Chap. 2) which express variation of different quantities belonging to extrinsic geometry of a fixed foliation under variation of the Riemannian structure of the ambient manifold. Also, some special solutions (called extrinsic geometric solitons here, EGS, for short, see Sect. 3.8) of the EGF equation are of great interest because, in several cases, they provide Riemannian structures with very particular geometric properties of the leaves.

xii

Preface

Throughout the book, (M n+1 , gt ) is a Riemannian manifold with a codimension one transversely oriented foliation F , ∇t the Levi-Civita connection of gt , 2 gt (∇tX Y, Z) = X (gt (Y, Z)) + Y (gt (X , Z)) − Z(gt (X ,Y )) +gt ([X ,Y ], Z) − gt ([X , Z],Y ) − gt ([Y, Z], X ) for all the vector fields X ,Y, Z on M, N the positively oriented unit normal to F with respect to any gt , A : X ∈ T F → −∇tX N the Weingarten operator of the leaves, which we extend to a (1, 1)-tensor field on T M by A(N) = 0. Observe that the difference of two connections is always a tensor, hence Πt := ∂t ∇t is a (1, 2)-tensor field on (M, gt ). Differentiating with respect to t the above classical formula yields the known formula, which allows us to express Πt by: 2 gt (Πt (X,Y ), Z) = (∇tX S)(Y, Z) + (∇Yt S)(X , Z) − (∇tZ S)(X ,Y ), where S = ∂t gt is time-dependent symmetric (0,2)-tensor field and X ,Y, Z ∈ T M. The definition of the F -truncated (r, k)-tensor field Sˆ (where r = 0, 1, and denotes the T F -component) will be helpful in Chaps. 2 and 3, ˆ 1 , . . . , Xk ) = S(Xˆ1 , . . . , Xˆk ) S(X

(Xi ∈ T M).

Acknowledgments The authors would like to thank their colleagues, David Blanc and Izu Vaisman (Mathematical Department, University of Haifa), Krzysztof Andrzejewski, Wojciech Kozłowski, Kamil Niedziałomski and Szymon Walczak (Faculty of Mathematics and Computer Science, University of Łód´z) for helpful corrections concerning the manuscript. The authors warmly thank Elizabeth Loew and Ann Kostant for support in the publishing process. The authors are also greatly indebted to Marie-Curie actions support of their research by grants EU-FP7PEOPLE-2007-IEF, No. 219696 and EU-FP7-PEOPLE-2010-RG, No. 276919.

Haifa – Łodz

Vladimir Rovenski Paweł Walczak

Contents

1

Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Hypersurfaces of Riemannian Manifolds ... . . . . . . . . . . . . . . . . . . . 1.2.2 Invariants of the Weingarten Operator .. . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Leaf-Wise Divergence of Operators Ak and Tr (A) . . . . . . . . . . . . 1.2.4 Leaf-Wise Divergence of Vector Fields h(A)Z and Tr (A)Z . . . 1.3 Integral Formulae for Codimension-One Foliations .. . . . . . . . . . . . . . . . . . 1.3.1 New Integral Formulae . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Some Consequences of Integral Formulae . . . . . . . . . . . . . . . . . . . . 1.3.3 Foliations Whose Leaves Have Constant σ2 . . . . . . . . . . . . . . . . . .

1 1 4 4 5 8 11 14 14 15 17

2 Variational Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Biregular Foliated Coordinates.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Foliations with a Time-Dependent Metric .. . . . . . . . . . . . . . . . . . . . 2.2.3 A Differential Operator .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Variational Formulae for Codimension-One Foliations . . . . . . . . . . . . . . . 2.3.1 Variations of Extrinsic Geometric Quantities . . . . . . . . . . . . . . . . . 2.3.2 Variations of General Functionals.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Variations of Particular Functionals.. . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Applications and Examples .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Variational Formulae for Umbilical Foliations . . . . . . . . . . . . . . . . 2.4.2 The Energy and Bending of the Unit Normal Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19 19 20 20 24 26 31 31 35 43 47 47

3 Extrinsic Geometric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Systems of PDEs Related to EGFs . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 55

49

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Contents

3.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 58 3.3.1 Diffeomorphism Invariance of EGFs . . . . . .. . . . . . . . . . . . . . . . . . . . 58 3.3.2 Quasi-Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 59 3.3.3 Generalized Companion Matrices . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 60 3.4 Existence and Uniqueness Results (Main Theorems) . . . . . . . . . . . . . . . . . 67 3.5 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69 3.5.1 Searching for Power Sums . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 70 3.5.2 Local Existence of Metrics (Proofs of the Main Theorems) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 72 3.5.3 Proofs of the Corollaries.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 75 3.6 Global Existence of EGFs (Time Estimation) . . . . .. . . . . . . . . . . . . . . . . . . . 77 3.7 Variational Formulae for EGFs . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 80 3.7.1 The Normalized EGFs . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 3.7.2 First Derivatives of Functionals .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 83 3.8 Extrinsic Geometric Solitons . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85 3.8.1 Introducing EGS . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 86 3.8.2 Canonical Form of EGS . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 3.8.3 Umbilical EGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 3.9 Applications and Examples .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 3.9.1 Extrinsic Ricci Flow .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 3.9.2 Extrinsic Ricci Solitons . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 3.9.3 EGS on Foliated Surfaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 3.9.4 EGS on Hypersurfaces of Revolution.. . . . .. . . . . . . . . . . . . . . . . . . . 106 References . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109 Index . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113

Acronyms

EGF EGS VF IF gt (M, gt ) F , Lα N A, b gˆ g⊥ bˆ j h(b) ∇t σk τk → − τ Γi kj λ ki Λlk (M) , M (M, F , N) R, Ric RN

Extrinsic Geometric Flow Extrinsic Geometric Soliton Variation Formula Integral Formula t-dependent Riemannian metric Riemannian manifold (with a t-dependent metric) A codimension-one foliation and its leaves Unit normal to (the leaves of) a foliation F Weingarten operator and 2-nd fundamental form for F with respect to N F -truncated metric tensor N-component of the metric g F -truncated tensor dual to A j The symmetric (0, 2) tensor expressed in terms of b The Levi-Civita connection for gt k-th elementary symmetric function of A k-th power sum (symmetric function) of the eigenvalues of A The vector function (τ1 , . . . , τn ) Christoffel symbols for the metric g the (musical) isomorphism : T M → T ∗ M, i.e., X = g(X, ·) The (musical) isomorphism : T ∗ M → T M The normal curvature of an umbilical foliation The principal curvatures of the leaves The bundle of F -truncated (k, l)-tensors on (M, g) The inner product of tensors The space of Riemannian metrics on M of finite volume with N being a unit normal to F The Riemannian curvature and the Ricci tensors = R(·, N)N – the Jacobi operator. Indeed, Tr RN = Ric(N, N)

xv

Chapter 1

Integral Formulae

Abstract The chapter presents a series of new Integral Formulae (IF) for a codimension-one foliation on a closed Riemannian manifold. The proof of IF is based on the Divergence Theorem. The IF start from the formula by Reeb, for foliations on space forms they generalize the classical ones by Asimov, Brito, Langevin, and Rosenberg. Our IF include also a set of arbitrary functions f j depending on the scalar invariants of the Weingarten operator. For a special choice of auxiliary functions the IF involve the Newton transformations of the Weingarten operator. We apply IF to umbilical foliations and foliations whose leaves have constant second-order mean curvature.

1.1 Introduction Here, for the readers’ convenience, we provide the following standard definition from foliation theory, see [12]. Definition 1.1. A family F = {Lα }α ∈A of connected subsets of a manifold M m is said to be an n-dimensional foliation, Fig. 1.1, if:

(1) α ∈A Lα = M m . (2) α = β ⇒ Lα Lβ = 0. / (3) For any point q ∈ M there exists a Cr -chart (local coordinate system) ϕq : Uq → Rm such that q ∈ Uq , ϕq (q) = 0, and if Uq Lα = 0/ the connected components of the sets ϕq (Uq Lα ) are given by equations xn+1 = cn+1 , . . . , xm = cm , where c j ’s are constants. The sets Lα are immersed submanifolds of M called leaves of F . The family of all the vectors tangent to the leaves is the integrable subbundle of T M denoted by T F . If M carries a Riemannian structure, T F ⊥ denotes the subbundle of all the vectors orthogonal to the leaves. A foliation F is said to be orientable (respectively, transversely orientable) if the bundle T F (respectively, T F ⊥ ) is orientable. V. Rovenski and P. Walczak, Topics in Extrinsic Geometry of Codimension-One Foliations, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4419-9908-5 1, © Vladimir Rovenski and Paweł Walczak 2011

1

2

1 Integral Formulae

Fig. 1.1 Foliation

From the definition earlier, it follows that the bundle T F is involutive, that is the Lie bracket [X,Y ] of two sections of T F is a section of T F again. Let us recall that the classical Frobenius Theorem says that any distribution (that is, a subbundle of T M) D on M is involutive if and only if D = T F for some foliation F . Clearly, any one-dimensional distribution is involutive, so tangent to a one-dimensional foliation. The [12, I, Part 2] is a short course on the codimension-one foliations. The [12, II, Part 3] studies compact 3-manifolds foliated by surfaces, a popular topic since the theorem by Novikov on the existence of Reeb components for foliated 3-sphere. Taut foliations are used as powerful tools for studying 3-manifold topology. Due to Thurston and Gabai, a taut foliation is a codimension-one foliation with the property that there is a single transverse circle intersecting every leaf. By a result of D. Sullivan, see [50], a codimension-one foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Let (M, g) be a Riemannian manifold with the metric g and the Levi-Civita connection ∇. In particular, we have ∇g = 0. The manifold M will be always closed (i.e., compact and without boundary). The Riemannian curvature tensor is given by: X,Y,V ∈ Γ (T M).

R(X,Y )V = (∇X ∇Y − ∇Y ∇X − ∇[X,Y ] )V,

Its trace (with respect to the metric g) is called the Ricci curvature tensor. If (ei ) are local orthonormal vector fields on M then Ric(X,Y ) = ∑i g(R(ei , X)Y, ei ),

X,Y ∈ Γ (T M).

The trace, Scal = ∑i Ric(ei , ei ), is called the scalar curvature. One of the tools to describe the Riemannian curvature is the sectional curvature, Kσ = g(R(X,Y )Y, X), where X,Y is the orthonormal basis of a two-dimensional subspace σ ⊂ T M. It is the Gaussian curvature of surfaces. If M is a space of constant curvature c then R(X,Y )V = c(g(Y,V )X − g(X,V )Y ),

Kσ = c.

Let N be a unit normal vector field of a transversally oriented codimensionone foliation F . Indeed, Ric(N, N) = Tr RN , where RN = R(·, N)N is the Jacobi operator.

1.1 Introduction

3

By integral formula (IF) we mean the vanishing of the integral over M of an expression depending on the Weingarten operator of F , the Riemannian curvature tensor of (M, g), and their scalar invariants. The total higher-order mean curvatures and power sums are the integrals Iσ , j =

M

σ j d vol,

Iτ , j =

M

τ j d vol,

j = 1, 2, . . .

(1.1)

see the definition of σ j and τ j in Section 1.2.2. Coming back to the main topic of this chapter, let us recall that the first known integral formula (for codimension-one foliations) belongs to Reeb [41], M

H d vol = 0,

(1.2)

where H is the mean curvature of leaves. By (1.2) we have Iτ ,1 = Iσ ,1 = 0. The proof of (1.2) is based on the Divergence Theorem, and the identity div N = −nH. Consequently, if a foliation has a projectable mean curvature function then, according to (1.2), one of its leaves has mean curvature equal to zero, so this leaf is minimal. Also, (1.2) and its counterparts for foliated domains with boundary provide the only obstructions for a function f on a compact foliated manifold (M, F ) to become the mean curvature σ1 of F with respect to some Riemannian metric on M. The conditions which are necessary and sufficient in this case read [33]: either f ≡ 0 or there exist points x, y ∈ M for which f (x) f (y) < 0, and, in any positive (respectively, negative) foliated domain D ⊂ M there exists a point x such that f (x) > 0 (respectively, f (x) < 0). The next well-known integral formula in the series for total σ ’s is M

2 σ2 − Ric(N, N) d vol = 0.

(1.3)

For n = 1 and dim M = 2, we have σ2 = 0, and (1.3) reduces to the integral of Gaussian curvature, M K d vol = 0, and means just that the Euler characteristic of M 2 with a unit vector field N is zero. The formula (1.3) follows from a result in [54] (for the foliation case, see also [39]) and has many applications. For example, one can see directly that it implies nonexistence of umbilical foliations on closed manifolds of negative curvature: if the integrand is strictly positive, so is the value of the integral. In [30], (1.3), together with some standard tools of analysis (Hölder inequality and so on), was used to show that codimension-one foliations of closed negatively Ricci-curved manifolds are “far” (in some sense) from being umbilical. Brito et al. [9], extending the result by D. Asimov [4] for gaussian curvature, have shown that the integrals Iσ , j on a compact space form M n+1 (c) do not depend on F : they depend on n, j, c and the vol(M) only, Iσ , j =

c j/2

n/2

j/2

0,

vol(M),

n and j even either n or j odd.

(1.4)

4

1 Integral Formulae

Example 1.1. Here is an amazing consequence of (1.4) for any sufficiently smooth codimension-one foliation on the round unit sphere S3 (communicated to the authors by D. Asimov in 2008). By S. Novikov’s theorem, any such foliation contains a leaf diffeomorphic to a torus. So, by Gauss-Bonnet theorem, there is a point with zero Gaussian leaf curvature KF . By (1.4) for n = j = 2 and c = 1, the average of σ2 is 1. Because KF = 1 + σ2 , there is a point, where KF > 1 + 1 = 2. Hence, the set of values of the function KF : S3 → R contains the interval [0, 2 + ε ] for some ε > 0. In this direction, following [3] and [44], we present a series of new IF (for symmetric functions σ ’s and τ ’s), which start from (1.2) and (1.3). For manifolds of constant curvature the IF are reduced (by a simple choice of functions f j ) to (1.4). IF are useful for several problems: prescribing mean curvatures σ j (or other symmetric functions of the principal curvatures ki ) of foliations, minimizing volume and energy defined for vector fields on manifolds, and existence of foliations whose leaves enjoy a given geometric property such as being minimal, umbilical, etc. (see, e.g., [3], [16], [42], [46], [47], [52], [54] and the bibliographies therein).

1.2 Preliminaries We shall use the Divergence Theorem

M div( f ) d vol

= 0 and the identity

div( f X) = f div X + X( f ) for smooth functions f and vector fields X on M. In some cases we calculate the leaf-wise divergence divF (along F ). The trace of (1, 1)-tensors will be calculated along F . We denote by ⊥ and the normal and tangent to F components of vectors, respectively. In what follows, we briefly write ∇N N = Z.

1.2.1 Hypersurfaces of Riemannian Manifolds When we have an isometric immersion (of manifolds) from L into M, we say that L is an immersed submanifold of M. If, in addition, the inclusion is an embedding then L is said to be an embedded submanifold of M. At each point q ∈ L, the inner product g(·, ·) on Tq M induces an inner product on Tq L that we call the induced Riemannian metric and denote it by the same symbol. We will consider always submanifold of a Riemannian manifold with the metric that is induced in this way. We decompose ∇X Y , where X,Y ∈ Γ (T L), into its tangent part (∇X Y ) and its normal part (∇X Y )⊥ . Then the Levi-Civita connection ∇L of L is given by ∇LX Y = (∇X Y ) , and one defines the second fundamental tensor of L by B(X,Y ) = (∇X Y )⊥ .

1.2 Preliminaries

5

We shall consider codimension-one (transversally oriented) submanifolds, called hypersurfaces. Let N be a unit positive oriented normal vector field of L. We call b defined by: b(X,Y ) = g(B(X,Y ), N) the scalar second fundamental form. This gives the orthogonal decomposition: ∇X Y = ∇LX Y + B(X,Y ) = ∇LX Y + b(X,Y )N, called the Gauss formula. The tensor field A = −(∇X N) is called the Weingarten (shape) operator of L (in direction N), and is related to b by b(X,Y ) = g(AX,Y ). The symmetry of B implies that A is self-adjoint. The orthogonal decomposition ∇LX N = (∇X N)⊥ − AX is known as the Weingarten formula. The formulae of Gauss and Weingarten can be seen as first-order differential equations. The covariant derivatives of the second fundamental form and of the Weingarten operator are given by the formulas: ⊥ (∇⊥ X B)(Y,V ) = ∇X B(Y,V ) − B(∇X Y,V ) − B(Y, ∇X V ),

(∇X A)Y = ∇X (AY ) − A(∇X Y ). Using the formulae of Gauss and Weingarten, one can obtain that ⊥ (R(X,Y )V )⊥ = (∇⊥ X B)(Y,V ) − (∇Y B)(X,V ),

(R(X,Y )V ) = RL (X,Y )V + g(AX,V)AY − g(AY,V )AX.

(1.5)

The first equation of (1.5) is called the Gauss equation, the second one the Codazzi equation. Notice that (R(X,Y )V )⊥ = 0 when M is a space of constant curvature. The difference Rmex (X,Y )V = RL (X,Y )V − (R(X,Y )V ) is called the extrinsic Riemannian curvature of L, see Sect. 3.9.1. The extrinsic sectional, Ricci, and scalar curvatures are given, respectively, by: K ex (X,Y ) = K L (X,Y ) − K(X,Y ) = g(AX, X)g(AY,Y ) − g(AX,Y)2 , Ricex (X,Y ) = g(AX, AY ) − (Tr A) g(AX,Y ), Scalex = Tr Ricex = (Tr A)2 − Tr (A2 ).

1.2.2 Invariants of the Weingarten Operator Power sums of the principal curvatures k1 , . . . , kn (the eigenvalues of A) are given by:

τ j = k1j + . . . + knj = Tr (A j ),

j ≥ 0.

6

1 Integral Formulae

The τ ’s can be expressed using the elementary symmetric functions σ1 , . . . , σn

σ j = ∑i

1 n), which in matrix form are: ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ σ1 τ1 1 ⎟ ⎜ τ1 ⎜σ2 ⎟ ⎜ τ2 ⎟ −2 ⎟. ⎜ ⎟ ⎜ ⎟ Tn ⎜ ⎠ ⎝. . .⎠ = ⎝. . .⎠ , where Tn = ⎝ . . . ... ... ... n n+1 σn τn τn−1 −τn−2 . . . (−1) τ1 (−1) n Hence, the σ ’s can be expressed in terms of the τ ’s using Tn−1 . Moreover, we have ⎞ σ1 1 0 ... ... ⎜ 2 σ2 σ1 1 0 . . .⎟ ⎟, τi = det ⎜ ⎠ ⎝ ··· ··· ··· ··· i σi σi−1 . . . σ2 σ1 ⎛

i = 1, 2, . . .

These formulae can be used to express the τn+i ’s as polynomials of τ1 , . . . , τn . Many authors investigated recently higher-order mean curvatures of hypersurfaces using the Newton transformations of the shape operator. Newton transformations of the shape operator have been applied successfully to foliations, see [3, 44] and [47]. Newton transformations Tr (A) are defined either inductively by: T0 (A) = id,

− A · Tr−1(A), Tr (A) = σr id

1 ≤ r ≤ n,

or explicitly as: r − σr−1 A + . . . + (−1)r Ar , Tr (A) = ∑i=0 (−1)i σr−i Ai = σr id

0 ≤ r ≤ n. (1.6)

Notice that A and Tr (A) commute. By the Cayley-Hamilton Theorem, Tn (A) = 0. Let λ = (λ1 , . . . , λm ) ∈ Zm + . The generalized mean curvatures σλ (A1 , . . . Am ) of a set of n × n matrices A1 , . . . Am are defined in [45] by: det(In + t1 A1 + . . . + tm Am ) = ∑ λ

1 +...+λm ≤n

σλ (A1 , . . . Am )t1λ1 · . . . · tmλm .

For example, when m = 1, we have det(In + tA) = ∑ r≤n σr (A)t r .

(1.7)

1.2 Preliminaries

7

Lemma 1.1 (see [47]). If B(t) (t ≥ 0) is a smooth family of n × n matrices then ˙ B) (for all r ≤ n). σ˙ r (B) = σ1, r−1 (B,

(1.8)

Proof. Let B(t) = B + t B˙ + o(t), hence ˙ + o(t). ∑ r ≤n σr (B(t)) sr = det(In + sB(t)) = det(In + s(B + t B)) By the definition (1.7), we have ˙ = det(In + st B˙ + sB) det(In + s(B + t B)) ˙ B) sr t + o(t). = ∑ r ≤n σr (B) sr + ∑ r 0 we have

σ r,l (B,C) = σr (B) σl (C) − ∑ i=1

min(r, l)

σ r−i, l−i, i (B, C, BC).

(1.9)

In particular,

σr,1 (B,C) = ∑i=0 (−1)i σr−i (B) Tr (BiC) = Tr (Tr−1 (B)C). r

(1.10)

Proof. Comparing the determinant det(In + tB + sC + ts BC) with the product of the determinants det(In + tB) det(In + sC) we get the equality

∑ r,l=1 σr (B) σl (C)t r sl = ∑ a,b,c=1 σa,b,c(B,C, BC)t a sb (st)c n

n

n min(r,l) = ∑ r, l=1 t r sl σr, l (B,C) + ∑ i=1 σr−i,l−i,i (B,C, BC) ,

which implies relation (1.9). From (1.9) with l = 1 we conclude that σr,1 (B,C) = σr (B) σ1 (C) − σr−1,1(B, BC). From the above, by induction, we obtain (1.10).

Remark 1.2. Let B(t) be a smooth family of n-by-n matrices with the symmetric ˙ j−1 for j > 1, by induction functions τ j = Tr B j . Using the identity B˙ j = BB˙ j−1 + BB j j i−1 j−i ˙ we find B˙ = ∑i=1 B BB . By the property Tr (AB) = Tr (BA) and the fact that for matrices the trace commutes with derivatives, we conclude that ˙ τ˙ j (B) = Tr (B˙ j ) = j Tr (B j−1 B).

(1.11)

By (1.8) and (1.10), the elementary symmetric functions of matrices B(t) satisfy ˙ σ˙ r = Tr (Tr−1 (B)B),

r = 1, 2, . . . , n.

(1.12)

8

1 Integral Formulae

Lemma 1.3 (see [3]). For the Weingarten operator A we have Tr Tr (A) = (n − r) σr , Tr (A · Tr (A)) = (r + 1) σr+1 , Tr (A2 · Tr (A)) = σ1 σr+1 − (r + 2) σr+2 , Tr (Tr−1 (A)(∇F X A)) = X(σr ),

X ∈ Γ (D).

Proof. The first three algebraic properties follow from Newton formulae of Remark 1.1. The last identity follows from (1.12).

In this chapter, we shall consider also integrals of the quantities involving the vector field h(A)Z, where h is of the form described later. Let b j be a symmetric → (0, 2)-tensor dual to A j . Given functions f j = f˜j (− τ ), define a symmetric (0, 2)tensor h(b) and its dual, a self-adjoint (1, 1)-tensor h(A), by h(b) = ∑ j=0 f j b j , n−1

h(A) = ∑ j=0 f j A j . n−1

(1.13)

The choice of h(b) appears to be natural: the powers b j are the only (0, 2)-tensors which can be obtained algebraically from the second fundamental form b, while τ1 , . . . , τn (or, equivalently, σ1 , . . . , σn ) generate all the scalar invariants of extrinsic geometry. For example, the Newton transformation Tr (A) depends on all A j ( j ≤ r).

1.2.3 Leaf-Wise Divergence of Operators Ak and Tr (A) Let {ei } (i ≤ n) be a local orthonormal frame of F . If S is a (1, j + 1)-tensor field S on M, the divergence div S is the (1, j)-tensor div S(Y1 , . . . ,Y j ) = divF S(Y1 , . . . ,Y j ) + (∇N S)(N,Y1 , . . . ,Y j ), where the partial divergence of S (i.e., along F ) is a (1, j)-tensor divF S(Y1 , . . . ,Y j ) = ∑i≤n (∇ei S)(ei ,Y1 , . . . ,Y j ). The covariant derivative of the (1, j)-tensor S is the (1, j + 1)-tensor given by: (∇S)(X,Y1 , . . . ,Y j ) = (∇X S)(Y1 , . . . ,Y j ) = ∇X (S(Y1 , . . . ,Y j )) − ∑i≤ j S(Y1 , . . . , ∇X Yi , . . .Y j ). For example, if S is a (1, 1)-tensor (a linear operator) then div S is a vector field. Indeed, the divergence divX of a vector field X is the scalar function div(X ) on M,

1.2 Preliminaries

9

where X = g(X, ·) is the 1-form (i.e., the (0, 1)-tensor) dual to X. (The “musical” isomorphism : T M → T ∗ M sends a vector X = X i ∂i to X = Xi dxi = gi j X j dxi ). For any X,Y ∈ T M, define a linear operator R X,Y : T F → T F by R X,Y : V → (R(V, X)Y )

(V ∈ T F ),

(1.14)

where R is the Riemannian curvature tensor. For short, write R N = R N,N and call it the Jacobi operator. Lemma 1.4. The leafwise divergence of Ak , k > 0, satisfies the inductive formula divF (Ak ) = A(divF Ak−1 ) + (1/k)∇F τk − ∑ i≤n R(N, Ak−1 ei ) ei . Equivalently, divF (A )= ∑1≤ j≤k k

(1.15)

1 j−1 j−1 F k− j A ∇ τk− j+1 − ∑1≤i≤n AN R N, A ei ei . k− j+1 (1.16)

Moreover, for any vector field X ⊥ N, we have for k > 0 g(divF Ak , X) = ∑1≤ j≤k

1 . A j−1 X(τk− j+1 ) − Tr F Ak− j RA j−1 X,N N k− j+1 (1.17)

Proof. By the Codazzi equation, see (1.5)2, we have (∇X A)Y − (∇Y A)X = −R(X,Y )N.

(1.18)

In order to verify (1.16), we decompose Ak = A · Ak−1 for k > 1, and calculate divF Ak = ∑ i≤n (∇ei Ak ) ei = A(divF Ak−1 ) + ∑ i≤n (∇ei A)Ak−1 ei .

(1.19)

Using (1.18), integrability of T F which yields [ei , X]⊥ = 0, and symmetries of the curvature tensor, we find for X tangent to F ,

∑ i≤n g((∇ei A)(Ak−1 ei ), X) = ∑ i≤n g(Ak−1ei , (∇ei A)X) = ∑ i≤n g(Ak−1 ei , (∇X A) ei − R(ei , X)N = Tr (Ak−1 (∇X A)) − ∑ i≤n g(R(N, Ak−1 ei )ei , X). For X ⊥ N, the equality (1.19) gives us g(divF Ak , X) = g(A(divF Ak−1 ), X) + Tr (Ak−1 (∇X A)) − ∑ g(R(N, Ak−1 ei )ei , X). i≤n

10

1 Integral Formulae

The above and the identity 1 Tr (Ak−1 (∇X A)) = X(τk ) k (for k > 0) yield (1.15), see Remark 1.2. By induction, (1.16) follows from (1.15). Finally, we conclude that (1.17) is a consequence of (1.16) and g

∑ A j−1R(N, Ak− j ei )e i , X

=g

i≤n

= Tr F (A

k− j

∑ Ak− j RA j−1 X,N ei , ei

i≤n

RA j−1 X,N ).

Notice that by Lemma 1.4 we have

n−1 divF h(A) = ∑ k=0 Ak ∇F fk + fk divF Ak . Using Lemma 1.4 with f j = (−1) j σr− j ( j ≤ r), we deduce the following claim for Newton transformations of A. Lemma 1.5. The leafwise divergence of Tr (A) satisfies the inductive formula divF Tr (A) = −A(divF Tr−1 (A)) + ∑ i≤n R(N, Tr−1 (A)ei )ei ,

r > 0.

Equivalently, divF Tr (A) for r > 0 is given by the formula: divF Tr (A) = ∑1≤ j≤r ∑ i≤n (−A) j−1 R(N, Tr− j (A)ei )e i .

(1.20)

Moreover, for any vector field X ∈ Γ (F ), we have g(divF Tr (A), X) =

Tr F Tr− j (A)R(−A) j−1 X, N −g([Tr− j (A)∇N N, (−A) j−1 X], N) .

∑1≤ j≤r

(1.21)

Proof. Using the inductive definition of Tr (A), we have divF Tr (A) = ∇F σr − A(divF Tr−1 (A)) − ∑ i≤n (∇ei A)Tr−1 (A)ei . Similarly to the proof of Lemma 1.4, using Codazzi’s equation (1.18), we obtain:

∑ i≤n g((∇ei A)Tr−1 (A)ei , X) = ∑ i≤n g(Tr−1 (A)ei , (∇ei A)X)

= ∑ i≤n g Tr−1 (A)ei , (∇X A)ei − R(ei , X)N = Tr (Tr−1 (A)(∇X A))− ∑ i≤n g(R(N,Tr−1 (A)ei )ei , X).

1.2 Preliminaries

11

By Remark 1.2, we have X(σr ) = Tr (Tr−1 (A)∇X A) for any X ∈ Γ (F ). Hence, the inductive formula (1.20) holds. Then (1.20) follows directly. Finally, from the above, it follows that g(divF Tr (A), X) = ∑1≤ j≤r ∑ i≤n g((−A) j−1 R(N, Tr− j (A)ei )e i , X) for every vector field X ∈ Γ (F ). One can obtain (1.21) from the above, using the operator (1.14) and the equality

∑ i≤n g((−A) j−1 R(N, Tr− j (A)ei )e i , X) = Tr F (Tr− j (A)R(−AN ) j−1 X,N ).

1.2.4 Leaf-Wise Divergence of Vector Fields h(A)Z and Tr (A)Z The next lemma is important for Propositions 1.1 and 1.2. Lemma 1.6. Let {ei } be a local orthonormal frame of F around q ∈ M such that ∇F X ei (q) = 0 (X ∈ (T M)q ). Then, at the point q, for Z = ∇N N we have g(∇ei Z, e j ) = (A2 )i j + g(R(ei , N)N, e j ) − (∇N A)i j + g(Z, ei )g(Z, e j ).

(1.22)

Proof. First, observe that − g(Z, ∇ei e j ) = g(∇ei Z, e j ) + g(∇ei N, ∇N e j ) + g(N, ∇ei ∇N e j ).

