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0: Let W : X --+ Y be a local Eminimizer, and suppose that the range of W is contained in a compact convex
set V C Y of diameter < it/(2V) such that geodesics in V are uniquely determined by their endpoints. Then
-oc, this follows from Axiom 2 when writing f as an increasing pointwise limit of finite continuous functions on 8V. For
general f, the upper integral H1(x) := f'" f dwv is harmonic in x E V,
2. Harmonic spaces, Dirichlet spaces, geodesic spaces
17
being the pointwise infimum of the harmonic functions Hg as g ranges over the lower directed family of all l.s.c. (i.e., lower semicontinuous) functions g >, f, g > -oo on 8V. Similarly, H f(x) := f. f dw2 is harmonic; and since H f - H f >, 0, with equality at x0, we conclude from the above minimum principle that H f = H f. Hence f is indeed integrable with respect to wx for every x E V, and its integral is harmonic in V.)
For a given regular domain V the harmonic measures wy, x E V, are pairwise absolutely continuous. (In fact, a set e C 8V has measure 0 with
respect to wy if the function f equal to oo in e and to 0 in 8V \ e, is integrable with respect to w, v, and this property is independent of x, see above.)
For given x E X, wy converges weakly to the Dirac measure ex as the regular set V shrinks to {x}. (Given f E C,(X) and c > 0, we have in fact If - f (x) I < c in some neighbourhood W of x, and so I f f dwy - f (x) _ f (f - f (x)) dws' I < c for any regular neighbourhood V C W of x.) Using the above results one may establish Mokobodzki's theorem: The family of all positive harmonic functions on a domain U C X having the value 1 at a given point of U is equicontinuous. (For the proof see [Br 1969, p. 22].)
A consequence of this is the following Harnack inequality: For any harmonic function u > 0 in a domain U C X and any compact set K C U, we have
max u(x) < c min u(x), zEK
xEK
where c is independent of u. (Indeed, for every x E U and e > 0 there is, by equicontinuity, a neighbourhood V of x such that max u(y) , 0 on X having no other harmonic minorant > 0 than the function 0.
The potentials on X form a convex cone. Any superharmonic function s >, 0 on X majorized by a potential is itself a potential. A classical theorem of F. Riesz extends as follows to any harmonic space (X,71): Every superharmonic function s on X which has a subharmonic minorant can be written uniquely as s = p + h, where p is a potential on X and h is harmonic on X ; namely, h is the greatest subharmonic minorant of s, see [Br 1969, p. 38]. In particular, every subharmonic minorant of a potential is < 0. The boundary minimum principle: If a superharmonic function s on an open set U C X satisfies lim infs(x) > 0 Uaz- .y
for every y E 8U,
and if moreover s >, -p on U for some potential p on X, then s >, 0 on U. If U is compact and OU 54 0, the condition s > -p can be omitted. If U (and hence X) is non-compact, the condition s >, -p can be replaced by lim inf s(x) > 0 as x E U tends to the point at infinity for X [Br 1959].
Definition 2.4. A subset e of X is polar if every point of X has an open neighbourhood U on which there is a superharmonic function which equals no at every point of e fl U.
2. Harmonic spaces, Dirichlet spaces, geodesic spaces
19
In a Riemannian manifold M" every set which is locally of finite (n - 2)dimensional Hausdorff measure is polar. The polar sets are the potential theoretic null sets. Let e be a polar subset of a harmonic space X. Then, as shown in [Br 1959, p. 1.34]: For any regular set V C X and any x E V, we have wy (e) = 0. If X is connected then so is X \ e. (See also [CC 1972, p. 145].)
If two superharmonic functions on X coincide off e then they are identical.
If e is closed, if s is superharmonic on X \ e and if
liminfs(x) > -oo
for every y E e,
then the extension of s by this lim inf is superharmonic on X. In particular, such a set e is a removable singularity set for locally bounded harmonic functions. A P-harmonic, or strongly harmonic, space is a harmonic space on which there exists a potential p > 0. Every open subset of a P-harmonic space is a P-harmonic space. Every harmonic space can be covered by P-harmonic open subspaces [CC 1972, §2.3].