(1.23)

We have (∇N A)i j = g(Z, ∇ei e j ) + g(N, ∇N ∇ei e j ). Therefore, we obtain at q ∈ M (A2 )i j + g(R(ei , N)N, e j ) − (∇N A)i j = (A2 )i j − g(R(ei , N)e j , N) + N(∇ei ∇N e j ) = (A2 )i j − g(Z, ∇ei e j ) − g(∇ei ∇N e j , N) +g(∇[ei ,N] e j , N).

(1.24)

Using (1.23), conditions at q, and ∇ei N = g(∇ei N, ek )ek ,

∇N ei = g(∇N ei , N)N,

(A2 )i j = g(∇ei N, ek )g(∇ek e j , N),

we can simplify the RHS in the last line in (1.24) as: g(∇ei Z, e j ) − g(Z, ei )g(Z, e j ). From the above it follows (1.22).

12

1 Integral Formulae

Let h(A) be of the form described earlier. Proposition 1.1. Let F be a foliation with a unit normal N on a Riemannian manifold (M, g). Then fk N(τk+1 ) divF (h(A)Z) = g(divF h(A), Z) + ∑ fk τk+2 − k+1 k0 gi j dxi dx j .

(2.2)

As usual, let gi j denote the entries of the matrix inverse to (gi j ) and gi j,k be the derivative of gi j in the direction of ∂k .

2.2 Auxiliary Results

21

Lemma 2.2. For a metric (2.2) in biregular foliated coordinates (of a codimensionone foliation F ) on (M, g), one has √ N = ∂0 / g00

(the unit normal)

Γi0 = (1/2) ∑s gis,0 gs j ,

Γ00i = −(1/2) ∑s g00,s gsi ,

j

1 √ √ bi j = Γi 0j g00 = − gi j,0 / g00 2

(the second fundamental form),

√ −1 Aij = −Γi0j / g00 = √ gis,0 gs j 2 g00 ∑s s

sm (bm )i j = Ass12 . . . Asm−1 m Ai g js1

τi =

−1 √ 2 g00

i

Γi 0j = −gi j,0 /(2 g00),

(the Weingarten operator),

(the mth “power” of bi j ),

∑ {ra }, {sb} gr1 s2 ,0 . . . gri−1 si ,0 gs1 r1 . . . gsiri .

In particular, −1 τ1 = √ ∑ grs,0 grs , 2 g00 r, s

τ2 =

1 4 g00

∑

r1 , r2 , s1 , s2

gr1 s2 ,0 gr2 s1 ,0 gs1 r1 gs2 r2 ,

etc.

Proof. This is standard and left to the reader. For convenience, observe that the formula for A follows from that for b and Aij = ∑s bis gs j . Notice that (Am )ij = s j s sm m s ∑{sl } As2 As23 . . . Asm−1 m Ai . Formulae for bm follow from the above and (A )i gs j = g(Am ei , e j ) = (bm )i j . Formulae for τ ’s follow directly from the above and the equality τi = Tr (Ai ). For example, we apply Lemma 2.2 to the tensor h(b) of (1.13). Proposition 2.1. The tensor h(b) of (1.13) with generic functions f0 , f1 in biregular foliated coordinates has a form: h(b)ac = f0 (0) gac −

f0,τ1 (0) ij g ψ gac + f1 (0) ψac + o([ψac ]), √ i j ∑ i, j 2 g00

(2.3)

where ψi j = gi j,0 . If f2 is also generic and f0 (0) = f0,τ1 (0) = f1 (0) = 0 then the second-order approximation of h(b) is ⎛ f0,τ (0) h(b)ac = ⎝ 2 4 g00

∑

i, r, s1 , s2

gs1 i gs2 r ψrs1 ψis2 +

f0,τ1 τ1 (0) 8 g00

2 ⎞

∑ g i j ψi j

⎠ gac

i, j

f1,τ (0) f2 (0) − √1 gi j ψi j ψac + gi j ψai ψc j + o([ψac ]2 ). (2.4) ∑ i, j 2 g00 4 g00 ∑ i, j

22

2 Variational Formulae

Proof. Take biregular foliated coordinates on M (with the origin at q ∈ M) as in Lemma 2.1. By definition (1.13), the F -components of the tensor are n−1 → h(b)ac = ∑m=0 fm (− τ ) g(Am ea , ec ).

We neglect the third (and more) order terms with ψac = gac,0 . By Lemma 2.2 we have (A2 )ij = ∑r Arj Ari = 4 g100 ∑s1 ,s2 ,r ψrs1 gs1 j ψis2 gs2 r and, in fact, τi = o([ψac ]i−1 ) for i > 0. From above we obtain expansions for f j ( j = 0, 1, 2) as: → f0 (− τ ) = f0 (τ1 , τ2 , 0, . . . , 0) + o([ψac]2 ) = f0 (0) 2 1 1 − √ f0,τ1 (0) ∑ i, j gi j ψi j + f0,τ1 τ1 (0) ∑ i, j gi j ψi j 2 g00 8 g00 +

1 f0,τ2 (0) ∑ i, r, s , s ψrs1 ψis2 gs1 i gs2 r + o([ψac]2 ), 1 2 4 g00

1 → τ ) = f1 (0) − √ f1,τ1 (0) ∑ i, j gi j ψi j + o([ψac ]), f1 (− 2 g00 1 → τ ) = f2 (0) − √ f2,τ1 (0) ∑ i, j gi j ψi j + o([ψac ]). f2 (− 2 g00 Substituting into the function h(b) with bm and τ ’s from Lemma 2.2, h(b)ac = ∑m,s ,r i

j

→ (−1)m fm (− τ ) ψar1 ψs1 r2 . . . ψsm−1 c gs1 r1 . . . gsm−1 rm−1 , √ m (2 g00 )

we obtain the second-order approximation of h(b), see (2.4), h(b)ac ≈ f0 (0) gac −

f0,τ1 (0) ij g ψ gac + f1 (0) ψac √ i j 2 g00 ∑ i, j

2 f0,τ1 τ1 (0) f 0,τ2 (0) s1 i s2 r ij + g g ψrs1 ψis2 + ∑ i, j g ψi j gac 4 g00 ∑ i, r, s1 , s2 8 g00 f1,τ (0) f2 (0) ij − √1 g ψ ψac + gi j ψai ψc j + o([ψac ]2 ). i j ∑ i, j 2 g00 4 g00 ∑ i, j From above it follows the first-order approximation (2.3) of h(b). → Example 2.1. (a) If h = f (− τ ) bˆ and f (0) = 0 then (2.3) has a form: 1

h(b)ac = f (0) ψac + o([ψac ]).


23

(b) The extrinsic Ricci tensor (see (3.87), Sect. 3.9.1) is a second-degree polynomial Ricex (g)ac = −

1 1 ij ij g ψ ψ − g ψ ψac . √ ai c j i j ∑ ∑ i, j 4 g00 i, j 2 g00

(c) The components of the Newton transformation Tr (A) are given by (Tr (A))ij = j ... j j (r!)−1 ε i11...ir ri Aij11 . . . Aijrr . Hence Tr (b) is the rth degree polynomial −1

Tr (b)ac = (r!)

∑

... jr j ε ij11...i ra

−1 √ 2 g00

s

gi1 ,m1 . . . gir ,mr g jc ψm1 , j1 . . . ψmr , jr .

(d) For a F -conformal metric gi j = eu(x) δi j (i, j > 0), we approximate h(b) as h(b)ac = f0 (0) δac + u,0

f0,τ (0) f1 (0) − n √1 2 g00

δac + o(u,0 ).

(2.5)

If f0 , f1 , f0,τ1 are zero at the origin 0, we use the second-order approximation h(b)ac = u,0 2

f 1,τ (0) f2 (0) − n √1 δac + o(u,02 ). 4 g00 2 g00

(2.6)

Let gˆ and g⊥ be the components of g along the distributions T F and T F ⊥ , respectively. Because T F ⊥ is one-dimensional, g⊥ is determined by the value g⊥ (N, N) = 1. The next lemma is standard. For the convenience of the reader we give its proof. Lemma 2.3. Let (M, g = gˆ ⊕ g⊥ ) be a Riemannian manifold with a codimensionone foliation F and a unit normal N. Define a metric g¯ = (e 2ϕ g) ˆ ⊕ g⊥ , where 1 ϕ ∈ C (M). Then the second fundamental forms and the Weingarten operators of F with respect to g¯ and g are related by b¯ = e 2φ (b − N(ϕ ) g), ˆ

A¯ = A − N(ϕ ) id.

Proof. By the formula for Levi-Civita connection and g(T F , N) = 0, we get ¯ X N,Y ) = 2 g( ¯ X N,Y ) = X(g(N,Y 2e2 φ g(∇ ¯∇ ¯ )) + N(g(X,Y ¯ )) − Y (g(N, ¯ X)) +g([X, ¯ N],Y ) − g([N,Y ¯ ], X) + g([Y, ¯ X], N) = 2 e2 φ N(φ ) g(X,Y ) + 2 e2 φ g(∇X N,Y ) for any vector fields X,Y ∈ T F . Hence ¯ X N,Y ) = N(φ ) g(X,Y ) + g(∇X N,Y ). g(∇ ¯ X N, it follows (2.7)2 . ¯ From this and the definition A(X) = −∇

(2.7)

24


¯ X Y, N), we compute the tensor b¯ on T F ¯ Using the definition b(X,Y ) = g( ¯∇ ¯ X Y, N) = e2 φ g(A(X) − N(φ )X,Y ) ¯ b(X,Y ) = g( ¯∇ = e2 φ b(X,Y ) − e2 φ g(X,Y )N(φ ) = e2 φ (b(X,Y ) − N(φ ) g(X,Y )) .

From this it follows (2.7)1 . Remark 2.1. If N(ϕ ) = 0 (e.g., ϕ is constant) then by Lemma 2.3 we have (i) b¯ = e 2ϕ b; (ii) A¯ = A; (iii) τ¯ j = τ j (the power sums).

2.2.2 Foliations with a Time-Dependent Metric Consider a foliation (M, F ) with a time-dependent metric gt such that S = ∂t g is an F -truncated tensor. We denote the bundle of F -truncated (k, l)-tensors on M by

k (M). The inner product of tensors F, G ∈ Λ

k (M), denoted by ·, · on M, is given Λ l l by the following sum: d ...d

...ck F, G = ga1 b1 . . . gal bl gc1 d1 . . . gck dk Fac11...a Gb11 ...bkl . l

Recall that the musical isomorphism : T ∗ M → T M sends a covector ω = ωi dxi to ω = ω i ∂i = gi j ω j ∂i , and : T M → T ∗ M sends a vector X = X i ∂i to X = Xi dxi = gi j X j dxi . In other words, X = g(X, ·). We denote by B the (1, 1)-tensor field on M which is g-dual to a symmetric (0, 2)-tensor B, B(X,Y ) = g(B (X),Y ) for all vectors X,Y. For symmetric (0, 2)-tensors B, S we have B, S = Tr (B S ) = B , S .

(2.8)

Indeed, in a local coordinate basis (ei ) of T M we have gik Bi j = Bkj ,

gkl Sl j = Skj ,

B, S = Bi j Si j .

From the above, in view of identity gik gk j = δij (Kronecker’s delta), (2.8) follows, Tr (B S ) = Bkj Skj = gik Bi j gkl Sl j = Bi j Si j .


For example,

25

= n, g, ˆ g ˆ = Tr id

g, ˆ S2 = S, S,

where S2 is the symmetric (0, 2)-tensor dual to (S )2 . If S = s gˆ for some scalar ˆ and has the trace Tr S = ns. function s then S = s id Lemma 2.4. For a smooth family (Bt ) of symmetric (0, 2)-tensors on (M, gt , F ) and S = ∂t gt , where gt ∈ M (M, F , N), we have (∂t Bt ) = ∂t Bt + S · Bt ,

∂t (Tr gt Bt ) = Tr gt (∂t Bt ) − Bt , Sgt ,

∂t Bt , Sgt = ∂t Bt , Sgt − 2Bt , S2 gt + Bt , ∂t Sgt .

(2.9) (2.10)

Proof. Because ∂t gi j = Si j , we have

∂t gi j = −Si j := −Skl gik g jl . To establish (2.9)1 , we write (B )ki = gk j Bi j for any t and compute (∂t B )ki = ∂t (gk j Bi j ) = ∂t gk j Bi j + gk j ∂t Bi j = −Sk j Bi j + (∂t B)ik = −gik Si gil Blj + (∂t B)ik = −Si Bkj + (∂t B)ik . j

j

Notice that (2.9)2 is a consequence of (2.9)1 and the identity Tr(∂t Bt ) = ∂t (Tr Bt ). Then (2.10) directly follows from the calculation

∂t (Bi j Si j ) = (∂t B)i j Si j + Bi j (∂t S)i j = ∂t (gai gb j Bab )Si j + Bi j (∂t S)i j = (∂t g)ai gb j Bab Si j + gai (∂t g)b j Bab Si j + gaigb j (∂t B)ab Si j + Bi j (∂t S)i j = −Bab (Sai gb j Si j + gai Sb j Si j ) + (∂t B)i j Si j + Bi j (∂t S)i j = −Bab gal Sli Sib + gbl Slj Saj + (∂t B)i j Si j + Bi j (∂t S)i j = −2Bab(S2 )ab + (∂t B)i j Si j + Bi j (∂t S)i j .

Now let ∇t denote the Levi-Civita connection on (M, gt ), where t ∈ [0, ε ). Observe that the difference of two connections is always a tensor, hence Πt := ∂t ∇t is a (1, 2)-tensor field on (M, gt ). Differentiating with respect to t, the classical formula for the Levi-Civita connection yields the known formula (see Preface) 2 gt (Πt (X,Y ), Z) = (∇tX S)(Y, Z) + (∇Yt S)(X, Z) − (∇tZ S)(X,Y ),

(2.11)

where X,Y, Z ∈ T M. If the vector field Y = Y (t) is time-dependent then

∂t ∇tX Y = Πt (X,Y ) + ∇X (∂t Y ).

(2.12)

26


Notice the symmetry of the tensor: Πt (X,Y ) = Πt (Y, X). If the tensor S is F truncated then by (2.11) we have Πt (N, ·) ⊥ N. Let E be the pull-back of the tangent bundle T M under the projection M × [0, ε ) → M, i.e., the fiber of E over a point (x,t) is given by Ex,t = Tx M. There ˜ on E, which extends the Levi-Civita connection ∇ on M. is a natural connection ∇ ∂ . In order to define this connection, we need to specify the covariant t-derivative ∇ t Given any section X of the vector bundle E, we define ∂ X = ∂t X + 1 S (X), ∇ t 2

in particular,

∂ N = 0. ∇ t

(2.13)

˜ is compatible with the natural bundle metric on E, i.e., The connection ∇ g)(X,Y ) = 0, (∇ ∂t

X,Y ∈ T M.

To show this, we calculate ∂ g)(X,Y ) = ∂t (g(X,Y )) − g(∇ ∂ X,Y ) − g(X, ∇ ∂ Y ) = (∂t g)(X,Y ) − S(X,Y ) = 0. (∇ t t t ∂ ∂i = 0, while ∇ ∂ ∂t = 0 always This connection is not symmetric: in general ∇ t i for i > 0. However, each submanifold M × {t} is totally geodesic, so computing derivatives of spatial tangent vector fields gives the same result as computing for sections of T (M × [0, ε )). In particular, the corresponding Weingarten operators A˜ and A satisfy A˜ = A. Clearly, the torsion tensor XY − ∇ Y X − [X,Y ] Tor(X,Y ) := ∇ vanishes if both arguments are spatial, so the only nonzero components of Tor are ∂ ∂i − ∇ ∂ ∂t = 1 S (∂i ), Tor(∂t , ∂i ) = ∇ t i 2

i > 0.

2.2.3 A Differential Operator To shorten later formulas, we introduce the differential operator V (F) := τ1 F − ∇N F, where F ∈ Λ lk (M) is either a smooth (k, l)-tensor or a function on M. Lemma 2.5. For a foliation F on a closed Riemannian manifold (M, g) we have M

V (F) d vol = 0

for any F ∈ C1 (M).


27

Proof. Using the equality div N = −τ1 , we have div(FN) = F div N + N(F) = −τ1 F + N(F) = −V (F). By the Divergence Theorem, equality follows.

M div(FN) d vol

= 0. From the above the required

The next lemmas (concerning the above operator V ) are also global and can be proved for arbitrary (k, l)-tensor fields. First, we show that the operator ∇N is conjugate to V . Notice that ∇N commutes with traces of F -truncated (1, 1)-tensors: ∇N (Tr F) = Tr (∇N F)

for any (1, 1)-tensor field F.

Lemma 2.6. Let S, B be F -truncated symmetric (0, 2)-tensor fields on a closed M. Then M

B, ∇N S d vol =

M

V (B), S d vol.

(2.14)

In particular, for F ∈ C1 (M) and S = s gˆ we have M

F N(s) d vol =

M

s V (F) d vol .

(2.15)

Proof. Notice that the (1, 1)-tensor ∇N S is g-dual to ∇N S. To show (2.14), calculate with the help of (2.8) and Lemma 2.5, M

B, ∇N Sd vol = =

M

M

Tr (B ∇N S ) d vol =

M

N(Tr (B S )) − Tr((∇N B )S ) d vol

Tr (τ1 B S − (∇N B )S ) d vol =

M

V (B), S d vol.

Applying (2.14) to S = s g, ˆ we obtain (2.15).

Example 2.2. Let F and s be smooth functions on a closed M. One may use Lemma 2.6 to prove the following: M

M

F s N(s) d vol =

F s N(N(s)) d vol =

M

1 2

M

V (F)s2 d vol,

1 V (V (F)) s2 − F N(s)2 2

(2.16) d vol.

(2.17)

Indeed, for any F -truncated symmetric (0, 2)-tensor field S on M, we have M

FS, ∇N S d vol =

M

V (FS), S d vol =

M

V (F)S, S − FS, ∇N S d vol.

28


From the above for S = s g, ˆ one obtains (2.16). Using (2.15) with substitution F → s F and s → N(s), one has (2.17).

0 (M) → Λ

0 (M), define Given linear operator Φ : Λ 2 2

μ (Φ ) :=

inf

S ∈Λ 20 (M) M

Φ (∇N S), ∇N S d vol/

M

S, S d vol.

Lemma 2.7. Let a be supremum of the lengths of N-curves.

0 (M) then μ (Φ ) = 0. (i) If a = ∞ and Φ (∇N S), ∇N S ≥ 0 for any S ∈ Λ 2

0 (M) and some b ≥ 0 then (ii) If 0 ≤ Φ (∇N S), ∇N S ≤ b2 S, S for any S ∈ Λ 2

μ (Φ ) ∈ [0, π 2 b2 /a2 ]. Proof. Consider the well-known constrained variation problem J(s) =

a 0

(s (x))2 dx → min,

s(0) = s(a) = 0,

a 0

(s(x))2 dx = 1,

where s is smooth on (0, a). We claim that the minimum of J is π 2 /a2 . Indeed, ˜ = 0a F˜ dx with F˜ = (s )2 + μ (s2 − 1) and the Euler equation for the functional J(s) μ ∈ R is s + μ s = 0. Using the boundary conditions at x ∈ {0, a}, we find

μ = (π /a)2 ,

s = C sin(π x/a).

From the constraint we calculate C2 = 2/a. The solution (to the constrained variation problem) is s¯ =

2/a sin(π x/a),

min J = J(s) ¯ = π 2 /a2 .

Assuming s¯ = 0 on {x < 0} ∪ {x > 1}, we build a piece-wise smooth function s¯ on R. Then, there exist functions sn ∈ C1 (R) vanishing outside of (− 1n , a + 1n ) such that sn → s¯ and R (s (x))2 dx → π 2 /a2 when n → ∞, the claim is proved. Let the lengths of N-curves be unbounded and Φ (∇N S), ∇N S ≥ 0. Then for any a > 0 there exist an N-curve γ : [0, a] → M of length a and a smooth function s1 ≥ 0 on γ with the properties s1 (γ (0)) = s1 (γ (a)) = 0,

γ

N(s1 )2 < 2π 2 /a2 .

There is a “thin” biregular foliated chart U(q) ⊃ γ with q = γ (0), x = (x0 , x) ˆ ∈ [0, 1]n+1, and 01 g00 (x0 , 0) dx0 = a. Consider a smooth function s = s1 (x0 )s2 (x) ˆ (with 0 ≤ s2 ≤ 1) supported in this chart. Define a tensor field S = sS˜ on M with an arbitrary F -truncated symmetric (0, 2)-tensor field S˜ satisfying ∇N S˜ = 0 on γ , and


29

˜ S. ˜ As the volume form vol satisfies vol < Q(a) dx0 ∧ dxˆ along let F = supM Φ (S), γ for some Q(a) > 0 (depending on γ ), one may assume that vol < 2 Q(a) dx0 ∧ dxˆ ˆ on U(q) = I × U(q). Applying the above inequality and the Fubini Theorem, we have U(q)

Φ (∇N S), ∇N S d vol < F

< 2F

U(q)

N(s1 )2 d vol

U(q)

Q(a)∂0 (s1 )2 /g00 (x) dx0 ∧ dxˆ

1 2 = 2F Q(a) ∂0 (s1 ) dx0 dxˆ ˆ g00 (x) I U(q) π 2 1 < 4 Q(a) dx. ˆ F ˆ a g U(q) 00 (x)

ˆ One may take U(q) such that 4π 2 F Q(a) U(q)

1 ˆ U(q) g00 (x)

dxˆ < 1 for all x0 . Thus,

Φ (∇N S), ∇N S d vol < 1/a2 .

As a > 0 is arbitrary, it follows that μ (Φ ) = 0, that completes the proof of (i). Claim (ii) follows directly from the estimates above.

Remark 2.2. Concerning Lemma 2.7(i), recall [20] that there exist compact manifolds (Mn+1 , g), n > 2, foliated by closed curves whose lengths are unbounded.

0 (M) → Λ

0 (M) (i = 1, 2, 3), define Lemma 2.8. Given linear operators Φi : Λ 2 2

J(S) := M

(Φ1 (S), S + Φ2 (∇N S), ∇N S + Φ3(S), ∇N S) d vol.

If J ≥ 0 for any symmetric tensor S ∈ Λ20 (M) then Φ2 (∇N S), ∇N S ≥ 0. Moreover, if Φ2 (∇N S), ∇N S ≥ 0,

(Φ1 + V ◦ Φ3 )(S), S ≥ −μ (Φ2 )S, S,

(2.18)

0 (M) then J ≥ 0. for any symmetric tensor S ∈ Λ 2 Proof. By (2.14), we have M

Φ3 (S), ∇N S d vol =

M

V (Φ3 (S)), S d vol.

Certainly, (2.18) with any symmetric tensor S ∈ Λ 20 (M) are suffices for J ≥ 0.

30


Fig. 2.1 A saw-shaped function s1

In order to prove that (2.18)1 is necessary, we shall use S with the support in a biregular foliated chart U(q) adapted to F and N with coordinates x = (x0 , x) ˆ ∈ [0, 1]n+1, xˆ = (x1 , . . . xn ), see Sect. 2.2.1. Hence, x0 = c = const on the leaves xi = ci = const (i > 0) along N-curves, the coordinate vector fields ∂i = ∂xi (i > 0), are tangent to leaves and N is directed along ∂0 = ∂x0 . In these coordinates, the metric g has the form: g = g00 dx20 + ∑i, j>0 gi j dxi dx j √ with g00 = 1 for xˆ = 0, and N = β ∂0 for β = 1/ g00 (because N is the unit normal to F ). One may assume that volg U(q) < 1. Take s = s1 (x0 )s2 (x), ˆ where 0 ≤ si ≤ 1 and supp(s) ⊂ U(q). Notice that N(s) = N(s1 )s2 . ˜ S ˜ < 0 for To prove (2.18)1 assume the contrary: that J ≥ 0 but F2 := Φ2 (S), some symmetric (0, 2)-tensor S˜ at a point q. One may extend S˜ on a neighborhood U(q) (of q) with the property ∇N S˜ = 0 at q and assume that F2|U(q) < −δ for some δ > 0. Take a saw-shaped function s1 = s1 (x0 ) (Fig. 2.1) with a number of oscillations of slope ±1, such that the values of s1 belong to [0, ε ], where ε > 0 can be chosen as small as necessary. Define a symmetric (0, 2)-tensor field S = sS˜ on M. Then N(s1 )2 = |∂0 s1 |2 /g00 = 1/g00 almost everywhere on [0, 1], and M

Φ2 (∇N S), ∇N Sd vol =

where β1 =

M

2 M s2 /g00 d vol > 0.

s22 ˜ Sd ˜ vol < −δ Φ2 (S), g00

M

s22 d vol = −δ β1 , g00

We also have

M

Φ1 (S), S + Φ3 (S), ∇N S d vol ≤ F1 ε 2 + F3 ε ,

˜ S|, ˜ F3 = sup U(q) |Φ3 (S), ˜ S|. ˜ Hence, where F1 = sup U(q) |Φ1 (S), J(S) < F1 ε 2 + F3 ε − β1 δ . For ε > 0 small enough we obtain J(S) < 0, a contradiction. Given the function F on M, define

μ (F) :=

inf

s ∈C1 (M) M

FN(s)2 d vol

M

s2 d vol .

2.3 Variational Formulae for Codimension-One Foliations

31

Remark that (by Lemma 2.7) we have the following: (a) If a = ∞ and F ≥ 0, then μ (F) = 0. (b) If 0 ≤ F ≤ b2 for some b ≥ 0, then μ (F) ∈ [0,

π 2 b2 ]; a2

Here a is supremum of the lengths of N-curves. From Lemma 2.8 we obtain the following Corollary 2.1. Let Fi (i = 1, 2) be continuous functions on M and

J(s) := M

(F1 s2 + F2 N(s)2 ) d vol.

If J(s) ≥ 0 for any s ∈ C1 (M) then F2 ≥ 0. Moreover, if F2 ≥ 0 and F1 ≥ −μ (F2 ) then J ≥ 0.

2.3 Variational Formulae for Codimension-One Foliations 2.3.1 Variations of Extrinsic Geometric Quantities In order to calculate variations of the functional I f with respect to metrics gt ∈ M , we find the variational formula for A, and apply it to the Newton transformations Ti (A) and to symmetric functions τ j , σ j of A. For short, we shall omit the index t for the time-dependent tensors S, A, bˆ j and functions τi , σi . Let bˆ be the extension of b to the F -truncated symmetric (0, 2)-tensor field on M. ˆ Notice that b(N, ·) = 0 and ˆ b(X,Y ) = g(A(X),Y ). ˆ In other words, b(N, ·) = 0 and bˆ is dual to the extended Weingarten operator A. ˆ Denote by b j the symmetric (0, 2)-tensor fields on M dual to powers A j of extended Weingarten operator, bˆ 0 (X,Y ) = g(X,Y ˆ ),

bˆ j (X,Y ) = g(A ˆ j (X),Y ) ( j > 0,

X,Y ∈ T M).

Lemma 2.9. Let gt ∈ M be a family of F -truncated metrics and S = ∂t gt . Then the Weingarten operator A of F and the symmetric functions τi and σi of A evolve by 1 [A, S ] − ∇tN S , 2 1 ∂t σi = − Tr (Ti−1 (A)∇tN S ), 2

∂t A = i ∂t τi = − Tr (Ai−1 ∇tN S ), 2

(2.19) i > 0.

(2.20)

32


For S = s gˆ (s ∈ C1 (M)) we get 1 ˆ ∂t A = − N(s) id, 2 i ∂t τi = − τi−1 N(s), 2

1 ∂t σi = − (n − i + 1) σi−1N(s), 2

i > 0.

Proof. Using (2.11) and S(·, N) = 0 for F -truncated tensors, we obtain

∂t b(X,Y ) = ∂t gt (∇tX Y, N) = (∂t gt )(∇tX Y, N) + gt (∂t ∇tX Y, N) = S(∇tX Y, N) + (1/2) (∇tX S)(Y, N) + (∇Yt S)(X, N) − (∇tN S)(X,Y ) = (1/2) S(AX,Y ) + S(AY, X) − (∇tN S)(X,Y ) for all X,Y ∈ T F . Because S(AX,Y ) = gt (S AX,Y ) and (∇tN S)(X,Y ) = gt ((∇tN S )X,Y ), we have gt ((∂t A)X,Y ) = gt (∂t (AX),Y ) = ∂t b(X,Y ) − S(AX,Y) 1 [gt (S AY, X) − gt (S AX,Y ) − (∇tN S)(X,Y )] 2 1 = [gt ([A, S ]X,Y ) − gt ((∇tN S )X,Y )]. 2 =

Formula (2.19) follows from the above and the freedom of choice of X,Y ∈ T F . Multiplying (2.19) from the left by Ai−1 , we get 2Ai−1 ∂t A = Ai−1 [A, S ] − Ai−1 ∇tN S ,

i > 0.

Notice that Tr (Ai−1 ·[A, S ]) = 0, see Remark 2.3. Then, using the identity i Tr (Ai−1 ∂t A) = Tr (∂t Ai ) = ∂t τi , see (1.11) of the following Remark 1.2, we deduce (2.20)1. Substituting ∂t A from (2.19) into the formula (1.12) of Remark 1.2, we obtain

∂t σi =

1 1 Tr Ti−1 (A)([A, S ] − ∇tN S ) = − Tr (Ti−1 (A)∇tN S ), 2 2

and ˆ we have, respectively, ∇tN S = N(s) id, that proves (2.20)2. For S = s g, Tr (Ai−1 ∇tN S ) = τi−1 N(s),

Tr (Ti−1 (A)∇tN S ) = (n − i + 1) σi−1N(s).

From the above, the case S = s gˆ of lemma follows.


33

Remark 2.3. For any n-by-n matrices A, B, and C such that AB = BA one has Tr (A·[B,C]) = Tr (ABC) − Tr ((AC)B) = Tr (BAC) − Tr(B(AC)) = 0. Example 2.3. (a) We shall find the evolution of tensors Ai and ∇tN Ai and their dual. Using (2.19) and the definition S = ∂t gt , we generalize (2.19) and find for i > 0 2 ∂t Ai = ∑ j=0 A j ([A, S ] − ∇tN S )Ai−1− j = [Ai , S ] − ∑ j=0 A j (∇tN S )Ai−1− j . i−1

i−1

As Ai = (bˆ i ) , from the above and (2.9) we obtain the evolution of bˆ i (i > 0), (∂t bˆ i ) = ∂t Ai + S Ai =

1 i i−1 S A + Ai S − ∑ j=0 A j (∇tN S )Ai−1− j . 2

Observe that tracing ∂t Ai we get (2.20)1. From (2.19), using (2.11), (2.12), and the following calculations: 2 gt (∂t (∇tN Ai )X,Y ) = 2 gt (∂t (∇tN (Ai X) − Ai∇tN X), Y ) = 2 gt (∂t ∇tN (Ai X) − (∂t Ai )∇tN X − Ai ∂t (∇tN X), Y ) = 2 gt (∇tN ((∂t Ai )X) − (∂t Ai )∇tN X,Y ) + (∇tN S)(Ai X,Y ) +(∇tAi X S)(N,Y ) − (∇Yt S)(N, Ai X) − (∇tN S)(X, AiY ) −(∇tX S)(N, AiY ) + (∇tAiY S)(N, X) = 2 gt ((∇tN (∂t Ai ))X,Y ) − gt ([ [Ai , S ], A]X,Y ) +gt ([(∇tN S ), Ai ]X,Y ), we obtain 2 ∂t (∇tN Ai ) = 2 ∇tN (∂t Ai ) − [ [Ai , S ], A] + [∇tN S , Ai ]. From the above and the formula for ∂t Ai we find the evolution of ∇tN Ai (i > 0):

∂t (∇tN Ai ) =

1 t i i−1 [ ∇N A , S ] − [ [Ai , S ], A] − ∇tN ∑ j=0 A j (∇tN S )Ai−1− j . 2

Notice that

∂t (∇tN bˆ i )(X,Y ) = ∂t (gt (∇tN Ai )X,Y ) = S((∇tN Ai )X,Y ) + gt (∂t (∇tN Ai )X,Y ). Putting these facts together yield the evolution equation for ∇tN bˆ i (i > 0): (∂t ∇tN bˆ i ) =

1 t i i−1 S ∇N A + (∇tN Ai )S − [ [Ai , S ], A] − ∇tN ∑ j=0 A j (∇tN S )Ai−1− j . 2

34


(b) Next, we shall find the evolution of tensors Ti (A) and ∇tN Ti (A) and their duals. By Lemma 2.9 and the method of Example 2.3(a) we find the evolution of Ti (A), 2 ∂t Ti (A) = [Ti (A), S ] − ∑ j=1 (−1) j σi− j ∑ p=0 A p (∇tN S )A j−1−p i

j−1

− ∑ j=0 (−1) j Tr (Ti− j−1 (A)∇tN S )A j . i−1

For S = s gˆ (s ∈ C1 (M)) and i > 0 we certainly have i ∂t Ai = − Ai−1 N(s), 2

∂t Ti (A) =

1 i N(s)(n − i) ∑ j=1 (−1) j σi− j A j−1 . 2

Notice that det A = 0 provides

−1 . ∑ j=1 (−1) j σi− j A j−1 = (Ti (A) − σi id)A i

Similar to the result for ∇tN Ai in (a), we find the evolution of ∇tN Ti (A), ∂t (∇tN Ti (A)) = ∇tN (∂t Ti (A)) + (1/2) [∇tN S , Ti (A)] − [ [Ti (A), S ], A] 1 i j−1 = − ∇tN ∑ j=1 (−1) j σi− j ∑ p=0 A p (∇tN S )A j−1−p 2 i−1 + ∑ j=0 (−1) j Tr (Ti− j−1 (A)∇tN S )A j +

1 t [∇N Ti (A), S ] − [ [Ti (A), S ], A] . 2

As Ti (A) = (Ti (b)) , ∇tN Ti (A) = (∇tN Ti (b)) , from the above and (2.9) we find the evolution of Ti (b) and ∇tN Ti (b) for i > 0, (∂t Ti (b)) = ∂t Ti (A) + STi (A) =

1 S Ti (A) + Ti (A)S 2

− ∑ j=1 (−1) j σi− j ∑ p=0 A p (∇tN S )A j−1−p i−1 −∑ j=0 (−1) j Tr (Ti− j−1 (A)∇tN S )A j , i

j−1

(∂t ∇tN Ti (b)) = ∂t ∇tN Ti (A) + S∇tN Ti (A) 1 t = S ∇N Ti (A) + (∇tN Ti (A))S − [ [Ti (A), S ], A] 2 i−1 −∇tN ∑ j=0 (−1) j Tr (Ti− j−1 (A)∇tN S )A j + ∑ j=1 (−1) j σi− j ∑ p=0 A p (∇tN S )A j−1−p i

j−1

.


35

Notice that using the connection (2.13), we also have ∂ X,Y ) − b(X, ∇ ∂ Y ) = − 1 (∇t S)(X,Y ). ∂ b)(X,Y ) = ∂t b(X,Y ) − b(∇ (∇ t t t 2 N Proposition 2.2. Let gt ∈ M be a family of F -truncated metrics and S = ∂t gt . Then ∂ bˆ i = − 1 ∑i−1 A j (∇t S )Ai−1− j , ∇ N t 2 j=0 i−1 j t i−1− j ∂ (∇t bˆ i ) = − 1 [ [Ai , S ], A] + ∇t ∇ , N N ∑ j=0 A (∇N S )A t 2 ∂ Ti (b) = − 1 ∑i (−1) j σi− j ∑ j−1 A p (∇t S )A j−1−p ∇ N t j=1 p=0 2 i−1 + ∑ j=0 (−1) j Tr (Ti− j−1 (A)∇tN S )A j , i−1 j t j (∇t Ti (b)) = − 1 [ [Ti (A), S ], A] + ∇t ∇ ∂t N N ∑ j=0 (−1) Tr Ti− j−1 (A)∇N S A 2 i j−1 + ∑ j=1 (−1) j σi− j ∑ p=0 A p (∇tN S )A j−1−p . Proof. Using Example 2.3(a), (2.13), and equalities bˆ i (X,Y ) = ∂t bˆ i (X,Y ) − bˆ i (∇ ∂ X,Y ) − bˆ i (X, ∇ ∂ Y ), ∇ ∂t t t ∂ (∇t bˆ i )(X,Y ) = ∂t ((∇t bˆ i )(X,Y )) − (∇t bˆ i )(∇ ∂ X,Y ) − (∇t bˆ i )(X, ∇ ∂ Y ), ∇ N N N N t t t ∂ bˆ i and ∇ ∂ (∇t bˆ i ). Similarly, from Example 2.3(b) and the definition we find ∇ N t t (∇t Ti (b)) follow. Ti (b) and ∇ (2.13), the formulae for ∇ ∂t ∂t N

2.3.2 Variations of General Functionals Here we develop variational formulae for the functional I f (g) of (2.1), restricted to metrics in M and M1 , respectively. Let

π : M → M1 ,

π (g) = g¯ = vol(M, g)−2/n gˆ ⊕ g⊥

be the F -conformal projection. Metrics g¯t = (φt gˆt ) ⊕ gt⊥ with dilating factors φt = −2/n , belong to M1 , i.e., M d volt = 1. vol(M, gt )

36


For a family of metrics gt ∈ M , we denote g = g0 , S = ∂t gt , and S˙ = ∂t S. Recall [53] that the volume form of gt evolves as: 1 ∂t (volt ) = (Tr S ) volt . 2

(2.21)

We certainly have Tr S = Tr gt S = gˆt , S. For conformal metrics gt = eϕt g, where (ϕt ) is a smooth family of continuous (smooth whenever needed) functions on M, we obtain a conformal tensor St = st gˆt , where st = ∂t ϕt . Remark 2.4. (a) Let S and S˙ be F -truncated symmetric (0, 2)-tensor fields. If 1 gt = g + tS + t 2 S˙ 2 is a “quadratic in t” variation of the metric g = gˆ ⊕ g⊥ ∈ M then the metrics 1 2˙ ⊕ g⊥ g¯t = π (gt ) = φt gˆ + tS + t S 2 belong to M1 . For a conformal tensor S = s gˆ (s : M → R), the above metrics are 1 ˆ gt = g + ts + t 2 s˙ g, 2

1 g¯t = π (gt ) = φt 1 + ts + t 2 s˙ gˆ ⊕ g⊥ . 2

One may use the above approximations for finding the 1st and second variations of our functionals at t = 0 with respect to general families gt and g¯t , respectively. (b) Let f˜ : M → (0, ∞) be a smooth function constant on the leaves of F , and g(X,Y ˜ ) = f˜2 g(X,Y ),

X,Y ∈ T F .

If at least one of vectors X,Y is perpendicular to F , we set g(X,Y ˜ ) = g(X,Y ). A foliated Riemannian manifold (M, F , g) ˜ is called a warped foliation (with a warping function f˜), see [59]. They were studied from the point of view of the Gromov–Hausdorff convergence. The warped foliation generalizes the Bergers modification of a metric of S3 along the fibers of the Hopf fibration (called Berger spheres). For warped foliations gt = ((1 + ct)g) ˆ ⊕ g⊥ (c ∈ R), A and τ j do not depend on t, and we have volt = (1 + ct)n/2 vol,

φ = (1 + tc)−1,

g¯t = g.

Hence, see (2.1), n

I f (gt ) = (1 + ct) 2 I f (g),

n I f (g) = I f (g). 2


37

We conclude that if g is a critical metric for I f with respect to F -conformal variations gt then I f (g) = 0. In Theorem 2.1 and its corollaries below, we shall find the variations of functionals I f with respect to metrics g¯t ∈ M1 and gt ∈ M . In the gt -case, the variations/gradients are given by the same formulae as in the first one but with underlined terms deleted. This can be explained as follows: under the π∗ projection (from T M onto T M1 ), the gradient of the functional contains additional (underlined) component. So, ∇I f (g) =

1 f gˆ − V (B f ) (in 2

T M ),

while its projection onto T M1 , see (2.22), is ¯ f (g) = ∇I f (g) − 1 I f (g) gˆ = 1 ( f − I f (g)) gˆ − V (B f ). ∇I 2 2 The scalar product in T M is given by g˙1 , g˙2 = ¯ ˆ = 0. M Tr V (B f ) d vol = 0, hence ∇I f (g), g

M g˙1 , g˙2 d vol.

By Lemma 2.5,

Theorem 2.1. The gradient of the functional I f : M → R, see (2.1), and its projection via π∗ : T M → T M1 are given by: 1 ∇I f (g) = ( f − I f (g)) gˆ − V (B f ), 2

(2.22)

where B f = ∑ni=1 2i f,τi bˆ i−1 . The F M 1 - and F M - components of the gradients are F

∇ I f (g) =

1 1 ( f − I f (g)) − V (Tr B f ) g, ˆ 2 n

(2.23)

where Tr Bf = ∑ni=1 2i f,τi τi−1 . The second variation of I f (g¯t ) (when S = ∂t gt ) at a critical metric g = g¯0 and its restriction to the F -conformal variations (i.e., S = s g, ˆ s : M → R) are given by: I f (g¯t ) |t=0 = I f (g¯t )|t=0 =

M

M

(Φ1 (S), S + Φ2(∇N S), ∇N S + Φ3 (S), ∇N S) d vol,

Φ f N(s)2 d vol,

respectively, where

Φf =

1 n 1 n i(i − 1) τi−2 f, τi + ∑i, j=1 i j τi−1 τ j−1 f, τi τ j , ∑ i=2 4 4

(2.24)

38


0 (M) → Λ

0 (M) (i = 1, 2, 3) are defined by: and the linear operators Φi : Λ 2 2 1

− B [S , A], Φ1 (S ) = ( f − I f (g))(Tr S ) id Φ3 (S ) = −(Tr S )Bf , f 4 ij i n n i−2 Φ2 (S ) = ∑i, j=1 f,τi τ j Tr (Ai−1 S )A j−1 + ∑i=2 f,τi ∑ j=0 A j S Ai−2− j . 4 4 Proof. As the metrics g¯t and gt are F -conformal with constant scale φt , by n/2 Lemma 2.3 we have τ j (g¯t ) = τ j (gt ) and volt = φt volt . Differentiating the last equality and using (2.21), we obtain n n 1 ∂t volt = (φ 2 ) volt +φ 2 ∂t volt = Tr S − (Tr S ) d volt volt . 2 M Here, we used the fact that φ0 = 1 and

2 2 φ = − vol− n −1 n

1 n ∂t (d volt ) = − φ 2 +1 n M

M

(Tr S ) d volt = −

φ n

M

(Tr S ) d volt .

Differentiating the functional I f (g¯t ), we obtain 1 ∂t f + f Tr S − (Tr S ) d volt I f (g¯t ) = d volt 2 M M 1 = ∂t f + ( f − I f (g¯t )) Tr S d volt . 2 M

(2.25)

→ τ ), by Lemma 2.9, we have Now, we simplify (2.25): for f = f (− i n n ∂t f = ∑i=1 f,τi ∂t τi = − Tr ∇tN S ∑i=1 f,τi Ai−1 = − Tr Bf ∇tN S , (2.26) 2 where Bf = ∑ni=1 2i f,τi Ai−1 is dual to B f . From (2.26), by Lemma 2.6, we have M

(∂t f )d volt =

M

[Tr ((∇tN Bf )S ) − N(Tr (Bf S ))]d volt =

M

−V (B f ), St d volt .

Thus, (2.25) yields (2.22): I f (g¯t ) =

M

1 ( f − I f (g¯t )) gˆt − V (B f ), S 2

d volt .

(2.27)

t

For a (0, 2)-tensor B˜t := 12 ( f − I f (g¯t )) gˆt − V (B f ) in (2.27), by Lemma 2.4 we have

∂t B˜t , Sgt = ∂t B˜t , Sgt − 2B˜t , S2 gt + B˜t , ∂t Sgt , ∂t B˜t =

1 1 1 (∂t f ) gˆt − I f (g¯t ) gˆt + ( f − I f (g¯t ))S − ∂t V (B f ). 2 2 2


39

Differentiating (2.27) at a critical metric g¯0 and using B˜ 0 = 0, we get I f (g¯t ) |t=0 =

1 M

1 (∂t f ) gˆ + ( f − I f (g))S − ∂t V (B f ), S d vol. 2 2

(2.28)

One may compute I f (g¯t ) |t=0 explicitly using Lemma 2.9 and (2.26). We have

∂t V (B f ) = (∂t τ1 )B f + τ1 ∂t B f − ∂t ∇tN B f ,

where for t = 0

1 i f,τi ∂t bˆ i−1 + (∂t f,τi )bˆ i−1 = (S Bf + Bf S ) 2 2 i 1 i−2 j f ,τ A (∇N S )Ai−2− j − ∑i 2 2 i ∑ j=0 j n + ∑ j=1 f,τi τ j Tr A j−1 (∇N S ) Ai−1 , 2 i N( f,τi )∂t bˆ i−1 + f,τi ∂t ∇N bˆ i−1 + ∇N ((∂t f,τi )bˆ i−1 ) (∂t (∇N B f )) = ∑i 2 1 S ∇N Bf + (∇N Bf )S = 2 i 1 i−2 ∇N f ,τi ∑ j=0 A j (∇N S )Ai−2− j − ∑i 2 2 j − (1/2)[ [Bf , S ], A]. +∇N Ai−1 ∑ j f,τi τ j Tr (A j−1 ∇N S ) 2 (∂t B f ) = ∑i

Here we used the t-derivatives of bˆ i−1 and ∇tN bˆ i−1 of Example 2.3 and the equality

∂t ( f,τi ) = − ∑ j=1 n

j f,τ τ Tr (A j−1 ∇tN S ). 2 i j

Substituting the above into (2.28), and using the equalities

Tr ([ [B f , S ], A]S ) = 2 Tr (B f [S , A]S ) V (Bf ) = M

Tr (∇N S ) Tr (Bf S )d vol = = =

1

( f − I f (g)) id 2

M

M

at a critical metric,

V (Tr (Bf S )) g, ˆ Sd vol V (Tr (Bf S )) Tr S d vol

M

for commuting B f and A,

Tr (V (Bf )S ) Tr S − Tr(Bf ∇N S ) Tr S d vol,

40


(the last one is based on Lemma 2.6) we obtain I f (g¯t ) |t=0

=

M

1 ( f − I f (g))(Tr S )2 − Tr (Bf [S , A]S ) − Tr(Bf ∇N S ) Tr S 4 i 1 f,τi ∑ Tr (A j (∇N S )Ai−2− j S )+ Tr (Ai−1 S ) +τ1 ∑ 2 2 i j≤i−2

j j−1 ∑ 2 f,τi τ j Tr (A ∇N S ) j i 1 i−2 Tr S ∇N f,τi ∑ j=0 A j (∇N S )Ai−2− j − ∑i 2 2 j t i−1 j−1 + Tr S ∇N A ∑ j f,τi τ j Tr (A ∇N S ) d vol . 2

Simplifying terms with f,τi τ j and sums “ ∑i−2 j=0 ” (by Lemma 2.5), we finally obtain I f (g¯t ) | t

=0=

M

1 ( f − I f (g))(Tr S )2 − Tr(Bf [S , A]S ) − Tr(Bf ∇N S ) Tr S 4 + ∑i=2 ∑ j=0 n

i−2

+ ∑i, j=1 n

i f,τ Tr (A j (∇N S )Ai−2− j ∇N S ) 4 i

ij f,τi τ j Tr (Ai−1 ∇N S ) Tr (A j−1 ∇N S ) d vol . (2.29) 4

By (2.29), the integrand of I f (g¯t ) |t=0 has the form (2.24)1. Let S = s g. ˆ Although the result follows from the above (the RHS of the formula for Φ2 reads as Φ f N(s)2 and M (Φ1 + V ◦ Φ3 )(S), S d vol = 0), we shall prove it independently. In this case, Tr Bf = ∑ni=1 2i f,τi τi−1 , and (2.27) reads: I f (g¯t ) = =

s M

n 2

M

( f − I f (g¯t )) − V (Tr Bf ) d volt

1 1 ( f − I f (g¯t )) − V (Tr Bf ) g, ˆ s gˆ d volt . 2 n

(2.30)

From (2.30) with t = 0 (or, from (2.22)), (2.23) follows. Differentiating (2.30) at a critical metric g = g¯0 , or applying S = s gˆ to (2.28), we find I f (g¯t )|t=0 =

n M

n s ∂t f + s2 ( f − I f (g)) − s Tr g (∂t V (B f )) d vol. 2 2


41

From the above formula and (2.23) at a critical metric, we have n n n s ∂t f + s2 ( f − I f (g¯t )) d vol = V (Tr Bf ) s2 − (Tr Bf )s N(s) d vol 2 2 2 M M n V (Tr Bf ) s2 d vol . = 1− 4 M By Lemma 2.9, we have i ∂t (Tr Bf )|t=0 = ∑i ((∂t f,τi )τi−1 + f,τi ∂t τi−1 ) = −Φ f N(s). 2 Using this, (2.9)2 and the identity

∂t V (φt ) = (∂t τ1 ) φt + V (∂t φt ),

∀ φt ∈ C1 (M),

we calculate M

s Tr g (∂t V (B f )) d vol = = =

M

s ∂t (Tr gt V (B f ))|t=0 + s Tr g V (B f ) d vol

M

M

V (Tr Bf ) s2 + s ∂t (V (Tr Bf ))|t=0 d vol V (Tr Bf ) s2 + s V (−Φ f N(s)) +(∂t τ1 ) Tr Bf

=

M

d vol

V (Tr Bf ) s2 − V (Φ f ) s N(s) + Φ f sN(N(s))

n − (Tr Bf ) s N(s) d vol 2 n 1− V (Tr Bf )s2 − Φ f N(s)2 d vol . = 4 M

The formula (2.24)2 follows from the above.

Example 2.4 (Totally geodesic foliations). Let F be a totally geodesic foliation on (M, g) of unit volume. Then A = 0,

− → τ = 0,

Bf =

1 f ,τ (0)g, ˆ 2 1

V (B f ) = 0,

I f (g) = f (0).

By (2.22) of Theorem 2.1, g is a critical metric for the functional I f with respect to 2 variations g¯t ∈ M1 . We also have Φ f = n2 f,τ2 (0) + n4 f,τ1 τ1 (0) and 1

Φ3 (S ) = − f,τ1 (0)(Tr S ) id, 2 1

+ 1 f,τ (0)S . Φ2 (S ) = f,τ1 τ1 (0)(Tr S ) id 4 2 2

Φ1 (S ) = 0,

42


Using Lemmas 2.5 and 2.6, we calculate 1 1 f,τ1 τ1 (0)(N(Tr S ))2 + f,τ2 (0) Tr ((∇N S )2 ) d vol . I f (g¯t )|t=0 = 2 M 4 We conclude that I f ≥ 0, when f,τ1 τ1 (0) ≥ 0 and f,τ2 (0) ≥ 0. Question. Under what conditions on a smooth function f (τ1 , . . . , τn ) is the form Φ2 (S), S =

n

n i−2 ij i f,τi τ j Tr (Ai−1 S ) Tr (A j−1 S ) + ∑ f,τi ∑ Tr (A j S Ai−2− j S ) i, j=1 4 i=2 4 j=0

∑

positive definite for all F -truncated symmetric (0, 2)-tensors S?

the Example. For f = τ2 , we have Φ2 (S), S = 12 S, S. Notice that for S = s id condition reads:

∑i, j=1 i j f,τi τ j τi−1 τ j−1 + ∑i=2 i(i − 2) f,τi τi−2 > 0. n

n

Remark 2.5. Consider the function n−1 n−1 F = ∑i, j=1 H˜ i j τi τ j + 2 ∑i=1 bi τi + c, where c = n2 f,τ1 τ1 +2 n f,τ2 ,

while the (n − 1)×(n − 1) matrix H˜ and (n − 1)-vector b are given by: H˜ i j = (i + 1)( j + 1) f,τi+1τ j+1

(1 ≤ i, j < n),

1 bi = n(i + 1) f,τ1 τi+1 − n2 f,τ1 τn δi,n−1 + (i + 2)(i + 1) f,τi+2 2

(1 ≤ i < n).

Critical points of F are solutions τ˜ = (τ1 , . . . , τn−1 ) to the system H˜ τ˜ = −b. If H˜ is positive definite and bT (H˜ −1 )T (H˜ − 2 id) b > −c for all τ˜ , then F > 0. Indeed, under above conditions, Φ f of Theorem 2.1 is strictly positive and for any g ∈ M1 the functional I f , when restricted on F -conformal metrics of unit volume, has at most one critical point. Proposition 2.3. Let a metric g on a closed foliated manifold (M, F ) be a stable local maximum on the space M1 for the functional I f (with a fixed f ∈ C2 (Rn )). If M Φ2 (∇N S), ∇N Sd vol ≥ 0 for any F -truncated symmetric (0, 2)-tensor S, and )(ki − k j ) ≥ 0 ∑m=1 m f,τm (kim−1 − km−1 j n

for any principal curvatures ki = k j , then F is umbilical. Proof. One may take S with the property Tr S = 0. Then, by Theorem 2.1, I f (g) =

M

− Tr (Bf [S , A] S ) + Φ2 (∇N S), ∇N S d vol.

(2.31)


43

Let S = (si j ) in the frame of principal directions (for A). One may show that )(ki − k j ) − Tr (Am [S , A] S ) = ∑i< j s2i j (kim−1 − km−1 j for all m > 0. Hence, by the condition (2.31), for Bf = ∑nm=1 f ,τm Am−1 we have − Tr (Bf [S , A] S ) ≥ 0. and the equality holds only when ki = k j for all i = j. As the second variation I f (g) ≤ 0, from the above we conclude that F is umbilical. ≥ 0, for m even Notice that the condition f ,τm yields the inequality (2.31). = 0, for m odd

2.3.3 Variations of Particular Functionals The following functionals on M for particular cases of f were introduced in (1.1): Iτ ,k (g) =

M

τk d volg ,

Iσ ,k (g) =

M

σk d volg ,

k = 1, 2, . . . .

From [41], see (1.2), it is known that Iτ ,1 = Iσ ,1 = 0 for any F and g on a closed M. From Theorem 2.1 with f = τk it follows Corollary 2.2. The gradient of the functional Iτ ,k : M → R for k > 1 and its projection via π∗ : T M → T M1 are given by: ∇Iτ ,k (g) =

1 k (τk − Iτ ,k (g)) gˆ − V (bˆ k−1 ). 2 2

The F M - and F M 1 - components of above gradient are given, respectively, by ∇F Iτ ,k (g) =

1 k τk − Iτ ,k (g) − V (τk−1 ) g. ˆ 2 n

The second variation of Iτ ,k at a critical metric g = g¯0 , and its restriction to the F -conformal variations S = s gˆ (s : M → R) are given by (2.24), where S = ∂t g¯t , f = τk , Φ f = 14 k(k − 1)τk−2 , and 1

− k Ak−1 [S , A], (τk − Iτ ,k (g))(Tr S ) id 4 2 k k k−2 Φ2 (S ) = ∑ j=0 A j S Ak−2− j , Φ3 (S ) = − (Tr S )Ak−1 . 4 2

Φ1 (S ) =

44


Proof. As in the proof of Theorem 2.1 (see (2.25) with f = τk ), we obtain Iτ ,k (g¯t ) =

M

1 ∂t τk + (τk − Iτ ,k (g¯t )) Tr S d volt . 2

By (2.20)1 and Lemma 2.6, we have at t = 0 M

(∂t τk ) d vol = −

k 2

k Ak−1 , ∇N S d vol = − V (Ak−1 ), S d vol.

2

M

M

The above (or Theorem 2.1 with B f = 2k bˆ k−1 and Tr Bf = 2k τk−1 ) yield the formula for the gradient ∇Iτ ,k (g). We shall comment about Iτ,k (g¯t ) |t=0 . By (2.28), we obtain Iτ,k (g¯t ) |t=0 =

1 2

M

(∂t τk ) gˆ + (τk − Iτ ,k (g)) S − k ∂t V (bˆ k−1 ), S d vol .

(2.32)

Observe that by (2.20)1 the first term in (2.32) yields: 1 2

M

k 4

(∂t τk ) g, ˆ S d vol = −

M

(Tr S ) Tr (Ak−1 ∇N S ) d vol .

We have ∂t V (bˆ k−1 ) = (∂t τ1 )bˆ k−1 + τ1 ∂t bˆ k−1 − ∂t ∇tN bˆ k−1 , where by Example 2.3(a), M

M

∂t bˆ k−1 , S d vol =

∂t (∇N bˆ k−1 ), S d vol =

M

M

Tr (S Ak−1 S ) 1 k−2 d vol, − Tr S ∑ j=0 A j (∇N S )Ak−2− j 2 Tr (S (∇tN Ak−1 )S ) − Tr(Ak−1 [S , A]S ) 1 t k−2 j t k−2− j d vol. − Tr S ∇N ∑ j=0 A (∇N S )A 2

Using the above, and the identities M

Tr (∇N S ) Tr (S Ak−1 ) d vol = kV (A

k−1

M

Tr S V (Ak−1 ) − Ak−1∇N S Tr S d vol,

) = (τk − Iτ ,k (g)) id

(at a critical metric)


45

(or directly from (2.29) with f = τk ) we obtain I f (g¯t ) |t=0 =

1

k (τk − Iτ ,k (g))(Tr S )2 − Tr (Ak−1 [ S , A]S ) 2 M 4 k k k−2 − Tr (Ak−1 ∇N S ) Tr S + Tr (∇N S ) ∑ j=0 A j (∇N S )Ak−2− j d vol. 2 4

From the above the required formulae for Φi (i = 1, 2, 3) follow.

Notice that for f = τ2 the form Φ2 (∇N S ), ∇N S is non-negative definite. By Lemma 2.7(i), if the lengths of N-curves are unbounded then μ (Φ2 ) = 0. In the next consequence of Theorem 2.1 (for f = σk ) we represent the operators Φk explicitly using Newtonian transformations. Corollary 2.3. The gradient of the functional Iσ ,k : M → R for k > 1 and its projection via π∗ : T M → T M1 are given by: 1 1 ∇Iσ ,k (g) = (σk − Iσ ,k (g)) gˆ − V (Tk−1 (b)). 2 2 The F M - and F M 1 - components of above gradients are given, respectively, by: ∇F Iσ ,k (g) =

1 2

n−k+1 V (σk−1 ) g. σk − Iσ ,k (g) − ˆ n

The second variation of Iσ ,k (k > 1) at a critical metric g = g¯0 , and its restriction to the F -conformal variations S = s gˆ (s : M → R) are given by (2.24), where S = ∂t g¯t , f = σk , Φ f = 14 (n − k + 1)(n − k + 2) σk−2, and 1

− 1 Tk−1 (A)[S , A], (σk − Iσ ,k (g))(Tr S ) id 4 2 1 k−2 j Φ2 (S ) = ∑ j=0 (−1) j Tr (Tk− j−2 (A)S )A j − σk− j−2 ∑ p=0 A p S A j−p , 4 1 Φ3 (S ) = − (Tr S ) Tk−1 (A). 2

Φ1 (S ) =

Proof. Using only Proposition 2.2 and the identity Tr Tk (A) = (n − k) σk we obtain B f = 12 Tk−1 (b). As in the proof of Theorem 2.1 (see (2.25) with f = σk ), we get Iσ ,k (g¯t ) =

M

1 ∂t σk + (σk − Iσ ,k(g¯t )) Tr S d volt . 2

By (2.20)2 and Lemma 2.6, we have at t = 0

1 (∂t σk ) d vol = − 2 M

M

1 Tk−1 (A), ∇N S d vol = − V (Tk−1 (A)), S d vol. 2 M

46


The above (or Theorem 2.1 with B f = 12 Tk−1 (b) and Tr Bf = 12 (n − k + 1)σk−1) yield the formula for the gradient ∇Iσ ,k (g). Concerning the second variation of Iσ ,k , as in the proof of Theorem 2.1 (with f = σk ) or by (2.28), one has Iσ ,k (g¯t ) |t=0 =

1 2

M

(∂t σk ) gˆ + (σk − Iσ ,k (g))S − ∂t V (Tk−1 (b)), S d vol. (2.33)

We have

∂t V (Tk−1 (b)) = (∂t τ1 )Tk−1 (b) + τ1 ∂t Tk−1 (b) − ∂t ∇tN Tk−1 (b), where ∂t Tk−1 (b) and ∂t ∇tN Tk−1 (b) are given in Example 2.3(b). Using the identities

(at a critical metric), V (Tk−1 (A)) = (σk − Iσ ,k (g))id 1 Tr ([ [Tk−1 (A), S ], A] S ) = Tr (Tk−1 (A)[S , A] S ), 2 from (2.33), as in the proof of Theorem 2.1, we obtain Iσ ,k (g¯t )|t=0 =

1 2

M

1 (σk − Iσ ,k (g))(Tr S )2 − Tr (Tk−1 (A)[S , A]S ) 2

1 k−2 (−1) j Tr (Tk− j−2 (A)∇N S ) Tr (A j ∇N S ) 2 ∑ j=0 1 k−1 j−1 + ∑ j=1 (−1) j σk− j−1 ∑ p=0 Tr (A p (∇tN S )A j−p−1 (∇N S )) d vol . 2

− Tr (Tk−1 (A)∇N S ) Tr S +

The formulae for Φi (i = 1, 2, 3) follow from the above.