Further development of potential theory on a Brelot harmonic space has led to generalizations of most qualitative results from classical potential theory. For example, for a connected P-harmonic space X, if any two non-zero potentials which are harmonic off one and the same point are proportional, then there exists a lower semicontinuous function C : X x X ]0, oo], finite and continuous off the diagonal in X x X, such that (1) For every positive measure p on X the function Gp defined on X by
Gp(x) = fX G (x, y) dµ(y),
xEX,
is a potential on X, unless Gp - oo. (2) Every potential p on X has the form p = Gµ for a unique positive measure p on X. The support of u is the complement of the biggest open set in which p is harmonic. Such a function G is called a Green kernel for X. See [Her 1962] and also [CC 1972, pp. 332f.].
Remark 2.1. A notion of harmonic space X, more general than that of Brelot, was introduced by Bauer [Bau 1966] (see also [Bau 1984]), who replaced Harnack's monotone convergence property by the weaker convergence
Part I. Domains, targets, examples
20
property of Doob [Doob 1956]: For any increasing sequence of functions u,, E 1-1(U) (U open in X), u := sup,, u, is in 1(U) provided that {u < oo} is dense in U. Bauer showed that the germs of solutions to the heat equation on R'+' form a harmonic space in his sense (this was his primary motivation), and that large parts of the potential theory on Brelot spaces carry over or can be adapted to Bauer spaces, even though the minimum principle for superharmonic functions may not hold. See also [CC 1972].
Dirichlet structures on a space General references: [BM 1995], [Fuk 1980], [FOT 1994]. Let X be a separable locally compact space, and p a Radon measure with
It > 0 on every non-void open subset of X, that is, suppp = X. Let V be a linear subspace of L2(X,µ).
Definition 2.5. A Dirichlet form E on X with domain V(E) = V is a symmetric bilinear map E : D x D - R which is nonnegative definite. Set E(f, f) = E(f ). Assume that (i) D is dense in L2 (X, p), and (ii) V is complete in the inner product
(f,g)a = (f,g)L2(x.,) +E(f,g) Thus (D,
is a Hilbert space. A normal contraction of R is a map
T(0)=0and DT(s)-T(t)I 0, the diffusion semigroup on L2(X, p) generated by A. It is strongly continuous: limo Pt f = f
for all f E L2(X, µ).
Part I. Domains, targets, examples
22
We represent Pt in terms of its symmetric kernel pt:
(Ptf,f)v= Jf x X \diag f(x)f(b)pt(x,dy)dp(x) If we write 2f(x)f(y) = If(x)I2 + If(y)12 -
Et(f)
If(x) - f(y)12 then
(f - Ptf,f)L2 = tt IIf Ili2 - 2t (2IIf IIi2 2t
- f fxxx I f (X) - f(y) 12pt (X, dy) dµ(x))
Jf x x I f (x) - f (y) I2pt (x, dal) dµ(x)
Et (f) increases as t > 0 decreases, and the D richlet form E has the representation
E(f) = to Et (f)
for f E D(E).
(See [BD 1959] and [De 1970].)
A core C of E is said to be p-separating if for all points x 76 y in X there is a function g E C with e(g,g) < it on X, and g(x) 96 g(y). Here e(g, g) < it means that e(g, g) is absolutely continuous with respect to p, with Radon-Nikodym derivative
e(g) := e(9,9) :=
deb 9)
0 in some neighbourhood of x such that u(x) > 0. Then the spaces ?{(U) of (continuous) local E-minimizers in open subsets U of X determine a Brelot harmonic structure on X.
Geodesic spaces General references: [Bal 1995], [BNik 1993], [GLP 19811, [Gro 1999], [P 1998].
Let (Y, dy) be a metric space and -y : I = [a, b] - Y a path. Its length is L(y) t=i
the supremum over all partitions it : a = to < . . < t,. = b of I. Say that y is rectifiable if L(y) < oc.
2. Harmonic spaces, Dirichlet spaces, geodesic spaces
25
A length space is a metric space (Y, dy) such that for any pair of points yo, Yi E Y,
dy (yo, yl) = inf {L(y) : y is a rectifiable path joining yo to yl I.
Such a metric is sometimes said to be intrinsic (or inner). Every length space (Y, dy) is locally connected. Indeed, every open ball U = {y E Y : dy(a, y) < r} is connected (even path connected). There is in
fact a rectifiable path y joining a and y such that L(y) < r, and so y has range in U because, for any z E y, dy (a, z) < dy(a, z) + dy(z, y) , L(y) < r.