Although the Ricci tensor is the notion of intrinsic geometry, the functional EN (g) =

M

Ric(N, N) d volg ,

g∈M

(the total normal Ricci curvature ) belongs to extrinsic geometry of a foliation F on (M, g): by known integral formula (1.3) we have EN = Iσ ,2 . Example 2.5. For the function f = σ2 , we have the following particular case of Corollary 2.3. The gradient of the functional EN : M → R and its projection via π∗ : T M → T M1 are given by ¯ N (g) = 1 (σ2 − EN (g)) gˆ − 1 V (T1 (b)). ∇E 2 2 Recall that T1 (b) = σ1 gˆ − bˆ 1 . The F M 1 - and F M - components of the above gradient are

2.4 Applications and Examples

47

1 n−1 ∇ EN (g) = σ2 − EN (g) − ˆ V (σ1 ) g. 2 n F

The second variation of EN at a critical metric g = g¯0 , and its restriction to the F -conformal variations are given by (2.24), where S = ∂t g¯t , f = σ2 , and 1

− 1 T1 (A)[S , A], (σ2 − EN (g))(Tr S ) id 4 2 1 1 1

− S , Φ2 (S ) = (Tr S ) id Φ3 (S ) = − (Tr S ) T1 (A). 4 4 2

Φ1 (S ) =

Notice that the form Φ2 (∇N S ), ∇N S is not definite. By Lemmas 2.7 and 2.8, if the lengths of N-curves are unbounded, there are no stable critical metrics for EN (g). Example 2.6. The property “being umbilical” (or being close, in some sense, to such), relates to the measure of “nonumbilicity” for foliations, see [30]. The measure of “nonumbilicity” for foliations is expressed by the functional (2.1) with f = ∑i< j (ki − k j )2 = n τ2 − τ12 . Metrics with minimal total “nonumbilicity” (if they exist) are critical for the functional UF (g) = M (n τ2 − τ12 ) d vol. Using notations of Theorem 2.1, one has B f = n bˆ 1 − τ1 g. ˆ (Notice that Tr B f = 0). In this case, 1 ∇UF = V (τ1 ) + (nτ2 − τ12 −UF (g)) gˆ − nV (bˆ 1 ). 2 We also have

1

− nA [S, A], (Tr S ) nτ2 − τ12 −UF (g) id 4 n

Φ2 (S ) = S , Φ3 (S ) = −(Tr S )(nA − τ1 id). 2

Φ1 (S ) =

Hence, Φ2 (∇N S), ∇N S ≥ 0. By Lemma 2.7(i), if the lengths of N-curves on M are unbounded then μ (Φ2 ).

2.4 Applications and Examples 2.4.1 Variational Formulae for Umbilical Foliations Let F be an umbilical foliation on (M, g) with the normal curvature λ : M → R. One may show that F -conformal variations gt ∈ M preserve this property (i.e., λ = H = 1n τ1 ).

48


Proposition 2.4. Let F be an umbilical foliation on (M, g0 ). If gt ∈ M (0 ≤ t < ε ) is an F -conformal variation of g0 then F is umbilical for any gt . Proof. The claim follows from Lemma 2.3 (see also Lemma 2.9 for S = s g). ˆ

Given function ψ ∈ C(R), consider the functional Iψ : M | U → R on the space U of all Riemannian metrics with respect to which F is umbilical, Iψ (g) =

M

ψ (λ ) d volg .

(2.34)

Corollary 2.4. Let F be an umbilical foliation on (M, g) with the normal curvature λ , and ψ ∈ C2 (R). The F M 1 - and F M - components of the gradient of the functional Iψ are given by: 1 1 V ( ∇F Iψ (g) = ψ (λ ) − Iψ (g) − ψ (λ )) g. ˆ (2.35) 2 n The second variation of Iψ at a critical metric g = g¯0 ∈ M1 , with respect to F -conformal (i.e., of T F M or T F M 1 ) variations g¯t ∈ M1 with S = s gˆ (s ∈ C1 (M)) is 1 Iψ (g¯t ) |t=0 = ψ (λ )N(s)2 d vol. (2.36) 4 M Proof. Because τi = nλ i , we set f = ψ (τ1 /n) and apply Theorem 2.1. In this case, 1 B f = 2n ψ g, ˆ and, see (2.35), Iψ (g¯t ) =

1 2

1 s(ψ − Iψ (g) − V (ψ )) d vol. n M

We prove (2.36) directly. To find Iψ we differentiate the above and get Iψ (g¯t ) =

1 2

M

s n ∂t (ψ ) − ∂t (V (ψ )) d vol.

(2.37)

Using ∂t λ = − 12 N(s), ∂t (ψ ) = − 12 ψ N(s), and so on, we compute

∂t (V (ψ )) =

1 (−nψ − nλ ψ + N(ψ )) s N(s) + ψ s N(N(s)) . 2

Hence, the components of the integral (2.37) are 1 n 2 2 s ∂t (V (ψ )) d vol = ψ N(s) − V (ψ )s d vol, 4 M M 2

M

s n ∂t (ψ ) d vol = −

This yields the formula for Iψ .

n 4

M

V (ψ ) s2 d vol.


49

Remark 2.6. If ψ > 0 then the functional Iψ : M1 → R restricted to umbilical metrics has at most one critical point. See also Remark 2.5.

2.4.2 The Energy and Bending of the Unit Normal Vector Field The energy of a unit vector field N on (M n+1 , g) can be expressed by the formula 1 EN (g) = (n + 1) vol(M, g) + 2

M

∇N2 d volg ,

see, for example, [10]. The last integral, BN (g) =

M

∇N2 d volg ,

up to the constant cn+1 = 1n vol(Sn+1 (1)), is called the total bending of N. The problems of minimizing EN (g) and BN (g) with respect to variations g¯t ∈ M1 are equivalent. Let e0 = N and e1 , . . . , en be a local orthonormal basis of (M, F , g). We calculate ∇N2 = ∑i=1 g(∇ei N, ∇ei N) + g(Z, Z) = τ2 + Z2, n

where Z = ∇N N for short. Thus, we decompose the bending into two parts, BN = Iτ ,2 + BN⊥ ,

where BN⊥ (g) =

M

Z2 d vol.

Notice that BN⊥ = 0 for Riemannian foliations, i.e., Z = 0. Lemma 2.10. The vector field Z = ∇tN N is evolved by gt ∈ M with S = ∂t gt as (i) ∂t Z = −S (Z),

(ii) ∂t Z = −s Z

for S = s g. ˆ

(2.38)

In particular, all variations gt ∈ M preserve Riemannian foliations. Proof. We use (2.11) to compute for any X ∈ T F gt (∂t Z, X) =

1 2(∇tN S)(X, N) − (∇tX S)(N, N) = −S(∇tN N, X) = −gt (S (Z), X). 2

From this, all of (2.38) follow. If Z = 0 at t = 0 then by uniqueness of a solution to the linear ODE (2.38)(i) along N-curves, we have Z = 0 for all t. By Lemma 2.10, we have ∂t (Z ) = 0. Indeed, we calculate (for any vector X)

∂t (Z )(X) = ∂t (g(Z, X)) = S(Z, X) + g(∂t Z, X) = 0. ˆ = Z2 . Notice that Z Z , S = S(Z, Z), in particular, Z Z , g

50


As the components of 1-form Z = g(Z, ·) are (Z )i = Z a gia , by definition of the tensor product, we have (Z Z )i j = Z a gia Z b g jb . From this we obtain Z Z , S = Z a gia Z b g jb Si j = Z a Z b Sab = S(Z, Z). By Lemma 2.10, we have ∂t (Z Z ) = 0. Indeed, we find (for any vectors X,Y )

∂t (Z Z )(X,Y ) = ∂t (g(Z,X)g(Z,Y )) = S(Z,X)g(Z,Y )+g(Z,X)S(Z,Y )+g(∂t Z,X)g(Z,Y ) +g(Z,X)g(∂t Z,Y ) = 0. Theorem 2.2. The gradient of the bending functional BN : M → R (and its projection via π∗ : T M → T M1 ) is given by ∇BN (g) =

1 Z2g + τ2 − BN (g) gˆ − Z Z − V (bˆ 1 ), 2

where Z = ∇tN N. The F M 1 - (and F M -) component of the gradient is ∇F BN (g) =

1 1 1 ˆ Z2g + τ2 − BN (g) − Z2g − V (τ1 ) g. 2 n n

The second variation of BN at a critical metric g = g¯0 , where S = ∂t g¯t , and its restriction to the F -conformal variations (i.e., S = s g, ˆ s : M → R) are, respectively, BN (g¯t ) |t=0 = B¯ N (g¯t )|t=0 =

M

(Φ1 (S), S + Φ2 (∇N S), ∇N S + Φ3 (S), ∇N S) d vol,

n n

2

M

2

− 1 Z2g s2 + N(s)2 d vol,

where

Φ1 (S ) = − Φ2 (S ) =

1

− A[S, A] − S2(Z, Z) id,

Z2g + τ2 − BN (g) (Tr S ) id 4

1 S, 2

Φ3 (S ) = Tr (A S ) id.

Proof. First, using (2.11) and case (i) in Lemma 2.10, we compute

∂t Z2gt = (∂t gt )(Z, Z) + 2 gt (∂t Z, Z) = S(Z, Z) − 2 S(Z, Z) = −S(Z, Z).


51

For f = Z2gt + τ2 we have

∂t f = −S(Z, Z) − Tr (A∇tN S ). Hence

M

(∂t f ) d vol = −

M

S(Z, Z) + V (bˆ 1 ), S d vol .

Then, similarly to (2.25) and (2.27), we obtain BN (g¯t ) =

1 M

2

Z2gt + τ2 − BN (g¯t ) gˆ − Z Z − V (bˆ 1 ), S d vol . (2.39)

In order to find the second variations, using Lemma 2.4, as for (2.28) we compute BN (g¯t ) |t=0

=

M

−

1 2 t (Zg + τ2 − BN (g))S, S + Tr(A∇N S ) (Tr S ) 2

1 S(Z, Z) − ∂t V (bˆ 1 ), S d vol. 2

We have ∂t V (bˆ 1 ) = (∂t τ1 )bˆ 1 + τ1 ∂t bˆ 1 − ∂t ∇tN bˆ 1 , where by Example 2.3(a), 1 2 τ1 Tr (S AS ) − N Tr (S ) d vol, 4 M M Tr S (∇tN A)S − Tr (A[S , A]S ) ∂t (∇N bˆ 1 ), S d vol =

τ1 ∂t bˆ 1 , S d vol =

M

M

Hence M

∂t V (bˆ 1 ), Sd vol =

1 1 2 t 2 − τ1 N(Tr (S )) + Tr ((∇N S ) ) d vol . 4 2

M

1 − N(Tr S ) Tr (AS ) + Tr (S V (A)S ) 2 1 t 2 + Tr (A[S , A]S ) − Tr ((∇N S ) ) d vol. 2

Finally, we obtain (see also Corollary 2.2 for k = 2) 1 BN (g¯t ) |t=0 = − Z2g + τ2 − BN (g) (Tr S )2 − Tr (A[S , A]S ) 4 M 1 −S2 (Z, Z) + Tr (A S )N(Tr S ) + Tr ((∇N S )2 ) d vol . 2 Formulae for F -conformal case follow directly from above.

Chapter 3

Extrinsic Geometric Flows

Abstract In the chapter we study the metrics gt satisfying the Extrinsic Geometric Flow equation (see Sect. 3.2). Sections 3.4 and 3.5 collect results about existence and uniqueness of solutions (Theorems 3.1 and 3.2) and their proofs. The key role in proofs play hyperbolic PDEs and the generalized companion matrix studied in Sect. 3.3. In Sect. 3.6, we estimate the maximal existence time. In Sect. 3.7 we use the first derivative of functionals (when they are monotonous) to show convergence of metrics in a weak sense (Theorem 3.3). In Sect. 3.8 we study soliton solutions of the geometric flow equation (Theorems 3.4 and 3.5), and characterize them in the cases of umbilical foliations and foliations on surfaces (Theorems 3.6–3.8). Section 3.9 is devoted to applications and examples, including the geometric flow produced by the extrinsic Ricci curvature tensor (Theorem 3.9).

3.1 Introduction Geometric Flow (GF) is an evolution equation associated to a functional on a manifold which has a geometric interpretation, usually related to some extrinsic or intrinsic curvature. They all correspond to dynamical systems in the infinite dimensional space of all possible metrics on a given manifold,

∂t gt = h(gt ). These evolutions try to move g0 toward a metric that is more natural for its underlined topology (e.g., with constant curvature). GF equations are quite difficult to solve in all generality, because of their nonlinearity. Although the short time existence of solutions is guaranteed by the parabolic or hyperbolic nature of the equations, their (long time) convergence to canonical metrics is analyzed under various conditions (e.g., in connection to the problem of formation of singularities). The theory of intrinsic (driven by curvature in various forms) and extrinsic (driven by extrinsic curvature of submanifolds) GFs is a modern subject of common critical interest in mathematics and physics. The most popular GF in V. Rovenski and P. Walczak, Topics in Extrinsic Geometry of Codimension-One Foliations, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4419-9908-5 3, © Vladimir Rovenski and Paweł Walczak 2011

53

54

3 Extrinsic Geometric Flows

mathematics are the Ricci flow and the Mean Curvature flow. Other examples are Gaussian curvature flow, Yamabe flow, Renormalization group flow, etc. The physical applications of GF include problems in quantum field theory as well as problems in fluid mechanics, general relativity, and string theory. Some of the most striking recent results in differential geometry and topology are related to the Ricci flow, that is the deformation gt of a given Riemannian metric g0 on a manifold M subject to the equation

∂t gt = −2 Rict , where Rict is the Ricci curvature tensor on the Riemannian manifold (M, gt ). This evolution equation has been introduced by Hamilton [23] (following earlier work of Eells and Sampson [19] on the harmonic map heat flow, see also [5]) who proposed its use in solving famous Poincaré and Geometrization Conjectures. Using the Ricci flow, Hamilton proved that every compact three-dimensional manifold with positive Ricci curvature is diffeomorphic to a spherical space form. Hamilton’s program was completed by Perelman who proved these conjectures in the series of preprints [35–37]. As some authors claim “Perelman’s proofs are concise and, at times, sketchy”, several authors made efforts to provide details that are missing in Perelman’s preprints, or to present “a proof more attractive to topologists”, see, for example, [7, 25]. Also, there exist several successful attempts to modify/simplify proofs of some elements constituting the proof of the Conjectures. Discussing these proofs is not our goal here (equations describing our extrinsic geometric flows (EGFs) are quite different from those related to the Ricci flow), these days interested readers can find enormous amount of articles (especially at arXiv) and books on Ricci flow, [2, 7, 8, 13, 14, 53], and so on. A famous (Rauch, Klingenberg, and Berger) theorem states that a complete simply connected n-dimensional Riemannian manifold, for which the sectional curvatures are strictly between 1 and 4, is homeomorphic to n-sphere. It has been a longstanding open conjecture as to whether or not the homeomorphism conclusion could be strengthened to a diffeomorphism. Only recently, BrendleSchoen proved this hypothesis, see [2, 8]. Their proof is based on analysis of Ricci flow including: Perelman’s monotonicity formulae, the blow-up analysis of singularities, and development of recent convergence theory for the Ricci flow. On the other hand, for at least 30 years there has been continuous interest in the study of Mean Curvature Flow, i.e., the variation of immersions F0 : M¯ → M of manifolds M¯ into Riemannian manifolds (M, g) subject to the mean curvature vector field, i.e., to the equation

∂t Ft = Ht , ¯ see [18,21]. Also, the second author where Ht is the mean curvature vector of Ft (M), [55] considered the foliated version of the Mean Curvature Flow: foliations which are invariant under the flow of the mean curvature vector of their leaves. The main goal of Chap. 3 is the study of Extrinsic Geometric Flows on foliations. They are defined as deformations of Riemannian metrics on a manifold M equipped

3.2 The Systems of PDEs Related to EGFs

55

with a codimension-one foliation subject to conditions expressed in terms of the second fundamental form of the leaves and its scalar invariants. The main results of the chapter are the local existence/uniqueness theorems, estimations of the existence time of solutions, the convergence in a weak sense to minimal, and totally geodesic foliations. Other results concern the geometry of extrinsic Ricci and Newton transformation flows, the introducing of geometric solitons and their classification among umbilical foliations and metrics on foliated surfaces.

3.2 The Systems of PDEs Related to EGFs We study two types of evolution of Riemannian structures, depending on functions f j (0 ≤ j < n) (at least one of them is not identically zero): (a) f j ∈ C2 (M × R), → (b) f j = f˜j (− τ , ·), where f˜j ∈ C2 (Rn+1 ). → τ ) with f˜j ∈ C2 (Rn ). Sometimes we will assume that f j = f˜j (− Definition 3.1. Given functions f j of type (a) or (b), a family gt , t ∈ [0, ε ), of Riemannian structures on (M, F ) will be called an Extrinsic Geometric Flow (EGF) if n−1 ∂t gt = ht , where ht = h(bt ) = ∑ f j bˆ tj . (3.1) j=0

Here, bˆ tj are symmetric (0, 2)-tensor fields on M gt -dual to (At ) j (See Sect. 1.2.2). The choice of the right-hand side in (3.1) for h(b) seems to be natural, the powers bˆ j are the only (0, 2)-tensors which can be obtained algebraically from the second fundamental form b, while τ1 , . . . , τn (or, equivalently, σ1 , . . . , σn ) generate all the scalar invariants of extrinsic geometry. Powers bˆ j with j > 1 in (3.1) are meaningful; for example, the EGFs produced by: – The extrinsic Ricci curvature tensor Ric ex (b), see (3.87), depends on bˆ 1 and bˆ 2 . – The Newton transformation Ti (b) = Ti (A) , see (1.6), depends on all bˆ j ( j ≤ i). In other words, the EGF is the evolution equation which deforms Riemannian metrics by evolving them along F in the direction of the tensor h(bt ). Indeed, any EGF preserves N to be unit and perpendicular to F , therefore, the F -component of the vector does not depend on t. One can interpret EGF as integral curves of a vector field g → h(b), h(b) being the right-hand side of (3.1), on the space C k (M, S2+ (M)) of Riemannian Ck -structures on M. Here, certainly, S2+ (M) is the bundle of positive definite symmetric (0, 2)tensors on M. This vector field may or may not depend on time. Although EGF of type (b) consists of first-order nonlinear PDEs, the corresponding power sums τi (i > 0) satisfy an infinite quasilinear system.

56


Lemma 3.1. Power sums {τi }i∈N of the EGF (3.1) of type (b) satisfy the system

∂t τi = −

jfj i n−1 N(τi+ j−1 ) + τi+ j−1N( f j ) , τi−1 N( f0 ) + ∑ j=1 2 i+ j−1

(3.2)

where N( f j ) = ∑ns=1 f j,τs N(τs ). Moreover, the evolution of σk (k ≤ n) is given by k−1 n−1 2 ∂t σk = − ∑i=0 (−1)i σk−i−1 N( f0 ) τi + ∑ j=1 N( f j ) τi+ j +

j f j N(τi+ j ) . i+ j (3.3)

Proof. For the EGF (3.1), the equality (2.20)1 with S = h(b) reduces to the system i n−1 ∂t τi = − Tr Ai−1 ∇tN ∑ j=0 f j A j , 2

i > 0.

The desired (3.2) follows from the above, using the identity N(τi+ j ) = Tr (∇tN Ai+ j ) =

i+ j Tr (Ai ∇tN A j ). j

(3.4)

We also prove (3.2) directly in what follows, see Remark 3.3 in Sect. 3.5.1. From (2.20)2 with S = h(b), we find n−1 2 ∂t σk = − Tr Tk−1 (A)∇tN ∑ j=0 f j A j .

From the above, using the identity (3.4), we obtain (3.3). Example 3.1. We define the kth Newton transformation flow as:

∂t gt = Tk (bt ).

(3.5)

In other words, we assume f j = (−1) j σk− j in (3.1). For the flow (3.5), the equality (2.20)1 with S = Tk (b) reduces to the system i ∂t τi = − Tr (Ai−1 ∇tN Tk (A)), 2

i > 0.

Using the identity (3.4), we obtain the system of PDEs of type (3.2), i k ∂t τi = − Tr Ai−1 ∇tN ∑ j=0 (−1) j σk− j A j 2 i k = − ∑ j=0 (−1) j N(σk− j )τi+ j−1 + σk− j Tr (Ai−1 ∇N (A j )) 2 i j k j =− N(σk ) τi−1 + ∑ j=1 (−1) N(σk− j ) τi+ j−1 + σk− j N(τi+ j−1 ) . 2 i+ j−1

3.2 The Systems of PDEs Related to EGFs

57

From (2.20)2 with S = Tk (b) we also obtain the system of PDEs for σ ’s 1 ∂t σi = − Tr (Ti−1 (A)∇tN Tk (A)), 2

0 < i ≤ n.

(3.6)

In particular, for i = 1, we have 1 1 ∂t σ1 = − Tr (∇tN Tk (A)) = − (n − k)N(σk ). 2 2 For k = 1, (3.6) reduces to the system (for 0 < i ≤ n) 1 1 1 ∂t σi = − Tr (Ti−1 (A)∇tN T1 (A)) = − (n − i + 1)σi−1N(σ1 ) + N(σi ). 2 2 2

(3.7)

For i = 1 we have the linear PDE 1 ∂t σ1 = − (n − 1) N(σ1 ) 2 representing the “unidirectional wave motion” σ1t (s) = σ10 (s−t(n − 1)/2) on any Ncurve γ : s → γ (s). Hence, the (triangular) system (3.7) of PDEs for σ ’s is solvable. EGFs preserve the following properties of foliations to be:

see Proposition 3.4. – umbilical (A = λ id), – totally geodesic (A = 0), see (2.19). – Riemannian (∇N N = 0), see Lemma 2.10. For a particular choice of functions f j , EGF might preserve a certain extrinsic geometric property (P) of foliations. For example, by (3.2), EGFs preserve minimal (τ1 = 0) foliations, when the generating functions satisfy the conditions f j (0, τ2 , . . . , τn ) ≡ 0 ( j = 1). Let gt be a solution of (3.1). From (3.2) with i = 1 we have n−1 → → → τ )) − ∑ j=1 f j (− τ )N(τ j ) + τ j N( f j (− τ )) . 2 ∂t τ1 = −n N( f0 (− One may show that if F is g0 -minimal then ∂t τ1 = 0, i.e., F is gt -minimal for all t. We propose the EGF (3.1) as a tool for studying the following question: Under what conditions on (M, F , g0 ) the EGF metrics gt converge to one for which F enjoys a given extrinsic geometric property (P), e.g., is umbilical, totally geodesic, Riemannian, minimal, etc.?

58


3.3 Auxiliary Results 3.3.1 Diffeomorphism Invariance of EGFs It is well known that the Riemannian curvature tensor is invariant under local isometries, and the Ricci curvature tensor Ric is preserved by diffeomorphisms φ ∈ Diff(M) in the sense that φ ∗ (Ric(g)) = Ric(φ ∗ g). Hence, Ric is an intrinsic geometric tensor. In the same way, the second fundamental form b of a foliation is invariant under diffeomorphisms preserving the foliation; thus, b is an extrinsic geometric tensor for the foliation. More precisely, we have the following Proposition 3.1. Let (Mi , Fi , gi ) (i = 1, 2) be Riemannian manifolds with foliations, and φ : M1 → M2 a diffeomorphism such that F1 = φ −1 (F2 ) and g1 = (σ · φ ∗ gˆ2 ) ⊕ φ ∗ (g⊥ 2 ) (for some positive σ ∈ R). Then the Weingarten operators Ai , → the second fundamental forms b(i) , and − τ (i) (the set of τ -s for Ai ) satisfy b (1) = σ · φ ∗ b (2) ,

− → → τ (1) = − τ (2) ◦ φ .

A1 = φ∗−1 A2 φ∗ ,

(3.8)

→ ˆ − For any tensor h(b) of the form h(b) = ∑n−1 j=0 f j ( τ ) b j , see (3.1), we have h(b (1) ) = σ · φ ∗ (h(b (2) )).

(3.9)

Proof. This follows from the following computation (see also Lemma 2.3). Notice that for any vector X ∈ T F1 , the vector φ∗ X = d φ (X) is tangent to F2 . From the definition of the metric g1 , we have g1 (X,Y ) = σ · g2 (φ∗ X, φ∗Y ) for X,Y ∈ T F1 . Let N2 be a unit normal of F2 . Then N1 = φ∗−1 N2 is a unit normal of F1 . In fact, g1 (X, N1 ) = g2 (φ∗ X, N2 ) = 0

(X ∈ T F1 ),

g1 (N1 , N1 ) = φ ∗ (g2 )(φ∗−1 N2 , φ∗−1 N2 ) = g2 (φ∗ φ∗−1 N2 , φ∗ φ∗−1 N2 ) = g2 (N2 , N2 ). For the second fundamental form b (2) : T F2 × T F2 → R of F2 , we obtain a symmetric (0, 2)-tensor field φ ∗ b (2) : T F1 × T F1 → R. Denote by ∇(i) the LeviCivita connection of (Mi , gi ). For arbitrary vector fields X,Y ⊂ T F1 we have (2)

(2)

(φ ∗ b (2) )(X,Y ) = b (2) (φ∗ X, φ∗Y ) = −g2 (∇φ∗ X N2 , φ∗Y ) = −g2 (∇φ∗ X φ∗ N1 , φ∗Y ) (1)

(1)

= −g2 (φ∗ ∇X N1 , φ∗Y ) = −φ ∗ (g2 )(∇X N1 ,Y ) = σ −1 · b (1)(X,Y ). Hence, the second fundamental form of a foliation is an extrinsic geometric tensor. Consequently, we have

φ∗ (A1 X) = A2 (φ∗ X) (X ∈ T F1 )

⇒

A1 = φ∗ A2 φ∗−1 .


59

Taking the trace, we get the identity for τ ’s. For h(b (i) ) we have → τ (1) ) bˆ j ∑ j=0 f j (− n−1

(1)

n−1 (2) → (X,Y ) = σ ∑ j=0 f j (− τ (2) ◦ φ ) bˆ j (φ∗ X, φ∗Y ).

From the above, (3.9) follows.

¯ F¯ ) is a diffeomorphism such that gt = φ ∗ g¯t , F = φ −1 (F¯ ), So if φ : (M, F ) → (M, ¯ F¯ ), by Proposition 3.1 with σ = 1, we obtain and g¯t is a solution to EGF on (M,

∂t gt = φ ∗ (∂t g¯t ) = φ ∗ h(b¯ t ) = h(bt ). Hence, gt is also a solution to EGF on (M, F ). Therefore, EGF with h(b) = → ˆ − ∑n−1 j=0 f j ( τ ) b j is invariant under the group D(F , N) of diffeomorphisms preserving both, the foliation F and the unit normal N. We can also translate EGFs along the time coordinate; If gt satisfies to the EGF equation (3.1), then so does gt−t0 for any t0 . Furthermore, EGF has a scale invariance along F ; If gt is a solution to EGF, and μ > 0 is real, then g¯t := ⊥ is also a solution. (μ gˆt/μ ) ⊕ gt/ μ

3.3.2 Quasi-Linear PDEs Now, we recall some facts about hyperbolic systems of quasi-linear PDEs. Let A = (ai j (x,t, u)) be an n × n matrix, b = (bi (x,t, u)) – an n-vector. A firstorder quasilinear system of PDEs, n equations in n unknown functions u = (u1 , . . . , un ) and two variables x,t ∈ R, has the form:

∂t u + A(x,t, u) ∂x u = b(x,t, u).

(3.10)

When the coefficient matrix A and the vector b are functions of x and t only, the system is just linear; if b alone depends also on u, the system is said to be semilinear. The initial value problem for (3.10) with given smooth data u0 , A and b consists in finding smooth function u(x,t) satisfying (3.10) and u(x, 0) = u0 (x). Definition 3.2. The system (3.10) is hyperbolic in the t-direction at (x,t, u) (in an appropriate domain of the arguments of A and b) if the right eigenvectors of A are real and span Rn . For a solution u(x,t) to (3.10), the corresponding eigenvalues λi (x,t, u) are called the characteristic speeds. The system is strictly hyperbolic if the functions λi (x,t, u) are distinct. For the hyperbolic system (3.10), the vector field ∂t + λi (x,t, u)∂x is called the i-characteristic field, and its integral curves are called i-characteristics, see Fig. 3.1. Remark that the hyperbolicity of A is equivalent to any of the properties: “A has real eigenvalues λ1 ≤ . . . ≤ λn and simple elementary divisors (i.e., A has no Jordan cells of order greater than one)” and “A is diagonal in some affine basis”. Hence, the

60


Fig. 3.1 Method of characteristics

hyperbolic matrix A can be represented as A = RDR−1 , where R is a nonsingular n × n matrix and D is a diagonal matrix. The columns ri of R are the right eigenvectors of A, whereas the rows of L−1 are left eigenvectors of A. A hyperbolic system reduces to the ODEs for its characteristic fields. Indeed, multiplying (3.10) by ri T and using du/dt = ∂t u + λi ∂x u, we obtain the ODE ri T du/dt = ri T b along the characteristic dx/dt = λi (x,t, u). Theorem A [24]. Let the quasi-linear system (3.10) be such that 1) It is hyperbolic in the t-direction in Ω = {|x| ≤ a, 0 ≤ t ≤ s, u∞ ≤ r} for some s, r > 0. 2) the matrix A and the vector b are C1 -regular in Ω . If the initial condition u(0, x) = u0 (x),

x ∈ [−a, a]

(3.11)

has C1 -regular u0 in [−a, a] and u0 ∞ < r then (3.10) – (3.11) admit a unique C1 regular solution u(x,t) in the domain Ω¯ = {(x,t) : |x| + Kt ≤ a, 0 ≤ t ≤ ε }, with K = max{|λi (x,t, u)| : (x,t, u) ∈ Ω , 1 ≤ i ≤ n}. Example 3.2. For any function ψ ∈ C1 (R), we can multiply the equation

∂t u + ψ (u) ∂x u = 0,

(3.12)

by ψ (u), and obtain ∂t ψ + ψ · ∂x ψ = 0 (the inviscid Burgers’ equation). Thus, the behavior of the solutions to (3.12) (for t before the first singular value) is not expected to be much different from that of Burgers’ equation.

3.3.3 Generalized Companion Matrices Let Pn = λ n − p1 λ n−1 − . . . − pn−1 λ − pn be a polynomial over R and λ1 ≤ λ2 ≤ . . . ≤ λn be the roots of Pn for n > 0. Hence, pi = (−1)i−1 σi , where σi are elementary symmetric functions of the roots λi .


61

Definition 3.3. Let c1 = 1 and ci = 0 (i > 1) be arbitrary numbers. The generalized companion matrices of Pn are defined by: ⎛ ⎜ ⎜ ⎜ Cˆ = ⎜ ⎜ ⎜ ⎝

0 0 ··· 0

cn−1 cn

0 ··· 0

cn pn cn−1 pn−1

⎞ ⎛ ⎞ 0 c1 p1 c2 p2 . . . cn−1 pn−1 cn pn cn−2 ⎟ c ⎜ 1 0 ⎟ 0 ··· 0 0 ⎟ cn−1 ⎜ c2 ⎟ ⎟ ⎟ c2 ˇ =⎜ ··· ··· ⎟ and C 0 · · · 0 0 ⎜ ⎟. ⎟ c 3 ⎜ c1 ⎟ ⎟ ⎝ ··· ··· ··· ··· ··· ··· ⎠ c2 ⎠ cn−1 0 ··· 0 0 cn . . . c2 p2 c1 p1 0

··· ··· ··· 0

ˆ where x = (x1 , . . . , xn ). Inverting the order Notice that Cˆ acts on Rn as x → Cx, ˇ If all ci ’s are of indices, i.e., taking (xn , . . . , x1 ), one may describe this action by C. ˆ equal to 1, the matrix C reduces to the standard companion matrix C of Pn . Explicit formulae (polynomials) for entries in powers of C and some applications to the theory of the symmetric functions are given in [17]. The reader can verify that the −1 ) and c˜ = cn−i+1 : inverse of the matrix Cˆ (when pn = 0) is Cˇ for Pñ = −λ n p−1 i n Pn (λ cn ⎛

−

pn−1 pn cn

⎜ ⎜ cn−1 ⎜ 0 Cˆ −1 = ⎜ ⎜ ⎜ ··· ⎝ 0

−

cn−1 pn−2 c n pn

0 cn−1 cn−2

··· ···

The following matrix (the matrix Cˆ with ci = ⎛

. . . − ccn2 pp1n ··· 0 0 ··· ··· ··· c2 0 c1

n n−i+1 ) plays a

1 2

0 0 ··· 0

c1 1 ⎞ cn pn

0

··· ··· ··· ···

0 0 ··· 0

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

key role in this chapter:

⎜ 2 0 ⎜ 3 ⎜ Bn,1 = ⎜ ··· ··· ⎜ ⎝ 0 0 n (−1)n−1 n1 σn (−1)n−2 n2 σn−1 . . . − n−1 σ2

⎞ 0 0 ⎟ ⎟ ⎟ ··· ⎟. n−1 ⎟ ⎠ n σ1

(3.13)

Lemma 3.2. Generalized companion matrices have the following properties: ˇ is Pn . a) The characteristic polynomial of Cˆ (or C) cn cn n−1 2 b) v j = (1, cn−1 λ j , cn−2 λ j , . . . , cn λ j ) is the eigenvector of Cˆ for the eigenvalue λ j , respectively, w j = (cn λ n−1 , . . . , cn λ j2 , cn λ j , 1) is the eigenvector of Cˇ for λ j . j

cn−2

cn−1

ˆ = V D, where V = { cn λ i−1 }1≤i, j≤n is the Vandermonde type matrix, and c) CV cn−i+1 j D = diag(λ1 , . . . , λn ) a diagonal matrix. (If all λi ’s are distinct then obviously ˆ = D). V −1CV ˆ = Pn , hence the eigenvalProof. (a) We show by induction on n that det |λ idn −C| ˆ ues λi of C are the roots of Pn . Expanding by co-factors down the first column, we obtain

62


ˆ = λ Pn−1 − (−1)n−1 cn pn ∏ n−1 − ci /ci+1 , det |λ idn −C| i=1 where Pn−1 = λ n−1 − p1 λ n−2 − . . . − pn−2 λ − pn−1 (by the induction n−1 ci assumption) is the certain polynomial of degree n − 1. As cn ∏ i=1 ci+1 = 1 and Pn + pn = λ Pn−1 , the claim follows. (b) A direct computation shows that ˆ v j = 0, (λ j idn −C)

ˇ w j = 0. (λ j idn −C)

ˆ = V D, where D = diag(λ1 , . . . , λn ) is a diagonal matrix. If {λ j } are (c) Hence CV ˆ = D. pairwise distinct then detV = 0 and obviously V −1CV Consider now the infinite system of linear PDEs with functions f j ∈ C2 (R2 )

∂t τi +

i n−1 j f j (t, x) ∂x τi+ j−1 = 0, 2 ∑ j=1 i + j − 1

i = 1, 2, . . . ,

(3.14)

where τi (i ∈ N) are the power sums of smooth functions λi (t, x) (1 ≤ i ≤ n). Let σ j ( j ≤ n) be the elementary symmetric functions of {λi (t, x)}. Proposition 3.2. The matrix of the n-truncated system (3.14) (where τn+i ’s are eliminated using suitable polynomials of τ1 , . . . , τn , as described in Remark 1.1) is the following polynomial of the generalized companion matrix (3.13): n−1 m B˜ = ∑m=1 fm · (Bn,1 )m−1 . 2

(3.15)

n−1 The eigenvalues of B˜ are λ˜ i = 12 ∑m=1 m fm λim−1 , while the corresponding eigenvectors (if λi ’s are pairwise different) read vi = (1, 2 λi , 3 λi2 , . . . , n λin−1 ).

Proof. Let Bn,m be the matrix of the n-truncated system (3.14) with f j = δ j,m+1 ,

∂t τi +

i m+1 · ∂x τi+m = 0, 2 i+m

i = 1, 2, . . . ,

→ → i.e., ∂t − τ + Bn,m ∂x − τ = 0. In particular, Bn,0 = 12 id. We need to prove only Bn,m =

m+1 (Bn,1 )m , 2

0 ≤ m < n,

(3.16)

n−1 fm Bn,m−1 . Notice that (3.16) follows directly from the equality therefore, B˜ = ∑m=1

Bn,m =

m+1 Bn,m−1 Bn,1 . m

(3.17)

The formulae (3.17) are true for m = 1. We shall show that all the (i, j)-entries of the matrices m+1 m Bn,m−1 Bn,1 and Bn,m coincide.


63

Replacing ∂x τn+ j in (3.14) by linear combinations of ∂x τi (i ≤ n) due to (3.19) in what follows, we find the (i, j) entry of Bn,m ⎧ ⎪ ⎪ ⎨

i (m + 1) j δ if i + m ≤ n, 2(i + m) i+m (n,m) bi j = ⎪ i (m + 1) ⎪ ⎩ (−1)n− j βn, i+m−n, j if i + m > n, 2j

(3.18)

for some β ’s (studied later in Lemma 3.3). The reader can verify using (3.20)1 that for m = 1 the formulae (3.18) determine the matrix Bn,1 of (3.13). Let us prove (3.17). First, assume i + m − 1 − n ≤ 0. Then, using (3.18), we have m + 1 n (n,m−1) (n,1) b bs j m ∑s=1 is

m+1 n im s s δi+m−1 δ j−1 ∑ s=1 m 2(i + m − 1) s+1 s

= =

m+1 i+m−1 i δ j−1 2 i + m − 1 i+m−1 i + m

=

i(m + 1) j δ 2(i + m) i+m

=

bi j

j=s+1=i+m

(n,m)

.

Now, let i + m − 1 − n = m˜ > 0. Then, assuming j > 1 and using (3.13), (3.18) and (3.20) – (3.21), we have m + 1 n (n,m−1) (n,1) b bs j m ∑s=1 is

m + 1 (n,m−1) (n,1) (n,m−1) (n,1) bn j bi, j−1 b j−1, j + bin m im j−1 (3.18) m + 1 (−1)n− j+1 = βn,m, ˜ j−1 m 2( j − 1) j im n− j n βn,m,n σn− j+1 + ˜ (−1) 2n j =

(3.21) i(m + 1) (−1)n− j (βn+1,m,n+1 σn− j+1 − βn+1,m, = ˜ ˜ j) 2j = The case j = 1 is similar.

i(m + 1) (3.18) (n,m) (−1)n− j βn,m+1, . ˜ j = bi j 2j

Remark 3.1. The generalized companion matrix Bn,1 of (3.13) is hyperbolic if and only if λi ’s are pairwise different. In this case, by Proposition 3.2, the matrices Bn,m (m > 0) of (3.16) and the matrix B˜ of (3.15) are also hyperbolic.

64


Lemma 3.3. The coefficients βn,m,i of the decomposition 1 1 n ∂x τn+m = ∑i=1 (−1)n−i βn,m,i ∂x τi , n+m i

m>0

(3.19)

satisfy the following recurrence relations:

βn,1,i = σn−i+1 ,

βn,m,i = βn+1,m−1,n+1σn−i+1 − βn+1,m−1,i

βn,m,i = βn+ j, m, i+ j

(1 ≤ i ≤ n,

(m > 1),

m, j > 0).

(3.20) (3.21)

Remark 3.2. In view of (3.21), relation (3.20)2 reduces to

βn,m,i = βn,m−1,n σn−i+1 − βn,m−1,i−1

(m > 1).

For small values of m, m = 1, 2, we obtain from (3.19) the relations n−i 1 n (−1) ∂x τn+1 = ∑i=1 σn−i+1 ∂x τi , n+1 i

(3.22)

n−i 1 n (−1) (σ1 σn−i+1 − σn−i+2 ) ∂x τi . ∂x τn+2 = ∑i=1 n+2 i

(3.23)

By Proposition 3.2, the last row of Bn,1 (respectively, of Bn,2 ) consists of the coefficients at ∂x τi ’s on the RHS of (3.22), (respectively, of (3.23)), and so on. Proof (of Lemma 3.3). Let m = 1. The definition τi = ∑ j λ j i yields the equality

∂x τi = i ∑ j λ j i−1 ∂x λ j . Using this, we find

∑i=1 n

(−1)n−i n σn−i+1 ∂x τi = ∑ j ∑i=1 (−1)n−i σn−i+1 λ ji−1 ∂x λ j . i

Define the polynomial Pn (x) = λ n − σ1 (x)λ n−1 + . . . + (−1)n σn (x). As λ j (x) are the roots of Pn , we obtain the identity n−i 1 n (−1) ∂x τn+1 − ∑i=1 σn−i+1 ∂x τi n+1 i = ∑ j λ jn − σ1 λ jn−1 + . . . + (−1)n σn ∂x λ j = 0

that proves (3.22). Hence, βn,1,i = σn−i+1 . In order to prove the recurrence relation in (3.20), assume temporarily that λn+1 = ε , n˜ = n + 1, and m˜ = m − 1. Hence, 1 1 ∂x τn+m = ∂x τn+ ˜ m| ˜ ε =0 . n+m n˜ + m˜


65

Then, we may put ε = 0 and replace ∂x τn+1 (x) via (3.22) to obtain n−i ˜ 1 n˜ (−1) ∂x τn+ βn,˜ m,i ˜ m˜ = ∑i=1 ˜ ∂x τi n˜ + m˜ i n−i+1 1 n (−1) ∂x τn+1 + ∑i=1 βn+1,m−1,i ∂x τi n+1 i n−i n (−1) ε= 0 = ∑i=1 βn+1,m−1,n+1 σn−i+1 − βn+1,m−1,i ∂x τi i

= βn+1,m−1,n+1

that completes the proof of (3.20). For m = 2, we deduce from (3.20) the equality

βn,2,i = βn+1,1,n+1 σn−i+1 − βn+1,1,i = σ1 σn−i+1 − σn−i+2 which proves (3.23). Finally, we prove (3.21) by induction on m. For m = 1, using (3.20)1 , we have

βn+ j, 1, i+ j = σ(n+ j)−(i+ j)+1 = σn−i+1 = βn,1,i . Assuming (3.21) for m − 1 and using (3.20)2, we deduce it for m:

βn+ j, m, i+ j = β(n+ j)+1, m−1, (n+ j)+1σ(n+ j)−(i+ j)+1 − β(n+ j)+1, m−1, i+ j = βn+1, m−1, n+1σn−i+1 − βn+1, m−1, i = βn,m,i .

This completes the proof of (3.21). Example 3.3. For f j = δ j1 , (3.14) reduces to the linear system 1 ∂t τi + ∂x τi = 0, 2 whose solution is a simple wave along the x-axis: τi = τi0 (t − 2 x). Consider the slightly more complicated cases. 1. For f j = δ j2 , (3.14) reduces to the system

∂t τi +

i ∂x τi+1 = 0, i+1

i = 1, 2, . . .

→ → The n-truncated system (3.24) reads: ∂t − τ + Bn,1 ∂x − τ = 0. For n = 2, using (3.22), we have just two PDEs 1 ∂t τ1 = − ∂x τ2 , 2

2 ∂t τ2 = − ∂x τ3 = (τ12 − τ2 )∂x τ1 − τ1 ∂x τ2 . 3

(3.24)

66


1 0 2 has the characteristic polynomial P2 = λ 2 − −2 σ2 σ1 σ1 λ + σ2 . If the roots λ1 and λ2 of P2 are distinct, the eigenvectors of B2,1 are equal to v j = (1, 2λ j ), j = 1, 2. If λ1 = λ2 = 0 then B2,1 has one eigenvector only, hence the system (3.24) is not hyperbolic in the t-direction. For n = 3, (3.24) reduces to the quasilinear system of three PDEs with the matrix ⎛ ⎞ 1 0 0 2 2 ⎠. B3,1 = ⎝ 0 0 3 3 3 σ3 − 2 σ2 σ1

The matrix B2,1 =

The characteristic polynomial of B3,1 is P3 = λ 3 − σ1 λ 2 + σ2 λ − σ3 , the eigenvalues are λ j , and the eigenvectors are v j = (1, 2 λ j , 3 λ j2 ), j = 1, 2, 3. 2. For f j = δ j3 , (3.14) reduces to the system

∂t τi +

3i ∂x τi+2 = 0, 2(i + 2)

i = 1, 2, . . .

(3.25)

The matrix of this n-truncated system is Bn,2 = 32 (Bn,1 )2 . For n = 3, (3.25) reduces to the system of three quasilinear PDEs 1 ∂t τ1 = − ∂x τ3 , 2

3 ∂t τ2 = − ∂x τ4 , 4

∂t τ3 = −

9 ∂x τ5 , 10

where ∂x τ4 and ∂x τ5 should be expressed using (3.22) and (3.23). Hence, the matrix of this system is ⎛ ⎞ 1 0 0 2 3 ⎜ ⎟ B3,2 = (B3,1 )2 = ⎝ 3 σ3 − 32 σ2 σ1 ⎠. 2 9 9 3 2 2 σ1 σ3 4 (σ3 − σ1 σ2 ) 2 (σ1 − σ2 ) Its eigenvalues are 32 λ j2 , and the eigenvectors are the same as for B3,1 . Similarly, for n = 4, (3.25) reduces to the quasilinear system with the matrix ⎞ ⎛ 1 0 0 0 2 ⎟ ⎜ 3 0 0 0 3 ⎟ ⎜ 4 B4,2 = (B4,1 )2 = ⎜ ⎟ 9 9 9 3 2 ⎠ ⎝ − 2 σ4 − 2 σ2 4 σ3 8 σ1 −6 σ1 σ4 3(σ1 σ3 − σ4 ) 2(σ3 − σ1 σ2 )

3 2 2 (σ1 − σ2 )

with eigenvalues λ˜ j = 32 λ j2 , and eigenvectors v j = (1, 2 λ j , 3 λ j2 , 4λ j3 ), j = 1, 2, 3, 4. 3. For f j = δ j4 , (3.14) reduces to the system

∂t τi +

2i ∂x τi+3 = 0, i+3

i = 1, 2, . . .

3.4 Existence and Uniqueness Results (Main Theorems)

67

with the corresponding matrix Bn,3 = 2(Bn,1 )3 . For example, ⎛

0

0

⎜ ⎜ −4 σ4 2σ3 ⎜ B4,3 = ⎜ ⎜ −6 σ1 σ4 3(σ1 σ3 − σ4 ) ⎝ 8σ4 (σ2 − σ12 ) 4(σ12 σ3 − σ2 σ3 − σ1 σ4 )

0

1 2

− 43 σ2

σ1

⎞

⎟ ⎟ ⎟ ⎟ 3 2 ⎟ 2(σ3 − σ1 σ2 ) ⎠ 2 (σ1 − σ2 ) 8 2 − σ + σ σ − σ 2 σ ) 2(σ − 2σ σ + σ 3 ) ( σ 4 1 3 2 3 1 2 2 1 1 3

has the eigenvalues 2 λ j3 . If λi < λ j (i < j), the four linearly independent eigenvectors of B4,3 are v j = (1, 2 λ j , 3 λ j2 , 4 λ j3 ), j ≤ 4. This series of examples can be continued as long as one desires.

3.4 Existence and Uniqueness Results (Main Theorems) We now formulate our results concerning existence/uniqueness of EGFs and their corollaries. For brevity, we shall omit the index t for t-dependent tensors A, b, bˆ j and functions τi , σi . The following theorem concerns an EGF of type (a) and is essential in the proof of Theorem 3.2 (for an EGF of type (b)). Theorem 3.1. Let (M, g0 ) be a closed Riemannian manifold with a codimensionone foliation F . Given functions f˜j ∈ C2 (M × R), there exists a unique smooth solution gt of (3.1) of type (a) defined on some positive time interval [0, ε ). In particular, there exists a unique smooth solution gt , t ∈ [0, ε ) to the PDEs n−1 ∂t gt = ∑ j=0 a j (t) bˆ j ,

a j ∈ C2 (R).

By Proposition 3.2, the n-truncated system (3.2) (where τn+i ’s are replaced by → → suitable polynomials of τ1 , . . . , τn ) has the form ∂t − τ = B t ∂x − τ with the n × n matrix ˜ where B t = A˜ + B, A˜ = (A˜ i j ),

i n−1 A˜ i j = ∑m=0 τi+m−1 fm,τ j , 2

n−1 m B˜ = ∑m=1 fm · (Bn,1 )m−1 , 2

(3.26)

and Bn,1 is the generalized companion matrix to the characteristic polynomial of AN . → τ ,t) gˆ (i.e., f = δ F) then, by (3.26), Example 3.4. (a) If g satisfy ∂ g = F(− t

B˜ = 0,

t t

A˜ = (a˜i j ),

t

where a˜i j =

j

j0

i → τi−1 F,τ j (− τ , 0). 2

The system (3.2) reduces to i n → ∂t τi + τi−1 ∑ j=1 F,τ j (− τ ,t) N(τ j ) = 0, 2

i = 1, 2, . . .

(3.27)

68


The matrix A˜ of n-truncated system (3.27) is hyperbolic if for any q ∈ M either 2 Tr A˜ = ∑ i τi−1 F,τi = 0 or A˜ ≡ 0 on the N-curve through q.

(H1 )

i

→ τ ,t) bˆ 1 (i.e., f j = δ j1 F) then, again by (3.26), (b) If the gt satisfy ∂t gt = F(− 1 →

B˜ = F(− τ , 0) id, 2

A˜ = (a˜i j ),

where a˜i j =

i → τi F,τ j (− τ , 0). 2

The system (3.2) reduces to 1 → i n → ∂t τi + F(− τ ,t)N(τi ) + τi ∑ j=1 F,τ j (− τ ,t) N(τ j ) = 0, 2 2

i = 1, 2, . . . (3.28)

The matrix A˜ + B˜ of the n-truncated system (3.28) is hyperbolic if for any q ∈ M either 2 Tr A˜ = ∑ i τi F,τi = 0 or A˜ ≡ 0 along the N-curve through q.

(H2 )

i

Recall that an n-by-n matrix is hyperbolic (see Sect. 3.3.2) if its right eigenvectors are real and span Rn . The central result here is the following. Theorem 3.2 (Short time existence). Let (M, g0 ) be a closed Riemannian manifold with a foliation F and a unit normal N. If the matrices A˜ + B˜ and B˜ of (3.26) are hyperbolic for all q ∈ M and t = 0 then the EGF (3.1) of type (b) has a unique smooth solution gt defined on some positive time interval [0, ε ). The proof of Theorem 3.1 (see Sect. 3.5.2) follows methods of the theory of firstorder hyperbolic PDEs with one space variable. The proof of Theorem 3.2 (see Sect. 3.5.2) consists of the following steps: (1) Power sums τi are recovered on M (as a unique solution to a quasilinear hyperbolic system of PDEs) for some positive time interval [0, ε ), see Lemma 3.5 and Proposition 3.2. (2) Given (τi ) (of Step 1), the metric gt is recovered on M (as a unique solution to certain quasilinear system of PDEs), see Theorem 3.1. (3) The τi -s of the gt -principal curvatures of F (of Step 2) are shown to coincide with τi (of Step 1), see Theorem 3.1 and Lemma 3.6. Remark that the solution in Theorem 3.2 is unique if only A˜ + B˜ is hyperbolic. For f j = 0 ( j ≥ 2), Theorem 3.2 holds under the weaker condition that only the matrix A˜ is hyperbolic for all q ∈ M and t = 0. In what follows we denote by LZ the Lie derivative along a vector field Z. Corollary 3.1. Let (M, g0 ) be a closed Riemannian manifold with a codimensionone foliation F and unit normal N. If F ∈ C2 (Rn+1 ) and the condition (H2 ) is satisfied at t = 0 and any q ∈ M then there is a unique smooth solution to the EGF → ∂t gt = F(− τ ,t) bˆ 1 , t ∈ [0, ε ) (3.29)

3.5 The General Case

69

for some ε > 0. Furthermore, gt can be determined from the system LZt gt = 0 with → → Zt = ∂t + 12 F(− τ ,t) N, where − τ are the unique smooth solution to (3.28). Corollary 3.2. Let (M, g0 ) be a closed Riemannian manifold with a codimensionone foliation F and a unit normal N. If F ∈ C2 (Rn+1 ) and the condition (H1 ) is satisfied at t = 0 and any q ∈ M then there is a unique smooth solution to the EGF → ∂t gt = F(− τ ,t) gˆt ,

t ∈ [0, ε )

(3.30)

t

→ → for some ε > 0. Furthermore, gˆt = gˆ0 exp( 0 F(− τ ,t) dt), where the power sums − τ are the unique solution to (3.27). Example 3.5. For f1 = c = const and f j = 0 ( j = 1), i.e., for the EGF with h(b) = C bˆ 1 , C ∈ R, the system (3.2) (see also (3.28) for f = c) reduces to the linear PDE

∂t τi + (c/2)N(τi ) = 0. → → The above PDE, ∂t − τ = B˜t ∂x − τ , can be interpreted on M × R by saying that τi are constant along the orbits of the vector field X = ∂t + (C/2)N. If (ψt ) denotes the flow of (c/2)N on M then the flow (φt ) of X is given by:

φt (p, s) = (ψt (p), t + s) for q ∈ M, s ∈ R, therefore, φt maps the level surface Ms = M × {s} onto Mt+s , in particular, M0 onto Mt . This example implies Corollary 3.3. If f1 = const and f j = 0 for all j = 1 (for the EGF) then for all i and t, the following equality holds:

τit = τi0 ◦ φ−t . In particular, if τi0 = const for some i then τi = const for all t. A closed manifold M equipped with a foliation F admits a Riemannian structure g for which all the leaves are minimal (τ1 = 0 in our terminology) if and only if F is topologically taut, that is every leaf meets a loop transverse to the foliation [49]. Known proofs of existence of such metrics use the Hahn–Banach Theorem and are not constructive. The above observations show how to produce a 1-parameter family of metrics with τ 2 j+1 = 0 (with fixed j) starting from one of such metrics.

3.5 The General Case Here we prove the results of Sect. 3.4 about EGFs. First, in Sect. 3.5.1, we solve PDEs for τ ’s. Then, in Sect. 3.5.2, we use this to prove local existence and uniqueness of EGFs. Section 3.5.3 is devoted to proofs of corollaries.

70


3.5.1 Searching for Power Sums Let gt satisfy (3.1) and Nt be the gt -unit normal vector field to F on M. It is easy to see that ∂t Nt = 0, therefore Nt = N for all t, where N is the unit normal of F on (M, g0 ). In fact, for any vector field X tangent to F one has 0 = ∂t gt (X, Nt ) = h(bt )(X, Nt ) + gt (X, ∂t Nt ) = gt (X, ∂t Nt ), and similarly 0 = ∂t gt (Nt , Nt ) = h(bt )(Nt , Nt ) + 2gt (Nt , ∂t Nt ) = 2gt (Nt , ∂t Nt ). Now let ∇t be the Levi-Civita connection on (M, gt ). Then Πt = ∂t ∇t is a (1, 2)tensor field on M. Following (2.11), we write for all t-independent vector fields X,Y , and Z gt (Πt (X,Y ), Z) =

1 t (∇X ht )(Y, Z) + (∇Yt ht )(X, Z) − (∇tZ ht )(X,Y ) . 2

(3.31)

The next lemma is a consequence of Lemma 2.9; we shall prove it directly. Lemma 3.4. For an EGF of type (b), on the tangent bundle of F we have 1 n−1 N( f j ) gt (A j X,Y ) + f j gt (∇tN (A j )X,Y ) , (3.32) ∑ j=0 2 1 n−1 1 ∂t A = − ∑ j=0 N( f j )A j + f j ∇tN (A j ) = − ∇tN h(A). (3.33) 2 2

∂t b(X,Y ) = ht (AX,Y ) −

Proof. By definition, b(X,Y ) = gt (∇tX Y, N), and h(·, N) = 0. Using (3.31) and the identity h(AX,Y ) = h(AY, X), we obtain

∂t b(X,Y ) = ∂t gt (∇tX Y, N) = (∂t gt )(∇tX Y, N) + gt (∂t ∇tX Y, N) 1 = (∇tX ht )(Y, N) + (∇Yt ht )(X, N) − (∇tN ht )(X,Y ) + ht (∇tX Y, N) 2 1 = ht (AX,Y ) + ht (AY, X) − N(ht (X,Y )) 2 1 n−1 = ht (AX,Y ) − ∑ j=0 f j gt (∇tN (A j )X,Y ) + N( f j ) gt (A j X,Y ) . 2 This proves (3.32). Now, (3.33) follows from (3.32) and gt ((∂t A)X,Y ) = gt (∂t (AX),Y ) = ∂t (gt (AX,Y )) − (∂t gt )(AX,Y ) = ∂t b(X,Y ) − ht (AX,Y ).


71

Remark 3.3. We prove (3.2) directly, applying i Ai−1 to both sides of (3.33) we obtain the PDE i Ai−1 ∂t A = −

i n−1 N( f j )Ai+ j−1 + f j Ai−1 ∇tN A j . ∑ j=0 2

Taking the trace of both sides of the above equality, and using the identities

∂t τi = Tr (∂t Ai ) = i Tr (Ai−1 ∂t A)

(3.34)

and (3.4) for i, j > 0, we obtain (3.2). → From Proposition 3.2 it follows directly that the functions − τ = (τ1 , . . . , τn ) satisfy the system of n quasilinear PDEs, whose matrix can be built using a generalized companion matrix of the characteristic polynomial of A. Lemma 3.5. The n-truncated system (3.2) has the form → → ˜ ∂x − ∂t − τ + (A˜ + B) τ =0 with A˜ and B˜ given by (3.26). The next lemma deals with the evolution equation for an EGF of type (a). Lemma 3.6. Let gt be the solution to the EGF (3.1) of type (a). Then the Weingarten operator A of F with respect to gt satisfies

∂t A = −

1 ˜

n−1 N( f0 ) id + ∑ j=1 N( f˜j )A j + f˜j · ∇tN (A j ) , 2

(3.35)

and τi (i ≥ 1) (the power sums of the eigenvalues of A) satisfy the PDEs

∂t τi = −

i j n−1 τi−1 N( f˜0 ) + ∑ j=1 f˜j N(τi+ j−1 ) + τi+ j−1 N( f˜j ) . (3.36) 2 i+ j−1

The n-truncated system (3.36) has the form → ∂t − τ +

∑ j=1 j f˜j (Bn,1 ) j−1 n−1

→ N(− τ ) = a,

where Bn,1 is the generalized companion matrix (3.13), a = (a1 , . . . , an ), and ai = −

i n−1 N( f˜j ) τi+ j−1 2 ∑ j=0

(1 ≤ i ≤ n).

72


Proof. The proof of (3.35) is similar to that of (3.33). On T F we obtain 1 gt ((∂t A)X,Y ) = − N(ht (X,Y )) 2 1 ˜ n−1 = − N( f0 )gt (X,Y ) + ∑ j=1 N( f˜j )gt (A j X,Y ) 2

+ f˜j gt (∇tN (A j )X,Y ) .