Definition 2.7. A geodesic space is a length space (Y, dy) for which any yo,yi E Y can be joined by a rectifiable path y with dy(yo,yl) = L(y). A (minimizing) geodesic segment of (Y, dy), parametrized by path length (as will be understood in the absence of any other indication), is a rectifiable
path y : I - Y for which dy (y(s), y(t)) = Is - tI for all s, t E I. For a compact interval [a, b] this requirement reduces to dy(y(a), y(b)) _ lb - al. The path y in the above definition is therefore a geodesic segment.
A geodesic of (Y, dy) is a path y : J -. Y whose restriction to every sufficiently small compact subinterval of J is a geodesic segment. We shall allow J to be open, closed, or half-open. A function f : V - 1R on a set V C Y is convex if for all geodesic segments y : [a, b) --- Y with image in V,
f(y(ta + (1 - t)b)) < t f(y(a)) + (1 - t) f(y(b)),
0 < t ,1.
On an open set V C R a function f : V --+ R is convex if the restriction of f to each component of V is convex in the usual sense. Say that a subset V C Y is convex if any two points of V can be joined by a geodesic segment of (Y, dy) lying in V. A geodesic space (Y, dy) is said to be locally squared-convex if every point y E Y has an open convex neighbourhood V such that V x V is convex; i.e., for any two minimizing geodesic segments a, Q : [0,1] - V parametrized
proportionally to path length, the function f : [0,1] - R defined by f (t) = dy (a(t), Q(t)) is convex. In other words, (Y, dy) is locally squared-convex if the function dy : V x V -+ R is convex for every such V ([AB 1990]). If this applies to V = Y, (Y, dy) is said to be globally squared-convex.
Part I. Domains, targets, examples
26
The following result is Gromov's version of the Hadamard-Cartan theorem [AB 1990], [Gro 1987]: A simply connected complete locally squared-convex geodesic space (Y, dy) is globally squared-convex; hence any two of its points are joined by a unique geodesic, and that geodesic varies continuously with its endpoints.
In a geodesic space (Y, dy) fix a point yo; and denote by (Gyo (Y), D) the metric space of geodesic segments y of constant speed (i.e., parametrized proportionally to path length) and with common domain I = [0, 1], for which
7(0) = yo, and D(/3, ry) = sup {dy (13(t), ry(t)) : t E I} .
Then Gya (Y) is clearly contractible, and the map expyo : Gyo(Y)
Y
defined by expyo (ry) = -y(l) is continuous and surjective. If (Y, dy) is a complete locally squared-convex geodesic space, then (gyo(Y), D) is complete
and (2.3) is its universal cover. In particular, (Y, dy) has no conjugate points. That is, for all yo E Y, expyo maps some neighbourhood V of any ry E Gyo(Y) homeomorphically onto a neighbourhood of i'(1) [Gro 1987], [AB 1990, Theorem 4]. Furthermore, if H denotes a homotopy class of paths in Y joining yo to yl, then H contains exactly one geodesic. It is minimizing, and it depends continuously on yo, yl. Every point of Y has a convex neighbourhood in which geodesics are uniquely determined by their endpoints and vary continuously with these [AB 1990, §3]. Consider a triangle pqr in a geodesic space (Y, dy) with minimal geodesic sides. Fix a real number K. We wish to compare pqr with the triangle P 4 r, with corresponding sides of the same lengths, in the 2-dimensional space form R2(K) with constant sectional curvature K. Thus
dy(p,q)=Ip-4I, dy(p,r)=IP-TI, dy(q,r)=]4-rI, the "norm" referring to the distance in R2(K). There is such a comparison triangle, and it is unique subject to the condition on the perimeter dy (p, q) + dy (q, r) + dy (r, p) < 2tr/vrK-
if K > 0.
(2.4)
Write 21r/v'-K- = oo if K S 0.