Following the lines of the proof of (3.2) in Lemma 3.5, we can deduce from the above our system (3.36). By Proposition 3.2, the n-truncated system (3.36) has the required form. Lemmas 3.5 and 3.6 together with Theorem A provide existence and uniqueness → τ t satisfying conditions following from (3.1). results for the symmetric functions − In particular, this allows us to reduce the existence and uniqueness problems for EGFs of type (b) to those for type (a), as we do in the proof of Theorem 3.2 in the next section.

3.5.2 Local Existence of Metrics (Proofs of the Main Theorems) Given a Riemannian metric g on a foliated manifold (M, F ), the symmetric tensor h(b) defined by (3.1) can be expressed in terms of the first partial derivatives of g. Therefore, g → h(b) is a first-order partial differential operator. For any F -truncated symmetric (0, 2)-tensor S the equation h(b) = S

(3.37)

has the form of a nonlinear system of first-order PDEs. A particular case of (3.37) is the Einstein type relation h(b) = ρ gˆ for some function (or constant) ρ on M. There are several obvious obstructions to the existence of solutions to (3.37) even at a single point. For example, if h(b) = f2 j bˆ 2 j (for some integer j and a function f2 j ) and S is neither positive nor negative definite then (3.37) has no solutions at q. Proof (of Theorem 3.1). Take biregular foliated coordinates (x0 , x1 , . . . xn ) on Uq ⊂ √ M (with center at q); see Lemma 2.1, and the metric (2.2). Then, N = ∂0 / g00 is the unit normal to F . Set ψab = gab,0 . The system (3.1) (for f j of type (a)) along a trajectory γ : x → γ (x) of ∂0 has the form

∂t gi j = Fi j (gab , ψab ,t, x),

(3.38)

where Fi j := h(b)i j . In view of symmetry, ψab = ψba and Fi j = Fji , we shall assume 1 ≤ i ≤ j ≤ n and 1 ≤ a ≤ b ≤ n. For example, if fm = 0 (m > 1) then (3.38) is the hyperbolic (diagonal) system


73

1 −1/2 ∂t gi j = f0 (q,t)gi j − g00 f1 (q,t)ψi j , 2 that completes the proof in this case. Now let fm = 0 for some m > 1 (e.g., general fm ). We may assume that A∂ j = k j ∂ j , g(∂i , ∂ j ) = δi j (i, j > 0) and g00 = 1 at the point q for t = 0. (By Lemma 2.2, we have (bm )i j = (−1/2)m ψimj δi j at q for t = 0). Differentiating (3.38) with respect to x and t, we obtain

∂0 pi j = ∂0 Fi j + ∑

∂F

a,b

∂t pi j = ∂t Fi j + ∑

ij

∂ gab

pi j := ∂t gi j ,

∂ Fi j ∂0 ψab , ∂ ψab

∂t gab +

∂ Fi j ∂t ψab , ∂ ψab

∂F

a,b

where

∂0 gab +

ij

∂ gab

ψi j := gi j,0 ,

(3.39)

Fi j := h(b)i j .

Since g is of class C2 , we conclude that

∂t ψab =

∂ 2 gab = ∂0 pab . ∂ t ∂ x0

Hence (3.39) together with (3.38) may be written in the form

∂t gi j = Fi j ({gab }, {ψab},t, x), ∂ Fi j ∂ Fi j ∂0 ψab = ∂0 Fi j + ∑ ψab , ∂ ψ ∂ ab a,b a,b gab

∂t ψi j − ∑

∂ Fi j ∂ Fi j ∂0 pab = ∂t Fi j + ∑ pab . a,b ∂ ψab a,b ∂ gab

∂t pi j − ∑

(3.40)

The above quasilinear system consists of parts: (i) our original equation (3.40)1, (ii) the corresponding equation (3.40)2 for ∂t A, and (iii) the equation (3.40)3 for ∂t2 g following from the previous ones. In general, the following 12 n(n + 1) × 12 n(n + 1) matrix is not symmetric: dψ F =

∂F ij , ∂ ψab

i ≤ j, a ≤ b.

We claim that it is hyperbolic. If we change the local coordinate system on M, the components Fi j (i ≤ j) and ψab (a ≤ b) at q will be transformed by the same tensor low. Notice that the above dψ F is a (1, 1)-tensor on the vector bundle of symmetric

74


(0, 2)-tensors on T F . Hence, dψ F(q) can be seen as the linear endomorphism of the space of symmetric (0, 2)-tensors on Tq F . The hyperbolicity is a pointwise property, so can be considered at any point q ∈ M in a special bifoliated chart around q, for example, such that gi j = δi j and Ai j (q) = ki δi j at q. (Indeed, (ki ) are the principal curvatures of F at q for t = 0). In this chart, our calculations show that the matrix dψ F is diagonal, so has real eigenvalues (vectors) at q. Indeed, for t = 0 one may find at the point q:

∂ Fi j {i, j} = ∑ fm (q) μ (m)i j δ{a,b} , ∂ ψab m≥1

μ (m)i j =

β

∑

α +β =m−1

kiα k j .

The order of indices of dψ F is [1, 1], [1, 2], . . . , [1, n], [2, 2], [2, 3], . . ., [2, n], . . . , [n, n]. For example, for F = b2 (i.e., f j = δ j2 ) the above matrix in an orthonormal frame at any point is ⎡ 2 ψ11 2 ψ12 2 ψ13 0 0 0 ⎤ ⎢ ⎢ ⎢ ∂ (b2 )i j 1⎢ = ⎢ ∂ ψab 4⎢ ⎢ ⎢ ⎣

ψ12 ψ11 + ψ22

ψ23

ψ12

ψ13

0

ψ13

ψ23

ψ11 + ψ33

0

ψ12

ψ13

0

2 ψ12

0

2 ψ22

2 ψ23

0

0

ψ13

ψ12

0

0

2 ψ13

ψ23 ψ22 + ψ33 ψ23 0

2 ψ23

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2 ψ33

with the order of indices [1, 1], [1, 2], [1, 3], [2, 2], [2, 3], [3, 3]. At q for t = 0 (i.e., ψab = 0, a = b and ψaa = ka ) it is diagonal with the elements μ (2)ab = ka + kb . Let A0 = [ 12 , 1, 1, 12 , 1, 12 ] be the diagonal matrix. Then the matrix A1 = A0 dψ (b2 ) is symmetric, and our system (3.40)2 for h(b) = b2 is “symmetrizable”: A0 ∂t ψ − A1 ∂0 ψ = {free terms}. Therefore, (3.40) for the functions gi j (t, x), pi j (t, x) and ψi j (t, x) with i ≤ j is the hyperbolic system which is “symmetrizable” in the sense of [51, p. 370]. Indeed, multiplying n columns (corresponding to i = j) of the matrix dψ F in an orthonormal frame by the factor 12 , we obtain the symmetric matrix. By Theorem A (in Section 3.3.2), given q ∈ M there exists a unique solution to (3.40) which is defined in Uq along the N-curve through q for some time interval [0, εq ) and satisfies the initial conditions gi j (0, x) = (g0 )i j (x),

pi j (0, x) = h(b0 )i j (x),

ψi j (0, x) = (∂0 g)i j (x).

Again by Theorem A (see definitions of K and Ω¯ ), the value εq continuously depends on q ∈ M. The claim follows from the above and compactness of M. → Proof (of Theorem 3.2). Let A0 and − τ 0 be the values of extended Weingarten operator and power sums of the principal curvatures ki of (the leaves of) F determined on (M, F ) by a given metric g0 : (1)

(2)

(a) Uniqueness. Assume that gt , gt are two solutions to (3.1) with the same (1) (2) → → initial metric g0 . Functions − τ t,1 , − τ t,2 , corresponding to gt , gt , satisfy (3.2)


75

→ → and have the same initial value − τ 0 . By Lemma 3.5 and Theorem A, − τ t,1 = (1) (2) → − → − t,2 τ t = τ t on some positive time interval [0, ε1 ). Hence gt , gt satisfy (3.1) → of type (a) with known coefficients f˜j (p,t) := f j (− τ t (p),t). By Theorem 3.1, (1) (2) gt = gt on some positive time interval [0, ε2 ). → (b) Existence. By Lemma 3.5 and Theorem A, there is a unique solution − τ t to (3.2) on some positive time interval [0, ε ∗ ). By Theorem 3.1, the EGF (3.1) of type → (a) with known functions f˜j (·,t) := f j (− τ t ,t) has a unique solution gt∗ (g∗0 = g0 ) for 0 ≤ t < ε ∗ . The Weingarten operator At∗ (A∗0 = A0 ) of (M, F , gt∗ ) satisfies → → → (3.35), hence the power sums of its eigenvalues, − τ t,∗ (− τ 0,∗ = − τ 0 ), satisfy (3.36) with the same coefficient functions f˜j . By Lemma 3.6 and Theorem A, → → → the solution of this problem is unique, hence − τ t=− τ t,∗ , i.e., − τ t are power → sums of eigenvalues of At∗ . Finally, gt∗ is a solution to (3.1) such that − τ t are power sums of the principal curvatures of the leaves in this metric.

3.5.3 Proofs of the Corollaries In all the proofs given later, q is a point of M, γ : x → γ (x) (γ (0) = q, x ∈ R) is the N-curve, and N is the unit normal of F . Proof (of Corollary 3.1). By Lemma 3.5, we have (3.2), which in our case reduces to the system (3.28). Using this system, we build the initial value problem in the → → (x,t)-plane for the vector function − τ (x,t) = − τ (γ (x),t) 1 → → → → − → ˜ − ∂t − τ + F(− τ ,t) idn +A( τ ,t) ∂x → τ = 0, − τ (x, 0) = − τ 0 (γ (x)). (3.41) 2 → The matrix A˜ is equal to { i τ F (− τ ,t)} . Note that rank A˜ ≤ 1. By condition 2 i ,τ j

1≤i, j≤n

(H2 ), either the function i → λ˜ = Tr A˜ = ∑1≤i≤n τi F,τi (− τ , 0) 2 ˜ is nonzero for all x ∈ R, or A(x) ˜ (the eigenvalue of A) ≡ 0. Hence (3.41) is hyperbolic for small enough t. In first case, the eigenvector of A˜ |t=0 for λ˜ (x) is v1 = (F,τ1 , F,τ2 , . . . , F,τn ), and the kernel of A˜ |t=0 is spanned by n − 1 vectors v2 = (−2 τ2 , τ1 , 0, . . . 0), v3 = (−3 τ3 , 0, τ1 , 0, . . . 0), . . . vn = (−n τn , 0, . . . 0, τ1 ). → (If λ˜ (x) = 0 for some x ∈ R, and F,τ j (− τ , 0) = 0 for some j, then A˜ is nilpotent and hence (3.41) is not hyperbolic). By Theorem A, the initial value problem (3.41) has a unique solution on a domain [−δ , δ ] × [0, ε ) of the (x,t)-plane. Hence, there exists tq > 0 such that the solution → − τ (·,t) to (3.28) exists and is unique for t ∈ [0,tq ) on a neighborhood Uq ⊂ M centered at q. By compactness of M, we conclude that there is an ε > 0 such that → (3.29) admits a unique solution − τ (q,t) on M for t ∈ [0, ε ).

76


Using the properties of the Lie derivative of gt along N, one may show that (3.29) is equivalent to 1 →t τ ,t) N. LZt gt = 0 with Zt = ∂t + F(− 2 → Denote by (Φ ) the flow of 1 F(− τ ,t)N. The solution metrics g can be detert

t

2

mined by:

gt (Φt X, Φt Y ) = g0 (X,Y ),

gt (X, N) = 0,

gt (N, N) = 1

for all X and Y tangent to F . The solution exists and is unique as long as the solution to (3.28) does. Remark 3.4. One may also apply the method of characteristics to solve (3.41) → explicitly when F = F(− τ ). For λ˜ = 0, the system has two characteristics: dx˜ ˜ F =λ+ , dt 2

dx F = dt 2

corresponding to two eigenvalues λ˜ + F2 (of multiplicity 1) and n − 1). First, let us observe that the function

F 2

(of multiplicity

− u := v1T · → τ = ∑ j≤n F,τ j τ j is constant along the first family of characteristics: F F d d ˜ N(u) = 0 ⇔ u = const along x˜ = λ˜ + . u = ∂t u + λ + dt 2 dt 2

(3.42)

Let us find a function that is constant along the second family of characteristics. For each m > 1, due to the form of vm , calculate the sum of the first equation in (3.41) multiplied by −m τm with the mth equation multiplied by τ1 (along the trajectories F of dx dt = 2 ) and get

τ1 ∂t τm − mτm ∂t τ1 +

F τ1 N(τm ) − mτm N(τ1 ) = 0. 2

(The terms with ∑ j Fτ j N(τ j ) cancel). Using d dx F τm = ∂t τm + N(τm ) = ∂t τm + N(τm ), dt dt 2 we get (again, along the second family of characteristics)

τ1

d d τm − m τm τ1 = 0 (m ≥ 2). dt dt

(3.43)

The complete integral of (3.43) is

τm = Cm (x) τ1m

(m ≥ 2),

(3.44)

3.6 Global Existence of EGFs (Time Estimation)

77

→ where Cm (x) = τm (x, 0)/τ1m (x, 0) are known functions. As the functions − τ (x,t) and → − F( τ ) exist for t ∈ [0,tq ), the EGF under consideration exists and is unique on M × [0,tq). → τ (x,t) = (τ t , . . . , τ t ) the power sums of Proof (of Corollary 3.2). Denote by − 1

n

the principal curvatures of F at the point γ (x) in time t. By Lemma 3.5, we have (3.2), which in our case reduces to (3.27). Using this system, we build the initial value problem in the (x,t)-plane → → → ˜ − ∂t − τ + A( τ ,t) ∂x − τ = 0,

− → → τ (x, 0) = − τ 0 (γ (x)),

(3.45)

→ where A˜ is equal to { 2i τi−1 F,τ j (− τ ,t)}1≤i, j≤n. As before, rank A˜ ≤ 1. Consider i → λ˜ = Tr A˜ = ∑1≤i≤n τi−1 F,τi (− τ , 0) 2 ˜ ≡ 0. Hence, the ˜ By (H1 ), either λ˜ (x) = 0 for all x ∈ R or A(x) (the eigenvalue of A). system (3.45) is hyperbolic at (x, 0). As in the proof of Corollary 3.1 we conclude → that there is an ε > 0 such that − τ (q,t) on M exists and is unique for t ∈ [0, ε ). Certainly, a unique solution to (3.30) is smooth and has the required form.

3.6 Global Existence of EGFs (Time Estimation) Fix positive integers m and l, and put Jm,l = α ∈ Zn+ : ∑ j α j = m,

∑j jαj = l

.

→ For a vector α = (α1 , . . . , αn ) ∈ Zn+ put − τ α := τ1α1 . . . τnαn . Recall that a vector field on a manifold M is complete if any of its trajectory γ : t → γ (t) can be extended to the whole range R of parameter t. If M carries a Riemannian structure g and a vector field X has bounded length then the completeness of (M, g) is sufficient for the completeness of X. Proposition 3.3. Let (M, g0 ) be a Riemannian manifold with a codimension-one foliation F and a complete unit normal field N. Given cα ∈ R (α ∈ Jm,l ) and → → m, l ∈ N, define functions Ft = ∑ α ∈Jm,l cα (− τ t )α (where − τ t is, as usual, the vector of power sums of principal curvatures of the leaves) and set 2 inf N(F0 ) otherwise. l+1 M Then the EGF ∂t gt = Ft bˆ 1 , compare (3.29), has a unique smooth solution on M for t ∈ [ 0, T ) and does not possess one for t ∈ [ 0, T ]. T = ∞ i f N(F0 ) ≥ 0 on M and T = −

78


Proof. Notice that → → → → τ α ),τ j τ j = m − τ α , ∑ j j(− τ α ),τ j τ j = l − τ α. ∑ j (− → → If F = ∑ cα − τ α then N(F) = ∑ ∑ cα − τ ,ατ j N(τ j ) (the derivative of F along N). α

j α

→ ˜ 0). One has PDEs (3.41) in the (x,t)˜ τ (γ (x))) and F˜0 = F(·, Define F(x,t) = Ft (− plane. Characteristics of the first family, see (3.42), are lines and F˜ = const along → them. To show this, observe that (by definition of Ft (− τ )) → ˜ τ ) τ j = mF(x,t), u := ∑ j Ft,τ j (−

l ˜ j → λ˜ := ∑ j Ft,τ j (− τ )τ j = F(x,t). 2 2

Because F˜ = u/m is constant along the first family of characteristics in the (x,t)plane, these characteristics (lines) are given by the equation d 1 x = (l + 1)F˜ dt 2

⇔

1 x = ξ + (l + 1)F˜0 (ξ )t. 2

Notice that F˜ = ∑ α ∈J

m, l

→ cα ∑ j (− τ 0 ),ατ j N(τ 0j ).

If F˜0 > 0 on γ , the solution F˜ exists for all t ≥ 0 (see Example 3.2) and we set tq = ∞. If F˜0 is negative somewhere on γ then F˜ exists (and is continuous) for t ∈ [0,tq ) where tq = −2/[(l + 1) min F˜ 0 (x)]. x

The second family of characteristics, dtd x = F2 , also exists for t ∈ [0,tq ). To show this, assume the contrary: there are t0 ∈ (0,t p ) and a trajectory γ1 (t) of the second family of characteristics that cannot be continued for values t ≥ t0 . Therefore, the ˜ γ1 (t),t)/2 of γ1 approaches to infinity when t → t0 , a contradiction to inclination F( continuity of F˜ on the strip t ∈ [0,t0 ] in the (x,t)-plane. We shall apply EGFs to umbilical foliations (that is, those for which the Weingarten operator A is proportional to the identity at any point, among them totally geodesic foliations appear when A = 0) and to foliations on surfaces. First we remark (see also Proposition 2.4) that EGFs preserve the umbilicity of F . Proposition 3.4. Let (M, g0 ) be a Riemannian manifold with umbilical foliation F . If the EGF (3.1) of type (b) has a unique smooth solution gt (0 ≤ t < ε ) then F is umbilical for any gt .

3.6 Global Existence of EGFs (Time Estimation)

79

for some function λ0 on M. Let Proof. As F is g0 -umbilical, we have A0 = λ0 id ˜ λt (0 ≤ t < ε ) be a unique solution to the quasi-linear PDE 1 ∂t λt + N(ψ (λt ,t)) = 0, 2

(3.46)

(see Theorem A with n = 1) where

ψ (λ ,t) = ∑ j=0 f j (nλ , nλ 2 , . . . , nλ n ;t)λ j n−1

is the function of two variables. Consider the family of F -truncated metrics g˜t (0 ≤ t < ε˜ ) defined along F by g˜t = g0 exp( 0t ψ (λt ,t) dt), hence ∂t g˜t = ψ (λt ,t)gˆ˜t . By Lemma 2.3, the Weingarten operator of F with respect to g˜t is conformal,

for some function μt (μ0 = λ0 ). Hence F is g˜t -umbilical. A˜ t = μt id By Lemma 2.9 with S = sgˆt and s = − 12 ψ , we have 1

∂t A˜ t = − N(ψ (λt ,t)id. 2 Taking the trace, we obtain the PDE 1 ∂t μt + N(ψ (λt ,t)) = 0. 2 Comparing with (3.46) we conclude that μt = λt for all t. Due to the definition of ψ , metrics g˜t also satisfy to (3.1). By uniqueness of the solution, we have g˜t = gt , which completes the proof. Proposition 3.5. Let (M, g0 ) be a Riemannian manifold, and F a codimension-one umbilical foliation on M with the normal curvature λ0 and a complete unit normal n−1 → field N. Then the EGF (3.1) with h = ∑ f j (− τ ) bˆ j has a unique smooth solution gt j=0

on M for t ∈ [0, T ), and does not possess one for t ≥ T . Here T = ∞i f N(ψ (λ0 )) ≥ 0 on M, and T = −2/ inf N(ψ (λ0 )) otherwise, M

where

ψ (λ ) = ∑ j=0 f j (nλ , . . . , nλ n ) λ j . n−1

(3.47)

Moreover, F is gt -umbilical, gˆt = gˆ0 exp( 0t ψ (λt )dt), and λt is a unique smooth solution to the PDE 1 ∂t λt + N(ψ (λt )) = 0. (3.48) 2

80


Proof. Theorem A (with n = 1) provides the short-time existence and uniqueness of the solution λt to (3.48) for 0 ≤ t < T . Furthermore, the EGF can be expressed as ∂t gt = ψ (λt )gˆt , and the solution gˆt has the required form. Consider the function λ˜ (x,t) = λ (γ (x),t) in the (x,t)-plane along the trajectory γ (x), γ (0) = q, of N, and set λ˜ 0 (x) = λ (γ (x), 0). Equation (3.48) in this case has the form of a conservation law 1 ∂t λ˜ + ∂x (ψ (λ˜ )) = 0. 2 One may show the following. If ψ , λ˜ 0 ∈ C1 (R) and if the functions λ˜ 0 and ψ are either nondecreasing or nonincreasing, the problem 1 ∂t λ˜ + ∂x (ψ (λ˜ )) = 0, 2

λ˜ (x, 0) = λ˜ 0 (x),

t≥0

has a unique smooth solution defined implicitly by the parametric equations,

λ˜ (x,t) = λ˜ 0 (ξ ), If

d dx

1 x = ξ + ψ (λ˜ 0 (ξ ))t. 2

ψ (λ˜ 0 (x)) is negative elsewhere along γ then λ˜ (x,t) exists for t < tq = −2

inf

x∈R

d ˜ ψ (λ0 (x)). dx

Notice that for ψ = λ 2 , (3.48) reduces to Burgers’ equation, see Example 3.2.

3.7 Variational Formulae for EGFs In this section we shall use the EGF (3.1), whose f j (at least one of them is not identically zero) are again of one of the two types: (a) f j = f j (q) (q ∈ M),

→ (b) f j = f j (− τ ).

Under certain conditions the EGF admits a unique smooth solution gt defined for some time interval [0, ε ), see Theorem 3.2 given earlier. j Recall that a self-adjoint (1, 1)-tensor h(A) = ∑n−1 j=0 f j A is dual to h(b) of (3.1). We shall apply the results of Sect. 2.3.2 with f = Tr h(A) = ∑n−1 j=0 f j τ j , to the functional Jh : g →

!

M

Tr h(A) d volg ,

g ∈ M.

(3.49)

3.7 Variational Formulae for EGFs

81

For umbilical foliations F (see Sect. 2.4.1) this reduces to the functional Iψ : g →

! M

ψ (λ ) d volg ,

g ∈ M.

(3.50)

3.7.1 The Normalized EGFs A metric g on (M, F ) is a fixed point of the EGF (3.1) if it satisfies the condition → τ )A j = 0. ∑ j=0 f j (− n−1

For a generic setting of f j ’s (when f0 (0) = 0), the fixed points of the EGF (h(A) = 0) are totally geodesic (At ≡ 0) foliations only. Several classes of foliations appear as → fixed points of the flow ∂t gt = f (− τ ) gˆt for special choices of f , for example: (a) (b) (c) (d)

Foliations of constant τi , when f = τi − c (minimal for i = 1 and c = 0). Umbilical foliations, when f = n τ2 − τ12 = ∑ i< j (ki − k j )2 (see Example 2.6). Parabolic foliations, when f = σn . Totally geodesic foliations, when f (0) = 0, etc.

In order to extend the set of solutions (fixed points) we define the normalized EGF by:

∂t gt = h(bt ) − (ρt /n) gˆt

with ρt = Jh (gt ) / vol(M, gt ).

(3.51)

For the normalized EGF we have vol(M, gt ) = const, because, by (2.21), 1 d vol(M, gt ) = dt 2

! M

d volt = 0. Tr h(A) − (ρt /n) id

The EGF and its normalized companion provide some methods of evolving Riemannian metrics on foliated manifolds. Fixed points of the normalized EGF satisfy h(b0 ) = (ρ /n) gˆ0 , where ρ = Jh (g). Among them there are umbilical (b = λ g) foliations with λ = const on M. Let gt be a family of Riemannian metrics of finite volume on (M, F ). Metrics g¯t = (φt gˆt ) ⊕ gt⊥ with φt = vol(M, gt )−2/n have unit volume: M d volt = 1. ¯ of metrics gt and g¯t are Geometries (e.g., second fundamental forms b and b) related by Lemma 2.3. Unnormalized and normalized EGF differ only by rescaling along the space T F . Proposition 3.6. Let (M, F ) be a foliation, and gt a solution (of finite volume) to → ˆ − the EGF (3.1) with h(b) = ∑n−1 j=0 f j ( τ ) b j . Then the metrics g¯ t = (φt gˆt ) ⊕ g⊥,

where φt = vol(M, gt )−2/n ,

82


evolve according to the normalized EGF

∂ t g¯t = h(b¯ t ) − (ρt /n) gˆ¯t ,

where ρt = Jh (gt ) / vol(M, gt ).

(3.52)

¯ = h(A) (respectively, for metrics g¯t and gt ). Proof. By Lemma 2.3, τ¯ j = τ j and h(A) ¯ Hence, Tr h(A) = Tr h(A). From (2.21) with S = h(bt ) we get d d vol(M, gt ) = dt dt

! M

d volt =

1 2

! M

Tr h(A) d volt .

Thus, φt = vol(M, gt )−2/n is a smooth function. By Lemma 2.3, we have h(b¯ t ) = φt . h(bt ). Therefore,

∂ t g¯t = φt ∂t gt + φ t gˆt = h(b¯ t ) + φ t /φt gˆ¯t . n/2

Using (2.21) and d volt = φt n

∂t volt = ∂t (φt 2 volt ) =

d volt , we obtain

n n2 −1 1 n 1 φ φt φt volt + φt 2 Tr h(A) volt = n t + Tr h(A) volt . 2 2 2 φt

Let ρt be the average of Tr h(A), see (3.52). From the above we get 0=2

d dt

! M

d volt =

!

n M

φt φ + Tr h(A) d volt = n t + ρt . φt φt

This shows that ρt /n = −φ t /φt . Hence, g¯ evolves according to (3.52).

Example 3.6. (a) For an umbilical foliation F (see Sect. 2.4.1) we have

h(A) = ψ (λ ) id,

Tr h(A) = n ψ (λ ),

(3.53)

where λ is the normal curvature of F , and ψ (λ ) is given in (3.48). The corresponding to EGF ∂t gt = ψ (λt ) gˆt the normalized companion, see (3.51), is

∂ t gt = (ψ − ρt ) gˆt ,

where ρt = Iψ (gt ) / vol(M, gt ).

(3.54)

For ψ = λ , due to M λ d vol = 0, the EGF ∂t gt = λt gˆt consist of metrics of the same volume. (b) Obviously, the normalized EGF corresponding to h(A) = Ak is defined by:

∂t gt = bˆ tk − (ρt /n) gˆt

with

ρt = Iτ ,k (gt ) / vol(M, gt ).

(3.55)

(c) Similarly, the kth normalized Newton transformation flow is given by:

∂t gt = Tk (bt ) −

ρt gˆt n

with ρt = (n − k)Iσ ,k (gt ) / vol(M, gt ).

(3.56)

3.7 Variational Formulae for EGFs

83

3.7.2 First Derivatives of Functionals Using Theorem 2.1 with f = Tr h(A) and Corollary 2.4, we obtain Proposition 3.7. If gt ∈ M1 (0 ≤ t < ε ) is a solution to the normalized EGF (3.51), the first derivative of the functional (3.49) is given by: Jh (gt )

1 = − Jh2 (gt ) + 2

! M

1 2 t (Tr h(A)) − Bh, ∇N h(b) d volgt , 2

(3.57)

where Bh = ∑ni=1 2i (Tr h(A)),τi bˆ i−1 . Moreover, if gt ∈ M1 is a solution to (3.54) (hence F is umbilical) then the first derivative of the functional (3.50) is given by: 2 Iψ (gt ) = −n Iψ2 (gt ) +

!

n ψ 2 (λt ) − ψ (λt )N(ψ (λt )) d volgt .

M

(3.58)

Proof. From (2.22) and (2.35) with S = h(b) − (ρ /n) gˆ and f = Tr h(A), we have ! " 1

# 1 (Tr h(b) − Jh (gt )) gˆt − V (Bh ), h(b) − Jh (gt ) gˆt d volgt , n M 2 ! n 1 ψ (λt ) − Iψ (gt ) − V (ψ (λt )) (ψ (λt ) − Iψ (gt )) d volgt . Iψ (gt ) = 2 M n Jh (gt ) =

The claim follows. From Proposition 3.7 for h(b) = bˆ k and h(b) = Tk (b), respectively, we obtain

Corollary 3.4. If gt ∈ M1 (0 ≤ t < ε ) is one of the normalized EGFs (3.55) – (3.56), the first derivative (3.57) reduces, respectively, to 2 Iτ ,k (gt )

=

−Iτ2,k (gt ) +

!

k2 τk − τ1 τ2k−1 d vol, 2k − 1 2

M

2 Iσ ,k (gt ) = −(n − k)Iσ2 ,k (gt ) + (n − k)

!

M

σk 2 d vol −

!

M

% $ Tk−1 (b), ∇tN Tk (b) d vol.

Proof. For (3.55) with h(A) = Ak and S = bˆ k − (ρ /n) g, ˆ we have f = τk and Bh = kˆ b . Equality (3.57) reduces to k−1 2 Iτ ,k (gt ) =

1 2

! & M

' 1 τk − Iτ ,k (gt ) gˆt − k V (bˆ k−1 ), bˆ k − Iτ ,k (gt )gˆt d vol n

1 1 = − Iτ2,k (gt ) + 2 2

! M

τk 2 d vol −

k 2

! M

Tr (Ak−1 ∇N Ak )d vol .

From the above equality, using (3.4), (3.34), and Lemma 2.6, the formula for Iτ ,k (gt ) follows.