Say that pqr is K-dominated if it satisfies (2.4) and has the following triangle comparison property: dy (a(t), p) < I d (t) - 0 1
(2.5)
2. Harmonic spaces, Dirichlet spaces, geodesic spaces
27
for all a(t) on the geodesic segment qr, with a (t) the corresponding point on 4f. Definition 2.8. A geodesic space (Y, dy) has curvature < K < oo (written curv(Y, dy) < K) if every point y E Y has a neighbourhood V such that every geodesic triangle with vertices in V is K-dominated. Similarly for curv(Y, dy) >, K > -oo. ([A 1951; 1957], [BNik 1993].) A smooth complete Riemannian manifold has curvature < K, resp. > K, if it has sectional curvature < K, reap. > K. In a geodesic space with curv(Y, dy) < K for some 0 < K < oo every triangle satisfying (2.4) is K-dominated ifY has bi-point uniqueness, in the sense that geodesics in Y of length < 7r/VK_ are unique and vary continuously with their endpoints (the continuous dependence being automatic if Y is complete and locally compact) [A 1951; 19571. A geodesic space with curv(Y, dy) < 0 is locally squared-convex. The easy proof is based on characterizing local squared convexity as follows: Every y E Y has a neighbourhood V in which dy (m(p, q), m(p, r)) < dy (q, r) 2 of the for any geodesic triangle pqr in V; here m(p, q) denotes the midpoint geodesic segment pq. Then applying [BNik 1993, Proposition 2.2]: dy (m(p, q), m(p, r)) < I m(f, 4) - m(P, r) I = 114 - r I =
2dy(q, r)
2
A smooth Riemannian manifold Y with intrinsic distance dy has nonpositive sectional curvature if (and only if) (Y, dy) is locally squared convex,
[BGS 1985, §1]. The "if part" does not extend to geodesic spaces, cf. [AB 1990, p. 313]. A simply connected complete geodesic space (Y, dy) of nonpositive curva-
ture (i.e., curv(Y, dy) < 0) has globally nonpositive curvature, in the sense that every triangle in Y is 0-dominated. Indeed, Y is globally squaredconvex, by the above Hadamard-Cartan theorem; we may therefore apply the quoted result of Alexandrov, or argue directly as follows: For a given triangle pqr in Y with midpoint m of pq, let pt, qt, rt denote
the images of p, q, r under the geodesic homothety with centre at m and factor t > 0 small enough so that the triangle ptgtrt is 0-dominated. Then dy(r, m) < I r - mI. In fact,
t24(rt,m) < dy(pt,rt)+ t2 2
2d2Y(gt,rt) - 1dY(pt,gt)
(p, r) + 2 t2dY (q, r) - 4 t24. (p, q) = t2 it
- m12
Part I. Domains, targets, examples
28
because dy (pt, rt) < tdy (p, r) and dy (qi, rt) < tdy (q, r), by global squaredconvexity, while dy(pt,gi) = tdy(p,q). Conversely, every locally compact, complete geodesic space (Y, dy) of glob-
ally nonpositive curvature is simply connected. Indeed, geodesic segments in Y are uniquely determined by their endpoints and (by local compactness) depend uniquely on them. Any continuous path in Y is therefore (even geodesically) fixed-end homotopic to a unique constant-speed geodesic segment. In a complete geodesic space Y with globally nonpositive curvature consider a geodesic quadrilateral with constant-speed geodesic segments yl, y2 as a pair of opposite sides and vertices ordered yi (0), -yl (1), 72 (1), y2 (0). There
is a convexity inequality, basically due to Reshetnyak [Resh 1968] and applied in [KS 1993, Corollary 2.1.3]: d ,(yi(t),y2(t)) 3, the link of a vertex may not bound an n-disc.
Lip polyhedron Replacing `homeomorphism' by `Lip homeomorphism' throughout the two preceding sections leads us to the notion (and some properties) of a Lip polyhedron; i.e., a metric space X which is the image of the metric space IKJ
of some complex K under a Lip homeomorphism 9 : JKJ - X. The pair (K, 9) is then called a Lip triangulation (or briefly: a triangulation) of the Lip polyhedron X. Thus any polyhedron X with a triangulation (K, 0) becomes a Lip polyhedron when X is given the metric corresponding via 0 to the barycentric metric d on IKI (or any locally equivalent metric). For a Lip circuit X with singular set S, X \ S is a Lipschitz manifold (cf. Example 3.6).
By a null set in a Lip polyhedron X is understood a set Z C X such that Z meets every maximal simplex s (relative to some, and hence any triangulation T = (K, 9) of X) in a set whose pre-image under 9 has pdimensional Lebesgue measure 0, p = dims. Note at this point that every proper face of s is a null set, and that every Lip homeomorphism between domains of R' preserves the sets of p-dimensional Lebesgue measure 0, cf. e.g., [Fe 1969, Chapter 3]; see also Example 3.6 above. As usual, `almost everywhere' (a.e.) means: everywhere except in some null set. Remark 4.1. From Lemma 4.1 follows that every Lip polyhedron (X, g, dX )
can be mapped Lip homeomorphically and (simplexwise) affinely onto a closed subset of a Euclidean space. Analogously, let X denote a locally compact separable metric space, each point of which has a neighbourhood that can be bi-Lip embedded as a closed subset of some Euclidean space. Then there is a proper Lip homeomorphism of X onto a closed subset of some Euclidean space. [IV 1977, Remark 4.3 (3)]. For instance, every Riemannian Lipschitz manifold can be embedded in this way.