84


For (3.56) with h(A) = Tk (A) and S = Tk (b) − (ρ /n) g, ˆ we have f = (n − k)τk and Bh = 12 (n − k)Tk−1 (b). Equality (3.57) (divided by n − k) reduces to Iσ ,k (gt )

1 = − (n − k)Iσ2 ,k (gt ) + 2

! M

1 1 t 2 (n − k)σk − Tk−1 (b), ∇N Tk (b) d vol. 2 2

From the above the required formula for Iσ ,k (gt ) follows. Example 3.7. Let us recall the formula BN = Iτ ,2 +

! M

Z2g d vol

for the total bending of the unit normal N (Sect. 2.4.2) and the definition h(A) = j ∑n−1 j=0 f j A for the EGF’s evolution operator, see (3.1). Using (2.39), we obtain the following corollary. Let gt ∈ M1 be a solution to the normalized EGF (3.51), N – a unit normal to F , and Z = ∇tN N. By Theorem 2.2, the first derivative of the total bending is given by: ' ! & 1 1 BN (gt ) = (Zt2 + τ2 − BN (gt )) gˆt − Z Z − V (bˆ 1 ), h(b) − ρt gˆt d vol n M 2 ! 1 1 Tr h(A)(Zt2 + τ2 − BN (gt )) − h(b)(Z, Z) + ρt Zt2 = n M 2 1 n−1 + ∑ j=0 f j τ1 τ j+1 − d vol. N(τ j+1 ) j+1 In Propositions 3.3 and 3.5 we provided sufficient conditions for existing and uniqueness of the EGF gt for t ∈ (0, ∞). Using this and Proposition 3.7, we obtain Theorem 3.3. (a) If gt ∈ M1 (t ≥ 0) is a normalized EGF (3.51) on a closed manifold M and h(b) = τ1k bˆ 1 (k ∈ N) then gt approach (in a weak sense) a metric making F a minimal foliation. (b) If gt ∈ M1 (t ≥ 0) is a normalized EGF (3.53) with ψ (λ ) = λ k (k ∈ N) and umbilical F then gt approach (in a weak sense) a metric making F totally geodesic. Proof. (a) We have Tr h(A) = τ1k+1 and Bh = 12 (k + 1)τ1k gˆ (see Proposition 3.7). From (3.2) with i = 1 we conclude that 1 ∂t τ1 = − (k + 1)τ1k N(τ1 ). 2 By (3.57) and Lemma 2.5, the derivative of the functional Jt = satisfies the differential inequality 2 ! k 1 1 2k+2 2 τ d volt +Jt ≤ − Jt2 . 2 Jt = − 2 2k + 1 M 1 2

k+1 M τ1 d volt

3.8 Extrinsic Geometric Solitons

85

If F is minimal w.r.t. some gt¯ (i.e., τ1 ≡ 0 at t = t¯) then by Theorem A (with n = 1) F is minimal w.r.t. all gt (i.e., τ1 ≡ 0 for all t) that completes the proof in this case. Now, suppose that F is not minimal w.r.t. to any gt , then M τ12k+2 d volt > 0, and certainly Jt < 0, for all t. One may compare the above inequality Jt ≤ − 14 Jt2 with the Riccati ODE 1 y (t) = − y2 (t). 4 If J0 < 0 then Jt < 0 for all t ≥ 0, and lim Jt = −∞ for some T > 0, – a t→T

contradiction. Similarly, we find that Jt > 0 for all t ≥ 0. Comparing with the −1 Riccati ODE, we see that the solution can be estimated as Jt ≤ 4 t + 4/J0 . As gt are defined for t ∈ [0, ∞), we have Jt → 0 as t → ∞. (b) The proof is similar to (a). From (3.46) we conclude that 1 ∂t λ = − (k + 1)λ k N(λ ). 2 By (3.58), and Lemma 2.5, the derivative of the functional Iψ ,t := Iψ (gt ) satisfies ! (k − 1)2 λ 2k d volt +Iψ2 ,t ≤ −n Iψ2 ,t . 2 Iψ ,t = −n 2k − 1 M If F is totally geodesic w.r.t. some gt¯ (i.e., λ ≡ 0 at t = t¯) then by Theorem A (with n = 1) F is totally geodesic w.r.t. all gt , that completes the proof in this case. Now, suppose that F is not totally geodesic w.r.t. to any gt . Then 2k M λ d volt > 0, and certainly Iψ ,t < 0, for all t. One may compare the above 1 2 inequality Iψ ,t ≤ − 4 Iψ ,t with the Riccati ODE y (t) = − n2 y2 (t). If Iψ ,0 < 0 then Iψ ,t < 0 for all t ≥ 0, and lim Iψ ,t = −∞ for some T > 0 – a contradiction. Similarly t→T

we find that Iψ ,t > 0 (t ≥ 0). Comparing with the Riccati ODE, we see that the solution is estimated as −1 n −1 Iψ ,t ≤ . t + Iψ ,0 2 As gt are defined for t ≥ 0, we have Iψ ,t → 0 as t → ∞.

3.8 Extrinsic Geometric Solitons Special soliton solutions of geometric flow motivate the general analysis of the singularity formation. In this section we introduce soliton solutions to EGFs and study their geometry for umbilical foliations, foliations on surfaces, and in the case when the EGF is produced by the extrinsic Ricci curvature tensor.

86


Fig. 3.2 Horosphere foliation

3.8.1 Introducing EGS Let Diff(M) be the diffeomorphism group of M. Let us introduce the following notation: – D(F ) the subgroup of Diff(M) preserving F . – D(F , N) the subgroup of Diff(F ) preserving both F and N. Recall that the Fibration Theorem of D. Tischler (see, for example, [12]) states that the property of a closed manifold: (a) M admits a codimension one C1 -foliation invariant by a transverse flow; is equivalent to each of the following conditions: (b) M fibers over the circle S1 . (c) M supports a closed 1-form of a class C1 without singularities. Taking into account the theorem of R. Sacksteder, see again [12], in class C 2 one obtains the following condition equivalent to any of (a), (b), or (c): (d) M admits a codimension one foliation without holonomy. Definition 3.4. We say that a solution gt = gˆt ⊕ gt⊥ to (3.1) is a self-similar extrinsic geometric soliton (EGS) on (M, F , N) if there exist a smooth function σ (t) > 0 (σ (0) = 1), and a family of diffeomorphisms φt ∈ D(F , N), φ0 = idM , such that gˆt = σ (t) φt∗ gˆ0 . (3.59) A simple example of EGS appears on the hyperbolic space Hn+1 with horosphere (horocycle when n = 1) foliation, see Sect. 3.9.4. On the Poincaré (n + 1)-ball B the leaves of such Riemannian umbilical foliations are Euclidean n-spheres tangent to ∂ B (Fig. 3.2). Trajectories orthogonal to the above foliations form foliations by geodesics. Let R+ act on M (M, F , N) by scalings along T F . By Remark 2.1, the Weingarten operator A and the principal curvatures of F are invariant under uniform scaling of the metric on F . Therefore, EGFs may be regarded as dynamical systems on the quotient space M (M, F , N)/(D(F , N)× R+ ). Solutions to (3.1) of the form (3.59) correspond to fixed points of the above dynamical system.


87

Question 3.1. Given (M, F , N), N being a vector field transverse to F , codim F = 1, and f j (0 ≤ j < n) of class C2 , do there exist complete EGS metrics on M? If they exist, study their properties, classify them, etc. The question is closely related to the basic problem in the theory of foliations mentioned in the Introduction. We are looking for initial conditions that give rise to self-similar EGSs. Vector fields represent diffeomorphisms infinitesimally. Let X (M) be the Lie algebra of all vector fields on M with the bracket operation. Let us introduce also the following notation: – X (F ), the set of vector fields on M preserving F . – X (F , N), the set of vector fields on M preserving F and commuting with N. By the Jacobi identity, X (F ) and X (F , N) are subalgebras of the Lie algebra X (M). Moreover, for any X ∈ X (F ) (or X ∈ X (F , N)) there exists a family φt ∈ D(F ) (respectively, φt ∈ D(F , N)) such that X = d φt /dt at t = 0. If φt ∈ D(F , N) then ϕt∗ N = N ◦ ϕt . For X ∈ X (F , N) generated by φt , the above yields LX N = 0. Remark 3.5. (a) The following conditions are equivalent, [55]: X ∈ X (F ) ⇐⇒ [X, T F ] ⊂ T F . By the above and the definition of X (F , N), we conclude that X ∈ X (F , N) ⇐⇒ [X, T F ] ⊂ T F

and [X, N] = 0.

(3.60)

(b) By (a), a normal vector field X = e f N preserves F if and only if ∇F f = −∇N N. Here ∇F f is the F -gradient of a function f ∈ C1 (M). Indeed, using (3.60), we get 0 = g([X, Y ], N) = −e f g( ∇N N,Y ) + Y (e f ),

∀ Y ⊥ N.

In particular, N ∈ X (F , N) if and only if F is a Riemannian foliation. Recall that the Lie derivative of a (0, p)-tensor S with respect to a vector field X is given by (LX S)(Y1 , . . . ,Yp ) = X(S(Y1 , . . . ,Yp )) − ∑i=1 S(Y1 , . . . , LX Yi , . . . ,Yp ). p

(3.61)

Using Proposition 3.1, we obtain Proposition 3.8. Let gt be a self-similar EGS (see Definition 3.4). Then bt = σ (t) φt∗ b0 ,

−1 At = φ t∗ A0 φ t∗ ,

h(bt ) = σ (t) φt∗ h(b0 ).

→ − → τ t =− τ 0 ◦ φt ,

(3.62) (3.63)

88


Differentiating (3.59) yields h(bt ) = σ (t) φt∗ gˆ0 + σ (t)φt∗ (LX(t) gˆ0 ),

(3.64)

where X(t) ∈ X (F , N) is a time-dependent vector field generated by the family φt . Since h(bt ) = σ (t)φt∗ h(b0 ), one may omit the pull-back in (3.64), h(b0 ) = σ (t)/σ (t) gˆ0 + LX(t) gˆ0 .

(3.65)

Motivated by the above, we have the following Definition 3.5. A pair (g, X) consisting of a metric g = gˆ ⊕ g⊥ on (M, F , N), and a complete vector field X ∈ X (F , N) satisfying for some ε ∈ R the condition → where h(b) = ∑0≤ j 0 and a family of diffeomorphisms φt ∈ D(F , N) such that a family of metrics gt , defined by (3.59) on (M, F ), is a solution to (3.1). Proof. (a) Recall that σ (0) = 1 and φ0 = id. Let X = dtd φt |t=0 ∈ X (F , N) be the vector field generated by diffeomorphisms φt . Then we have h(b0 ) = ∂t gt|t=0 = ∂t gˆt|t=0 = σ (0)gˆ0 + LX gˆ0 . This implies that g0 and X satisfy (3.66) with ε = σ (0). (b) Suppose that a pair (g0 , X) satisfies (3.66). Put σ (t) = eε t ; hence σ (0) = ε . Let ψt ∈ D(F , N) with ψ0 = idM be a family of diffeomorphisms generated by X. A smooth family gt of F -truncated metrics on M, defined by gˆt = σ (t)ψt∗ gˆ0 , is of the form (3.59). Moreover, ∂t gt = σ (t)ψt∗ (gˆ0 ) + σ (t)ψt∗ (LX gˆ0 ) =

σ (t) ∗ σ (t) ∗ ψt (ε gˆ0 + LX gˆ0 ) = ψt (h(b0 )). ε ε

→ → By Proposition 3.8, we have f j (− τ 0 ◦ ψt ) = f j (− τ 0 ) for t ≥ 0, and

ψt∗ (h(b0 )) = ψt∗

→ τ 0 )bˆ 0j ∑ j=0 f j (− n−1

n−1 → = ∑ j=0 f j (− τ 0 ◦ ψt ) ψt∗ bˆ 0j

n−1 → = ∑ j=0 f j (− τ t ) σ −1 (t) bˆ tj = σ −1 (t) h(bt ).

Hence h(bt ) = σ (t) ψt∗ (h(b0 )), and we conclude that ∂t gt =

σ (t) σ (t) ε

h(bt ) = h(bt ).

Remark 3.7. If (X, g) is an EGS structure then from Theorem 3.4 and Proposition 3.8 it follows that all the τ ’s are constant along X: X(τi ) = 0,

1 ≤ i ≤ n.

(3.71)

Example 3.9. We will illustrate Theorem 3.4, case (b). Let (g0 , X) with X = 0 be an EGS structure on (M, F ). Then h(b 0 ) = ε gˆ0 for some ε ∈ R. The family of leaf wise conformal metrics gt = eε t gˆ0 ⊕ g⊥ 0 obviously satisfies the PDE ∂t gt = ε gˆt . → − → − t ε t 0 t 0 ˆ ˆ Using b j = e b j and τ = τ (see Remark 2.1), we obtain h(bt ) = eε t h(b0 ) = eε t ε gˆ0 = ε gˆt . Hence ∂t gt = h(bt ), and gt is a self-similar EGS.


91

Theorem 3.5 (Canonical form). Let a self-similar EGS (gt ) be unique among soliton solutions to (3.1) with initial metric g0 . Then there is a 1-parameter family of diffeomorphisms ψt ∈ D(F , N) and a constant ε ∈ {−1, 0, 1} such that gˆt = (1 + ε t) ψt∗ gˆ0 .

(3.72)

The cases ε = −1, 0, 1 in (3.72) correspond respectively to shrinking, steady, or expanding EGS. Proof. Similar to that of [14, Proposition 1.3]. For the convenience of the reader, we prove the Theorem in the case σ (0) = 0. From (3.65) it follows that: h(b0 ) = (log σ ) (t) gˆ0 + LX(t) gˆ0 ,

(3.73)

where X(t) ∈ X (F , N) is a family of vector fields such that X(t) = d φt /dt. Differentiating (3.73) with respect to t gives (log σ ) (t) gˆ0 + LX (t) gˆ0 = 0.

(3.74)

Let Y0 = −X (0)/(log σ ) (0). We then have LY0 gˆ0 = gˆ0 . Substituting this into (3.65), we have for all t h(b0 ) = L(log σ ) (t)Y0 +X(t) gˆ0 . Put X0 = (log σ ) (0)Y0 + X(0). Then h(b0 ) = LX0 gˆ0 . Let ψt ∈ D(F , N) be a family of diffeomorphisms generated by X0 . We will check that g˜t = (ψt∗ gˆ0 ) ⊕ (ψt∗ g0 )⊥

(3.75)

is the EGF with the same initial conditions g0 , and that it is a steady soliton (i.e., σ (t) = 1 for all t). Indeed, differentiating (3.75), we have by (3.63),

∂t g˜t = ψt∗ (LX0 gˆ0 ) = ψt∗ (h(b0 )) = h(ψt∗ b0 ) = h(b˜ t ). Thus gt = g˜t , by uniqueness assumption for EGS solutions to our flow with initial metric g0 . Replacing φt by ψt we get σ (t) ≡ 1 in (3.59).

3.8.3 Umbilical EGS Let F be an umbilical foliation on (M, g) with the normal curvature λ . We have

and τ j = nλ j . By Propositions 3.4 and 3.5, EGFs preserve the umbilicity A = λ id of F and is given by: ∂t gt = ψ (λt ) gˆt , (3.76)

92


where ψ is given by (3.47). In this case, λt obeys the quasilinear PDE (3.48), and the EGS structure equation (3.66) reduce to the PDE ˜ ψ (λ ) = ε + (2/n) divF X.

(3.77)

If (g, X) is an EGS structure with an umbilical metric, by (3.71) we have X(λ ) = 0.

(3.78)

Certainly, any one-dimensional foliation is totally umbilical. The EGF on a surface (M 2 , g), foliated by curves F , is given by (3.76), where λ = τ1 is the geodesic curvature of the curves-leaves, and ψ = f0 ∈ C2 (R). EGS equation (3.66) on (M 2 , F ) reduce to the PDE

ψ (λ ) = ε + 2 divF X,

(3.79)

see (3.77) with n = 1. Remark 3.8. Let F be an umbilical foliation with metric gt = (e2 ft gˆ0 ) ⊕ g⊥ 0,

(3.80)

where ft : M → R ( f0 = 0) are smooth functions. We claim that 2 ∂t ft = ψ λ0 − N( ft ) ,

(3.81)

where λ0 is the normal curvature of g0 . Indeed, by Lemma 2.3, At = A0 − N( ft ) id, hence, λt = λ0 − N( ft ). By Lemma 2.3, we also have bt = e2 ft (λ0 − N( ft )) gˆ0 = (λ0 − N( ft )) gˆt . Similarly, btj = (λ0 − N( ft )) j gˆt . Differentiating (3.80) yields

ψ (λt )gˆt = h(bt ) = ∂t gt = (2 ∂t ft ) gˆt . Hence, 2 ∂t ft = ψ (λt ) that gives us the nonlinear PDE (3.81). One can solve this equation explicitly for ft in the special case f = c1 λ + c2 of the problem ∂t gt = ψ (λt ) gˆt , when (3.81) becomes linear of the form 2 ∂t ft + N( ft ) = c1 λt + c2 . In general, the nonlinear PDE (3.81) is difficult to solve, and we apply EGF’s approach: first we find λt from (3.48) then find gt from (3.76). Example 3.10. Let F be an umbilical foliation on (M, g0 ) with λ = const (if M is closed then λ = 0 by the known IF (1.2), i.e., M λ f vol = 0), and X – an


93

infinitesimal homothety along leaves: LX gˆ = C g, ˆ where C ∈ R. Using h(b0 ) = ψ (λ )gˆ0 , see (3.76) and (3.5), we conclude that (g0 , X) is an EGS structure with ε = ψ (λ ) − C. Taking X = 0, we obtain a self-similar EGS, gt = (eψ (λ )t gˆ0 ) ⊕ g⊥ 0

with

φt = idM .

For the special case of a totally geodesic foliation (i.e., λ ≡ 0, if such g0 exists on (M, F )), the pair (g0 , X) is an EGS structure with ε = ψ (0) − C. In particular, the totally geodesic metrics gt = (eψ (0)t gˆ0 ) ⊕ g⊥ 0 provide a self-similar EGS with φt = idM . Conversely, let (g, X) be an EGS structure on (M, F ). If F is umbilical (with the normal curvature λ ) then X is a leaf-wise conformally Killing field: LX gˆ = (ψ (λ ) − ε ) g. ˆ If F is totally geodesic (hence ψ = f0 (0)), then X is an infinitesimal homothety along leaves, LX gˆ = C g, ˆ with the factor C = f0 (0) − ε . In particular, X is a leafwise Killing field, LX gˆ = 0, when ε = f0 (0). This happens, for example, when M is a surface of revolution in M n+1 (c) foliated by parallels, see Example 3.16. Example 3.11. Consider biregular foliated coordinates (x0 , x1 ) on a surface M 2 (see [15, Sect. 5.1]). Because the coordinate vectors ∂0 , ∂1 are directed along N and F , respectively, the metric has the form g = g00 dx20 + g11 dx21 . Recall that h(b) = ψ (λ ) g11 . By Lemma 2.5 with n = 1, we have 1 λ = − √ (log g11 ),0 . 2 g00 Let X = X 0 ∂0 + X 1 ∂1 ∈ X (F , N). Using g01 = 0, we obtain divF X = g(∇∂1 X, ∂1 ) = ( ∂1 (X 1 ) + X 0Γ011 + X 1Γ111 ) g11 , where Γ011 = 12 (log g11 ),0 and Γ111 = 12 (log g11 ),1 . Hence, (3.79) has the form

ψ (λ ) − ε = 2 ∂1 (X 1 )g11 + X 0 g11,0 + X 1 g11,1 . From the condition [X, ∂1 ] ⊥ ∂0 , see (3.60), and [X, ∂1 ] = −∂1 (X 0 )∂0 − ∂1 (X 1 )∂1 we obtain ∂1 (X 0 ) = 0. Next, from the condition [X, N] = 0, see (3.60) again, and ) X,

−1 g002 ∂0

*

− 21 − 21 −1 0 = X(g00 ) − g00 ∂0 (X ) ∂0 + g002 ∂0 (X 1 )∂1

94


we obtain

∂0 (X 1 ) = 0,

1 ∂0 (X 0 ) = − X(log g00 ). 2

Definition 3.6 (see [31]). Denote the torus Rn+1 /Zn+1 by Tn+1 . For v ∈ Rn+1 , let Rtv (x) := x + tv be the flow on Tn+1 induced by a “constant” vector field Xv . We say v ∈ Rn+1 is Diophantine if there is s > 0 such that inf{|u, v| · us > 0 : u ∈ Zn+1 \ {0}}, where , and · are the Euclidean inner product and the norm in Rn+1 . When v is Diophantine, we call Rv a Diophantine linear flow. Theorem 3.6. Let F be an umbilical foliation (with a unit normal N) on a torus (Tn+1 , g), n > 0. Suppose that X is a smooth unit vector field on T F with the properties (i) ∇X X ⊥ T F ,

(ii) R(X,Y )Y ∈ T F

(Y ∈ T F ).

(3.82)

If the X-flow is conjugate (by a homeomorphism) to a Diophantine linear flow Rv then for any function ψ of a class C2 in either (3.76) or (3.47), there exists f : Tn+1 → R such that (g, f X) is an EGS structure, that is, (3.77) holds with X replaced by f X. Proof. For an umbilical foliation with the normal curvature λ , the Weingarten

By (3.82)ii and the Codazzi equation (see [42]) operator is conformal: A = λ id. (∇X A)Y − (∇Y A)X = R(X,Y )N , we have λ = const along X-curves. By (3.82)i, the X-curves are F -geodesics. ˜ be an EGS structure with X˜ = f X. From (3.77) and the known identity Let (g, X) divF ( f X) = f divF X + X( f ) it follows that divF X˜ = X( f ). We are looking for a solution of PDE, see (3.79),

ψ (λ ) − ε = (2/n) X( f ). As X-flow is conjugate to a Diophantine linear flow, by the Kolmogorov Theorem (see [31]) the above PDE has a solution ( f , ε ) ∈ C∞ (Tn+1 ) × R. From Theorem 3.6 follows: Corollary 3.5. Let a unit vector field X on a torus (T2 , g) define a foliation F by curves of constant geodesic curvature λ . If X-flow is conjugated to a Diophantine liner flow Rv then for any function ψ of a class C2 in (3.76), (3.47), there exists f : T2 → R such that (g, f X) is an EGS structure, see (3.79).


95

Notice that, if ψ ∈ C2 (R) then the following function is differentiable of class C1 : +

μ=

− n2 (ψ (λ ) − ψ (0))/λ ,

λ = 0,

− n2 ψ (0),

λ = 0.

(3.83)

Theorem 3.7. Let F be an umbilical foliation on (M, g) with normal curvature λ , and the function ψ ∈ C2 (R) given in (3.47) satisfies ψ = 0. Then the following properties are equivalent: (1) The normal curvature of F satisfies N(λ ) = 0. (2) (g, μ N), for some function μ , is an EGS structure, compare (3.77), indeed, one may take μ as in (3.83) and ε = ψ (0). Proof. (1) ⇒ (2): The EGS equations (for an umbilical metric g and the vector field X = μ N) are, see Proposition 3.10 and (3.78),

ψ (λ ) − ε = −(2/n) μ λ ,

X(λ ) = 0.

(3.84)

For ε = ψ (0) and μ given in (3.83), the above (3.84) are satisfied. Hence, by Definition 3.5, the pair (g, μ N) satisfies (3.77). (2) ⇒ (1): Using Definition 3.5, (3.78) and ψ = 0, we have the equality μ N(λ ) = 0 with μ given in (3.83). Consider an open set

Ω = int{q ∈ M : μ (q) = 0}. Certainly, N(λ ) = 0 on M \ Ω . By (3.83), we have ψ (λ ) = ψ (0). Thus N(ψ (λ )) = ψ (λ )N(λ ) = 0 on Ω . Since ψ = 0, we have N(λ ) = 0 on Ω . From the above we conclude that N(λ ) = 0 on M. From Theorem 3.7 it follows directly Corollary 3.6. Let F be a foliation (by curves) on a surface (M 2 , g), ψ ∈ C2 (R) be as in (3.47) and satisfies ψ = 0. Then the following properties are equivalent: (1) The geodesic curvature λ of F satisfies N(λ ) = 0. (2) (g, μ N), for some function μ , is an EGS structure, compare (3.79), indeed, one may take μ as in (3.83) with n = 1 and ε = ψ (0). Example 3.12 (Non-Riemannian EGS on double-twisted products). Let M = M1 × ˜ of M2 be the product (with the metric g˜ = g1 ⊕ g2 and Levi-Civita connection ∇) 1 a closed Riemannian manifold (M1 , g1 ) and a circle M2 = S with the canonical metric g2 . Let fi : M → R (i = 1, 2) be positive differentiable functions, πi : M → Mi the canonical projections, πi∗ : T M → ker π3−i the vector bundle projections. The metric of a double-twisted product M1 ×( f1 , f2 ) M2 is given by: g(X,Y ) = f12 g1 (π1∗ X, π1∗Y ) + f22 g2 (π2∗ X, π2∗Y ),

X,Y ∈ T M,

96


i.e., g = ( f12 g1 ) ⊕ ( f22 g2 ). The Levi-Civita connection ∇ of g obeys the relation [38] ,XY + ∑ ∇X Y = ∇ g(πi∗ X, πi∗Y )Ui − g(X,Ui)πi∗Y − g(Y,Ui )πi∗ X , (3.85) i=1,2 where Ui = −∇(log fi ). Both foliations M1 × {q2 } and {q1 } × M2 are umbilical with mean curvature vectors H1 = π2∗U1 and H2 = π1∗U2 , respectively. This property characterizes the double-twisted product, see [38]. By Proposition 3.4, the EGFs preserve the above double-twisted product structure. The mean curvature of the foliation F := M1 × {q2 } is constant along the curves {q1 } × M2 (i.e., N-curves) if and only if π2∗U1 is a function of M1 . In this case, by Theorem 3.7, F admits an EGS structure with X || N. Remark that for

as in Lemma 2.3. f1 = eϕ and f2 = 1, by (3.85), we find AN = −N(ϕ ) id, Theorem 3.8. Let (g, X) be an EGS structure on a closed surface M 2 foliated by curves F of the geodesic curvature λ (Torus T 2 is the only possibility here if we assume orientability!) Let ψ ∈ C2 (R), see (3.47), satisfies ψ = 0. If X || N then Xcurves are closed and define a fibration π : M 2 → S1 , and F is the suspension of a diffeomorphism f : S1 → S1 . Moreover, if ψ (λ ) = −2λ + c then (g, X) is the EGS structure (with ε = c) for any metric g ∈ M satisfying N(λ ) = 0, otherwise λ = 0 (i.e., F is a geodesic foliation). Proof. Assume the contrary. Then the foliation FN has a limit cycle. As M is compact, there is a domain Ω ⊂ M2 bounded by closed N-curves (which are limit cycles). By (3.78), λ = const along N-curves. As there are limiting leaves, λ = const on Ω . From the relation div N = −λ and the Divergence Theorem ! Ω

div N d vol =

! ∂Ω

N, ν d ω ,

where ν is the outer normal to the boundary ∂ Ω (hence ν ⊥ N), we conclude that λ = 0 on Ω , hence FN is a Riemannian foliation – a contradiction. By the classification theorem for foliations on closed surfaces, see [22], all the Xcurves are closed and define a fibration π : M 2 → S1 . By (3.84) with μ = 1 we have the following. If ψ (λ ) = −2λ + c then λ = const on M. Hence, using the integral formula M λ d vol = 0, we conclude that λ = 0. In this case, by Lemma 2.3, any F geodesic metric on (M, F , N) has the form g¯ = (π −1 ◦ σ g) ˆ ⊕ g⊥ , where σ : S1 → R is a smooth function.

3.9 Applications and Examples 3.9.1 Extrinsic Ricci Flow The extrinsic Riemannian curvature tensor Rm ex of F is, roughly speaking, the difference of the curvature tensors of M and of the leaves. More precisely, by the Gauss formula, see (1.5), we have


97

Rm ex (X,Y )V = g(AY,V )AX − g(AX,V)AY. Here, we study (in small dimensions n > 1) the extrinsic Ricci flow

∂t gt = −2 Rictex ,

(3.86)

where the extrinsic Ricci tensor is given by: Ric ex (X,Y ) = Tr Rm ex (·, X)Y,

X,Y ∈ T F .

Hence, Ric ex = τ1 bˆ 1 − bˆ 2.

(3.87)

Therefore, −2 Ric ex relates to h(b) of (3.1) with f1 = −2τ1 , f2 = 2 (others f j = 0). For n = 2, we have Ric ex = σ2 g. ˆ Example 3.13. By Lemma 3.5, for the extrinsic Ricci flow (3.86) we have

∂t τi = i τi N(τ1 ) + τ1 N(τi ) −

2i N(τi+1 ), i+1

i > 0.

(3.88)

Putting τi = ∑ nj=1 (k j )i in (3.88), we obtain PDEs for the principal curvatures

∂t ki = N(ki (τ1 − ki )),

i = 1, . . . , n

(3.89)

which for an umbilical F (i.e., ki = λ ) reduce to the Burgers’ type PDE

∂t λ = 2 (n − 1) λ N(λ ). For n = 2, (3.89) takes the form of the system for k1 and k2 with equal RHS’s,

∂t ki = N(k1 k2 ),

i = 1, 2,

hence ∂t (k2 − k1 ) = 0 and the difference k2 − k1 does not depend on t. We may say that in this case the pointwise “distance from umbilicity” is constant in time. Corollary 3.7. There exists a unique solution gt , t ∈ [0, ε ) (for some ε > 0), to the extrinsic Ricci flow (3.86) in both of the following cases: (i) n = 2, and τ1 = 0; in this case, gˆt = gˆ0 exp 0t σ2 dt ; (ii) n = 3, and |σ1 |3 > 27|σ3 | > 0.

Notice that if a (positive or negative) definite operator A is not proportional to id then the inequalities of Corollary 3.7 (ii) are satisfied. Proof. It is sufficient to show that in conditions of our Corollary, (3.88) is hyperbolic in the t-direction.

98


Let n = 2. By equality τ3 = 32 τ1 τ2 − 12 τ13 , the matrix of 2-truncated system (3.88) −2τ 1 1 is C2 = . One can see directly, or applying condition (H1 ) to Ric ex = −2τ12 τ1 σ2 g, ˆ that C2 is strictly hyperbolic if τ1 = 0. (If τ1 = 0, C2 has real eigenvalues λ1 = 0, λ2 = −τ1 , and the left eigenvectors v1 = (−τ1 , 1) and v2 = (−2τ1 , 1). If τ1 ≡ 0, i.e., F is minimal, the matrix C 2 is nilpotent, hence it is not hyperbolic). By Corollary 3.2, gˆt = gˆ0 exp 0t σ2 dt , where σ2 exists for 0 ≤ t < T . Remark that τ12 − 2τ2 = −(k1 − k2 )2 = const along the first family of characteristics dx dt = 0, hence the function k1 − k2 does not depend on t. Along the second 2 family of characteristics dx dt = −τ1 we have k1 k2 = (τ1 − τ2 )/2 = const. For n = 3, the matrix of 3-truncated system (3.88) is ⎛

−2 τ1 C3 = ⎝ −2τ2 τ13 − τ3 − 3τ1τ2

1 −τ1 3 2 2 (τ2 − τ1 )

0 4 3

τ1

⎞ ⎠.

Replacing τ -s by σ -s, see Remark 1.1, we obtain the characteristic polynomial P3 = λ 3 + 2 σ1 λ 2 + σ12 λ + 4 σ3 . Substituting λ = y − 23 σ1 into P3 gives P3 = y3 + py + q,

where

p = −(1/3) σ12 and q = 4 σ3 − (2/27) σ13 .

Depending on the sign of the discriminant D = (q/2)2 + (p/3)3 , we have ⎧ ⎪ ⎪ > 0, ⎨ < 0, D ⎪ = 0, ⎪ ⎩

one real and two complex roots, three different real roots, one real root with multiplicity three in the case p = q = 0, or a single and a double real roots when ( 13 p)3 = −( 12 q)2 = 0.

4 σ3 (27 σ3 − σ13 ). The condition “three different real roots” is In our case, D = 27 3 D < 0 ⇔ |σ1 | > 27 |σ3 | > 0.

Corollary 3.8. Let (M, g0 ) be a Riemannian manifold with a codimension-one umbilical foliation F of the normal curvature λ0 and a complete unit normal field N. Set at t = 0 T = ∞i f N(λ0 ) ≤ 0 on M, and T = 1/[4(n − 1) sup N(λ0 )] otherwise. M

Then the extrinsic Ricci flow (3.86) has a unique smooth solution gt on M for t ∈ [0, T ), and does not possess one for t ≥ T . Proof. The function λ˜ t (x) = λ (γ (x),t) along the trajectory γ (x) (γ (0) = q), of N satisfies (3.48) with ψ (λ ) = 4(1 − n)λ 2 and initial value λ˜ 0 (x) = λ (γ (x), 0). The statement follows from Proposition 3.5.


99

3.9.2 Extrinsic Ricci Solitons Foliations satisfying Ric ex = 0 are fixed points of extrinsic Ricci flow. That is, they

= 0, or equivalently, have the property A(A − τ1 id) k j (k j − τ1 ) = 0 (1 ≤ j ≤ n) for the eigenvalues k j of A. Because τ1 = ∑ j k j , from above it follows k j = 0 for all j. Hence, all extrinsic Ricci flat foliations are totally geodesic. In order to extend the set of such solutions we shall apply the normalized EGFs, see (3.51). Definition 3.7. We call gt ∈ M on a closed manifold M a normalized extrinsic Ricci flow if

∂t gt = −2 Rictex +(ρtex /n) gˆt ,

ρtex = −4

! M

σ2 d volt

vol(M, gt ).

(3.90)

Following Definition 3.5, an extrinsic Ricci soliton structure is a pair (g, X) of a metric g on (M, F ), and a complete field X ∈ X (F , N) satisfying for some ε ∈ R ˆ −2 Ric ex = ε gˆ + LX g. Remark 3.9. To explain ρtex in (3.90), we find the extrinsic scalar curvature: Tr Ric ex = Tr (τ1 A − A2 ) = τ12 − τ2 = 2 σ2 .

Substituting this into (3.51) instead of M Tr h(A) d vol, we obtain ρtex of (3.90). Notice that by the known integral formula (1.3), we have ! M

Tr Ric ex d vol =

! M

Ric(N, N) d vol .

Remark 3.10. A foliation (M, g) will be called CPC (constant principal curvatures) if the principal curvatures of leaves are constant. (a) From (3.2) it follows that all the (either normalized of not) EGFs preserve CPC property of foliations. Indeed, let such a flow on (M, F ) starts from a CPC metric. From N(τi ) = 0, N( f j ) = 0 ( j < n), and (3.2) we conclude that τ ’s do not depend on t. (b) Let (G, g) be a compact Lie group with a left invariant metric g. Suppose that the corresponding Lie algebra has a codimension one subspace V such that [V,V ] ⊂ V . Then V determines a CPC foliation on (G, g). Theorem 3.9. Let (g, X) be an extrinsic Ricci soliton structure on (M n , F ) (n > 2), and X a leaf-wise conformal Killing field (i.e., LX gˆ = μ g). ˆ

100


(i) Then, there are at most two distinct principal curvatures at any point q ∈ M. (ii) Moreover, if μ is constant along the leaves then F is CPC foliation. Proof. (i) As

(Ric ex ) = −A(A − τ1 id)

= r id

with r = 1 (ε + μ ), and LX gˆ = μ g, ˆ we obtain the equality A(A − τ1 id) 2 that yields equalities for the principal curvatures k j , k j (k j − τ1 ) = r

∀ j.

Hence, each k j is a root of a quadratic polynomial P2 (k) = k2 − τ1 k − r. Its roots are real if and only if τ12 + 4 r ≥ 0. In the case r > −τ12 /4 we have two distinct roots 1 τ1 ± τ12 + 4 r . k¯ 1,2 = 2 Let n1 ∈ (0, n) eigenvalues of A are equal to k¯ 1 and others to k¯ 2 . From τ1 = n1 k¯ 1 + n2 k¯ 2 and n = n1 + n2 we obtain n2 − n1 =

(n − 2) τ1 ∈ Z. (τ12 + 4 r)1/2

(3.91)

√ r for s = nn−2 . If n2 = n1 then τ1 = 0 and k1,2 = ± r, otherwise, τ12 = s24−1 2 −n1 (ii) Assume that μ is constant along the leaves. Then r = const on any leaf. Since n > 2, a continuous function τ1 : M → R has values in a discrete set, hence it is constant. Then all k j ’s (from both sets) are constant on M. For n even, (3.91) admits a particular √ solution: τ1 = 0, and n/2 principal curvatures k j equal to √ r, others to − r.

3.9.3 EGS on Foliated Surfaces In this section, M2 is a two-dimensional manifold (a surface) equipped with a transversally orientable foliation F (by curves) and a 1-parameter family of Riemannian structures (gt ), N a unit normal to F w.r.t. gt , and λt the geodesic curvature of the leaves w.r.t. N and gt . Given ψ ∈ C2 (R2 ), the EGF gt of type (b) on (M 2 , F ) is a solution to the PDE ∂t gt = ψ (λt ,t) gˆt , where λt satisfies the PDE 1 ∂t λt + N(ψ (λt ,t)) = 0 2 (see the proof of Proposition 3.4). By Corollary 3.2, if ψλ (λ0 , 0) = 0, see the condition (H1 ) with n = 1, then there is a unique local smooth solution λt for 0 ≤ t < T


101

with initial value λ0 determined by g0 , and the equality gˆt = gˆ0 exp( 0t ψλ (λt ,t) dt) holds. We have 1 ∂t (d volt ) = ψ (λt ,t) d volt , 2 see (2.21) with S = ψ (λt ,t) gˆt . Hence, for closed M 2 , the volume volt := of gt satisfies the equation

∂t volt =

1 2

! M

ψ (λt ,t) d volt .

M d volt

(3.92)

In order to estimate T (i.e., the maximal time interval of Proposition 3.5), suppose that the function ψ does not depend on t and put T = ∞ if N(ψ (λ0 )) ≥ 0 on M, and T = −2/ infM N(ψ (λ0 )) otherwise. Then the EGF ∂t gt = ψ (λt ) gˆt has a unique smooth solution gt on M 2 for t ∈ [0, T ), and does not possess one for t ≥ T . If, in addition to condition (H1 ), see Sect. 3.4, the inequality ψ (λ0 )N(λ0 ) ≥ 0 holds then the solution exists for all t ≥ 0 (i.e., T = ∞). Proposition 3.11. The Gaussian curvature Kt of the EGF of type (b) on (M 2 , F , gt ) is given by the formula !t Kt = div exp − ψ (λt ,t) dt ∇0N N + N(λt ) − λt2 . (3.93) 0

Proof. By Lemma 2.10 with S = ψ (λt ,t) gˆt and n = 1 , we have

∂t (∇tN N) = −ψ (λt ,t)∇tN N. Integrating the above ODE yields ∇tN N

!t = exp − ψ (λt ,t) dt ∇0N N.

(3.94)

0

In our situation here, K = Ric(N, N), therefore the formula (3.93) is a consequence of (3.94) and Proposition 1.4 with r = 0 (see also [54]), div(∇N N + τ1 N) = Ric(N, N) + τ2 − τ1 2 . Indeed, one should apply the identities divN = −τ1 and div(τ1 N) = N(τ1 ) − τ12 . (a) Let ψ = 1. The solution to ∂t gt = gˆt is gˆt = et gˆ0 (t ≥ 0). From ∂t λ = 0 we get λt = λ0 . By (3.93), the Gaussian curvature is Kt = e−t div(∇0N N) + N(λ0 ) − λ02 . There exists the limit as t → ∞: K∞ = N(λ0 ) − λ02 .

102


(b) Let ψ = λ , i.e., ∂t g = bˆ 1 . Then λt (s) = λ0 (s + 12 t) along any N-curve γ (s) in the (t, s)-plane. From the EGF’s equation

∂t gt = λ0 (s + t/2)gˆt we obtain gˆt (s) = gˆ0 (s) exp

! 0

t

λ0 (s + ξ /2) dξ

(t ∈ R).

For closed M 2 , by (3.92) we have volt = const. (c) Let ψ = λ 2 . Then

∂t λ + λ N(λ ) = 0 (the Burgers’ equation). If N(λ0 ) ≥ 0 (for t = 0) then the solution λt exists for all t ≥ 0. Example 3.14. Let a function f ∈ C2 (−1, 1) has vertical asymptotes x = ±1. Consider the foliation F in the closed strip Π = [−1, 1] × R (equipped with the standard flat metric) whose leaves are L± = {x = ±1},

Ls (x) = {(x, f (x) + s), |x| < 1},

where s ∈ R.

The normal N at the origin is directed along y-axis. The tangent and normal to F unit vector fields (on the whole strip) are X = [cos α (x), sin α (x)],

N = [− sin α (x), cos α (x)],

where α (x) is the angle between the leaves Ls and the x-axis at the intersection points. That is, f and α are related by f (x) = tan α (x)

and

cos α = [1 + ( f )2 ]−1/2 ,

sin α = f [1 + ( f )2 ]−1/2 .

The curvature of Ls is

λ0 (x) = f (x)[1 + ( f (x))2 ]−3/2 = α (x) · | cos α (x)|,

|x| < 1.

N-curves through the critical points of f are vertical, and divide Π into substrips. Typical foliations in the strip |x| < 1 with one vertical trajectory x = 0 are the following: (a) f has exactly one strong minimum at x = 0. (b) f is monotone increasing with one critical point x = 0; Taking f=

1 x2 /(1−x2 ) e −1 10

or α (x) =

π x 2


103

for (a), we obtain the Reeb strip, see Fig. 3.4c. For (b) one may take f = tan

π x , 2

π 2 x . 2

or α (x) =

Let ψ = ψ (λt ), where λt (x) is known for a positive time interval [0, ε ), see Proposition 3.5. We use X ∈ T F and normal N to represent the standard frame e1 = cos α (x)X − sin α (x)N, in the (x, y)-plane. By gˆt = gˆ0 e gt (X, X) = e

t

0 ψ (λt (x)) dt

t

0 ψ (λt (x)) dt

,

e2 = sin α (x)X + cos α (x)N , we have

gt (X, N) = 0,

gt (N, N) = 1.

The gt -scalar products of the frame {e1 , e2 } are Et = gt (e1 , e1 ) = sin2 α + cos2 α e Ft = gt (e1 , e2 ) = sin α cos α [e

t

0 ψ (λt (x)) dt

t

0 ψ (λt (x)) dt

Gt = gt (e2 , e2 ) = cos2 α + sin2 α e

,

− 1],

t

0 ψ (λt (x)) dt

.

The Gaussian curvature of g = E dx2 + 2F dx dy + G dy2 is given by the known formula .⎞ . ⎛ .E ∂x E ∂y E . . . ∂ G− ∂ F ∂ E− ∂ F 1 −1 x y y x .⎠ . ⎝∂x √ +∂y √ − K= √ 2 )2 .F ∂x F ∂y F . 2 2 2 4(EG−F 2 EG−F EG − F EG − F .G ∂ G ∂ G. x

y

which in our case, when gt do not depend on the y-coordinate, reads: 0 1 ∂x Gt −1 . Kt = / ∂x / 2 Et Gt −Ft2 Et Gt −Ft2 We have Et Gt − Ft2 = e

t

0 ψ (λt (x)) dt

while the N-curves satisfy the ODEs

dx/dt = − sin α (x),

dy/dt = cos α (x).

From the first of the ODEs above, we deduce the implicit formula t=−

! φt (x) x

dx sin α (x)

for local diffeomorphisms φt (x) (|x| < 1, t ≥ 0) of the N-flow. Suppose that ψ (λ ) = λ . As λt (s) = λ0 (s + 2t ) is a simple wave along N-curves, we have

104


λt (x) = λ0 (φt/2 (x)). For example, λt (0) = λ0 (0) for all t ≥ 0. Substituting Et , Ft , and Gt into the above formula for Kt yields !t ! t 2 1 λt dt − cos(2α )(α )2 Kt = (cos(2α ) − 1) 2 λt dt + 8 0 0 ! t 1 t t 1 1 − e− 0 λt dt − sin(2α ) α λt dt 3 + e− 0 λt dt . + sin(2α ) α 2 4 0 (3.95) Because α (0) = 0 and λt (0) = λ0 (0), one has Kt (0) = −(α (0))2 1 − e−t λ0 (0) . In case(i), we get α (0) > 0 and λ0 (0) > 0, so Kt (0) < 0 for t > 0. As lim φt (x) = 0 for |x| ≤ 1, there also exists lim λt (x) = λ0 (0) > 0. Hence for any

t→∞

t→∞

x ∈ [−1, 1] there is tx > 0 such that Kt (x) < 0 for t > tx . In case (ii) with α (x) = (π /2)x2, we have α (0) = 0 and α (0) = π . Moreover, λ0 (0) = 0 and λ0 (0) = π = 0. Since lim φt (x) = 0 for all 0 < x ≤ 1, there also t→∞ exists lim λt (x) = lim λ0 (φt/2 (x)) = λ0 (0) = 0.

t→∞

t→∞

By (3.95) we have Kt (0) ≡ 0 and the series expansion 5 Kt (x) = − π 2 x3 2

! t 0

λt dt + Ot x4 .

We conclude that there exists t0 > 0 such that Kt (x) changes its sign for t > t0 when we cross the line x = 0. Example 3.15. (a) Consider a foliation F by circles Lρ = {ρ = c} in the annulus Ω = {c1 ≤ ρ ≤ c2 } for some c2 > c1 > 0 with polar coordinates (ρ , θ ). Then X = ∂θ and N = ∂ρ are tangent and normal vector fields to the foliation. The standard flat metric is given by: ds2 = dρ 2 + Gt (ρ ) dθ 2 . Notice that λ0 (ρ ) = 1/ρ . Because ∂t Gt = ψ (λt (ρ ),t) Gt , we have Gt = ρ exp 2

!

t 0

ψ (λt (ρ ),t) dt .


105

Fig. 3.3 (a)–(b) Graphs of λ (ρ ). (c) Reeb annulus

From the formula for the Gaussian curvature of Example 3.14 we have ∂ρ Gt 1 . Kt = − √ ∂ρ √ 2 Gt Gt Let ψ = λ . Then λt (s) = λ0 (s + 2t ) on the N-curves. For the foliation by circles ρ = c we have λt (ρ ) = (ρ + 2t )−1 . The Gaussian curvature satisfies Kt ≡ 0 for all t. (b) Consider the Reeb foliation in the ring Ω = {1 ≤ ρ ≤ 2} of R2 . The leaves Ls (s ∈ R) and the boundary circles L± are parameterized as Ls (x) = L± (θ ) =

1 (3 + x)[cos(2π ( f (x) + s)), sin(2π ( f (x) + s))], 2 1 (3 ± 1)[cos(2πθ ), sin(2πθ )], 2

where, for example, f =

1 x2 /(1−x2 ) − 1]. 10 [e

|x| < 1,

0 ≤ θ < 1,

In polar coordinates we have

1 Ls : ρ = (3 + x), θ = 2π ( f (x) + s). 2 The curvature of Ls (at t = 0) is ks (x) = 4 π

4 π 2 (3 + x)2 ( f (x))3 + 2 f (x) + (3 + x) f (x) , [1 + 4 π 2(3 + x)2 ( f (x))2 ]3/2

|x| < 1.

The unique solution to ks (x) = 0 is x0 ≈ −0.49. The function λ of the Reeb foliation is rotationally symmetric, and is given in polar coordinates (θ , ρ ) as λ (θ , ρ ) = ks (2ρ − 3). The unique solution to λ (ρ ) = 0 is ρ0 = (x0 + 3)/2 ≈ 1.25. Hence λ (ρ ) < 0 for 1 ≤ ρ < ρ0 and λ (ρ ) > 0 for ρ0 < ρ ≤ 2, see Fig. 3.3a–c. Among all N-trajectories, one is closed, ρ = ρ0 , others approach to it as t → ∞. Notice that there exists λ∞ (x) = lim λt (x) = ks (2ρ0 − 3). t→∞

106


Fig. 3.4 (a) Graph of γ . (b) Hypersurface of revolution (of γ ) with λ = const. (c) Reeb strip

3.9.4 EGS on Hypersurfaces of Revolution In case of rotational symmetric metrics on M = R × Sn, g = dx20 + ϕ 2 (x0 ) ds2n ,

where ds2n is the metric of curvature 1,

the n-parallels {x0 = c} form a Riemannian umbilical foliation F with the unit normal field N = ∂0 . Hence, the EGFs preserve rotational symmetric metrics. (Notice that there are no umbilical foliations with λ = const = 0 on closed, or of finite volume, see [46], manifolds). In this case, the EGFs with generating functions → f j = f j (− τ ) can be reduced to (3.76) with ψ of (3.47), and λt can be found from (3.48). Any leaf-wise Killing field X ⊥ ∂0 provides the EGS structure on M n+1 with the rotational symmetric metric g and foliation by parallels. The unit normal field N with g as above is also the EGS structure. Assuming gˆt = ϕt2 gˆ0 and using (3.76), we obtain λt = −(ϕt ),0 /ϕt2 and !t 1 1 ∂t ϕt = ψ (λt ) ϕt ⇒ ϕt = ϕ0 exp ψ (λt ) dt . 2 2 0 In the particular case of ψ (λ ) = λ , we get the linear PDE ∂t λ + 12 N(λ ) = 0 representing the “unidirectional wave motion” along any N-curve γ (s),

λt (s) = λ0 (s − t/2).

(3.96)

If λ0 = c ∈ R then λt = c and ϕt = ϕ0 exp( 2t ψ (c)). Rotationally symmetric metrics with λ = const exist on hyperbolic space Hn+1 with horosphere foliation, Fig. 3.2. For example, assume n = 1, and consider (M 2 , gt , F ) in biregular foliated coordinates (x0 , x1 ). Hence (gt )00 = 1 and (gt )11 = ϕt2 . By Lemma 2.2 for b11 = gt (A∂1 , ∂1 ) = λt ϕt2 , we have λt = −(ϕt ),0 /ϕt2 . The Gaussian curvature of M 2 is Kt = −(ϕt ),00 /ϕt .


107

Example 3.16 (EGS on hypersurfaces of revolution). Some of rotationally symmetric metrics come from hypersurfaces of revolution in space forms. Evolving them by EGF corresponding to the foliation by parallels yields deformations of hypersurfaces of revolution. Revolving the graph of x1 = f (x0 ) about the x0 -axis of Rn+1 , we get the hypersurface Mn : f 2 (x0 ) = ∑ni=1 x2i foliated by (n − 1)-spheres {x0 = c} (parallels) with the induced metric g = (1 + f (x0 ) ) dx20 + f 2 (x0 ) ∑i=1 dx2i . n

2

(3.97)

(a) Revolving a line γ0 : x1 = x0 tan β about the x0 -axis, we build the cone C0 : (x0 tan β )2 = ∑ni=1 x2i , with the metric g0 = dx20 + (x0 sin β )2 ∑i=1 dx2i . n

Hence

ϕ0 = x0 sin β ,

λ0 (x0 ) = −2/x0 .

Applying the EGF ∂t gt = λt gˆt , we obtain by (3.96) that

λt (x0 ) = −

2 . x0 − t/2

The rotationally symmetric metric t 2 n (sin2 β ) ∑i=1 dx2i gt = dx20 + x0 − 2 appears on the same cone translated across the x0 -axis, t 2 2 n tan β = ∑i=1 x2i . Ct : x0 − 2 Any leaf-wise Killing field X ⊥ N with the induced metric g provide an EGS structure on M n . (b) Let us find a curve y = f (x) > 0 such that the metric (3.97) on the hypersurface of revolution M n : ∑ni=1 x2i = f 2 (x0 ) has λ0 = const = 1. Using λ0 = f (x1 ) sin φ , 0 where tan φ = f (x0 ), we obtain the ODE

f

/

| f | 1 + ( f )2

=1⇒

df f , =/ dx 4+ f2

whose solution is / 4+ f2−2 / γ : x = log / + 4 + f 2 + C, 4+ f2+2

where C ∈ R.

108


The hypersurface M n ⊂ Rn+1 looks like a pseudosphere (for n = 2 see Fig. 3.4a,b), however for x0 → ∞ it is asymptotic to the cone (x0 + C)2 = ∑ni=1 x2i . The sectional curvature K satisfies K(∂0 , ∂1 ) = −(x21 + 2)−2 < 0,

lim K = 0.

x1 →±∞

As for a horosphere foliation on hyperbolic space, the unit normal N to parallels and any leaf-wise Killing field X ⊥ N provide EGS structures on (M n , g).

References

1. L.J. Alias, S. de Lira, J.M. Malacarne: Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. of Math. Jussieu, 5(4), (2006) 527–562. 2. B. Andrews B., C. Hopper: The Ricci Flow in Riemannian Geometry, LNM 2011, Springer, 2011, 296 pp. 3. K. Andrzejewski and P. Walczak: The Newton transformation and new integral formulae for foliated manifolds, Ann. Glob. Anal. Geom. 37 (2) (2009), 103–111. 4. D. Asimov: Average gaussian curvature of leaves of foliations, Bull. Amer. Math. Soc. 84(1) (1978), 131–133. 5. P. Baird P., J. Wood: Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monographs 29, Oxford University Press, 2003, 520 pp. 6. J.L.M. Barbosa, K. Kenmotsu, G. Oshikiri: Foliations by hypersurfaces with constant mean curvature. Math. Z. 207, (1991) 97–108. 7. L. Bessierès et al: Geometrization of 3-manifolds, EMS, 2010, 237 pp. 8. S. Brendle: Ricci Flow and the Sphere Theorem, Graduate Studies in Math., 111, AMS, 2010, 176 pp. 9. F. Brito, R. Langevin, H. Rosenberg: Intégrales de courbure sur des variétés feuilletées, J. Diff. Geom. 16 (1981), 19–50. 10. F. Brito and P. Walczak: On the energy of unit vector fields with isolated singularities. Annales Polonici Math. LXXIII 316 (2000), 269–274. 11. M. Brunella and E. Ghys: Umbilical foliations and transversely holomorphic flows, J. Diff. Geom. 41 (1995), No 3, 1–19. 12. A. Candel and L. Conlon: Foliations, I, II, AMS, Providence, 2000, 2003. 13. B. Chow and D. Knopf: The Ricci Flow: An Introduction, AMS, 2004. 14. B. Chow et al: The Ricci Flow: Techniques and Applications, Parts I, II, III, AMS, 2007, 2010. 15. T. Cecil and S. Chern (eds.): Tight and taut submanifolds. Papers in memory of Nicolaas H. Kuiper, Cambridge University Press, 1997. 16. M. Czarnecki and P. Walczak: Extrinsic geometry of foliations, 149–167, in “Foliations 2005”, World Scientific Publication, NJ, 2006 17. W. Chen and J. Louck: The Combinatorial Power of the Companion Matrix, Linear Algebra and Its Applications 232 (1996) 261–278. 18. K. Ecker: Regularity theory for Mean curvature flow. Birkhäuser, Boston, 2004, 165 pp. 19. J. Eells and J. Sampson: Harmonic Mappings of Riemannian Manifolds, Amer. J. Math. 86, 109–160 (1964). 20. A. Epstein and E. Vogt: A counterexample to the periodic orbit conjecture in codimension 3. Ann. of Math. (2-nd series) 108(3) (1978), 539–552.

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51. M. Taylor: Partial Differential Equations, III: Nonlinear Equations, 2nd Ed., Applied Math. Sciences 117, Springer 2011. 52. P. Tondeur: Geometry of foliations, Birkhäuser, Verlag, Basel, 1997 53. P. Topping: Lectures on the Ricci flow, LMS Lecture Notes 325, London Math. Society and Cambridge University Press, 2006. 54. P. Walczak: An integral formula for a Riemannian manifold with two orthogonal complementary distributions. Colloq. Math. 58 (1990), 243–252. 55. ——— : Mean curvature invariant foliations, Illinois J. Math. 37 (1993), 609–623. 56. ——— : Dynamics of foliations, groups and pseudogroups, Birkhäuser, Basel, 2004. 57. ——— : On foliations with leaves satisfying some geometrical conditions, Dissertationes Math. 226 (1983), 1–51. 58. ——— : Mean curvature functions for codimension-one foliations with all the leaves compact, Czech. Math. J. 34 (1984), 146–155. 59. S. Walczak: Warped compact foliations, Annales Polonici Math. 94(3) (2008), 231–243.

Index

B Berger spheres, 36 biregular foliated coordinates, 20, 30, 72, 93 C characteristics, 59 method, 76 connection ˜ on E, 26 ∇ Levi-Civita, xii, 4, 25, 58, 70, 96 curvature extrinsic Ricci, 97 extrinsic Riemannian, 5, 96 Gaussian, 4, 101, 105 mean, 6 normal Ricci, 46 Ricci, 2 Riemannian, 2 scalar, 2 sectional, 2 D diffeomorphism group, 86 preserving F , 86 preserving F and N, 86 distribution, 2 involutive, 2 divergence, 8 partial, 8 E equation Burgers, 60, 80, 97, 102 Codazzi, 5, 9, 94 Euler, 28

Gauss, 5 hyperbolic, 59 quasilinear, 59 extrinsic geometry, ix

F fixed point, 81, 86 flow Diophantine linear, 94 extrinsic geometric (EGF), xi, 55 Newton transformation, 56 normalized extrinsic geometric, 81 normalized extrinsic Ricci, 99 normalized Newton transformation, 82 foliation, 1 constant principal curvature (CPC), 99 extrinsic Ricci flat, 99 horospheric, 86, 108 minimal, x, 57 orientable, 1 parabolic, 81 Reeb, 103, 105 Riemannian, 57 totally geodesic, x, 57, 78 transversely orientable, 1 umbilical, x, 47, 57, 78 warped, 36 form second fundamental, ix volume, 36 formula Gauss, 5 integral, x, 3, 14 Newton, 6 variational, 31, 35 Weingarten, 5

V. Rovenski and P. Walczak, Topics in Extrinsic Geometry of Codimension-One Foliations, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4419-9908-5, © Vladimir Rovenski and Paweł Walczak 2011

113

114 formulae variational, xi

G gradient of the functional, 37

H hessian, 88 hypersurface, 5

I initial value problem, 59 integral formula, 46 isomorphism (musical) , 9, 24 , 24 K Kronecker’s delta, 24

L Lie bracket, 2 derivative, 87

M manifold closed, 2 Einstein, 16 of constant curvature, 3 matrix companion, 61 generalized companion, 61, 67 hyperbolic, 68 measure of non-umbilicity, 47 metric F -truncated, 20 critical, 37 double twisted product, 95 rotational symmetric, 106 time dependent, 24

N Newton transformation, 6

Index O operator V , 26 Jacobi, 2, 9 Weingarten, xii, 5

S soliton (EGS) expanding, 91 extrinsic geometric, 85, 86, 88 extrinsic Ricci, 99 gradient, 88 shrinking, 91 steady, 91 submanifold, 4 symmetric functions elementary, 6 power sums, 5

T tensor F -truncated, xii extrinsic geometric, 58 inner product, 24 second fundamental, ix time-derivative of connection, 70 torsion, 26 Theorem Cayley-Hamilton, 6 divergence, 4, 14, 27, 89 Frobenius, 2 Fubini, 29 Kolmogorov, 94 Sacksteder, 86 Tischler, 86 truncated conformal variation, 37

V vector field complete, 77 energy, 49 infinitesimal leaf-wise homothety, 93 leaf-wise conformal Killing, 93, 99 leaf-wise Killing, 93 mean curvature, x, 96 total bending, 49, 84

Topics in extrinsic geometry of codimension-one foliations

Topics in Extrinsic Geometry of Codimension-One Foliations

Extrinsic geometry of convex surfaces

Birational Geometry of Foliations

Foliations in Cauchy-Riemann geometry

Foliations: Geometry and Dynamics

Foliations: Geometry and Dynamics

Foliations: Geometry and dynamics

Topics in Differential Geometry

Topics in differential geometry

Topics in differential geometry

Topics in elementary geometry

Topics in Integral Geometry

Topics in Differential Geometry

Foliations and the geometry of 3-manifolds

Selected topics in integral geometry

Selected topics in algebraic geometry

Selected Topics in Convex Geometry

Selected Topics in Integral Geometry

Elementary topics in differential geometry

Selected Topics in Convex Geometry

Elementary Topics in Differential Geometry

Selected Topics in Integral Geometry

Geometric Theory of Foliations

Stochastic Geometry: Selected Topics

Foliations in Cauchy-Riemann Geometry (Mathematical Surveys and Monographs)

Geometric Theory of Foliations

Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs)

Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs)

Topics in algebraic geometry and geometric modeling

Topics in classical algebraic geometry. Part 1

Topics in extrinsic geometry of codimension-one foliations