4. Riemannian polyhedra
47
Example 4.2. Every domain X in RI, endowed with the metric induced from the Euclidean distance on R', is an admissible Lip polyhedron of homogeneous dimension n (without boundary), and indeed a normal Lip circuit. To see this, consider a usual locally finite Euclidean triangulation T of X, i.e., a covering of X by a sequence of Euclidean n-simplexes sJ, j E N, such that the intersection (if nonvoid) of any two of these is a face of each, and that every compact in X meets only finitely many sj. In particular, the s3 are pairwise non-overlapping. The set K of all vertices vi of T is then a simplicial complex such that {vi,,, , vi.) is an n-simplex of K if the convex hull conv{vi°, , vin } equals some sj. The map f : IKI - X taking
a formal finite linear combination E aivi E IKI to E aivi E R is a bijection, and f is a Lip homeomorphism, by the proof of Lemma 4.1. The Lip polyhedron X becomes a Riemannian polyhedron (see below) when given a Riemannian metric with measurable components and local elliptic bounds, cf. Example 3.6.
Subexample 4.2. The closed halfspace X = 1R! = {(x1,
, xn) E R' :
xn > 0} with the Euclidean metric is an admissible Lip polyhedron of homo-
geneous dimension n with boundary bX = {xn = 0}, and indeed a normal Lip circuit. This is proved as in the preceding example, using a locally finite Euclidean triangulation of lR+. Furthermore, X becomes a Riemannian polyhedron when endowed with a Riemannian metric (as a Lipschitz manifold with boundary).
Riemannian polyhedron A Riemannian polyhedron X = (X, g) will be defined as a Lip polyhedron X endowed with a covariant bounded measurable Riemannian metric tensor ge on each maximal simplex s, satisfying the ellipticity condition (4.1) below.
Let T = (K, 9) be a specific (Lip) triangulation of the Lip polyhedron X. We shall view IKI as embedded in a Euclidean space V via an affine Lip homeomorphism, cf. Remark 4.1. Suppose, to begin with, that X has homogeneous dimension n. Choose a measurable Riemannian metric g8 on the open Euclidean n-simplex 9-1(s°) of IKI C V. In terms e.g., of Euclidean coordinates x1, , xn of points
x = 9-1(p) E 9-1(8°) (in its affine span in V), ge thus assigns to almost every point p E s°, or to x E 9-1(s°), an n x n symmetric positive definite matrix 98(x) = (9ij(x))i,9=1 ... ,n
48
Part I. Domains, targets, examples
with measurable real entries; and there is a constant A, > 0 such that (with the usual summation convention) n
n
j(X)&j < As E(e )2
Ae-2
i=1
(4.1)
t=1
= (t1, ,tn) E Rn. This condition is independent of the choice of Euclidean frame on 0-1(s°). The second for a.e. x E 9-1(s°) and every
inequality in (4.1) amounts to the components of gs being bounded. The condition (4.1) also becomes independent of the chosen triangulation when we define, for any other (Lip) triangulation T' = (K', 9') of X, the Riemannian metric g,' a.e. on 9'-1(sj0) for each n-simplex s' of T' by covariance. Explicitly, if the open set U = s° n s'° in X is non-empty, define
(g,01e-(U) = (0-1 o (7)'(g9)Ie-'(u), noting that 0-1(U) C 0-1(s°), etc. As in the case of Riemannian Lipschitz manifolds (Example 3.6) we have here used Rademacher's theorem on the differentiability almost everywhere of Lipschitz continuous functions. When
s ranges over S(n)(X,T) (the n-simplexes of (X,T)), the open sets 9'-1(U) with U as above cover 9'-1(s'O) up to a null set, and so g,,(x') is altogether defined a.e. for x' E 0'-1(s'°) as an n x n symmetric positive definite matrix with bounded measurable entries g,, (x'). Furthermore,
A2
< 9 j (x i=1
A;,
n D02
i=1
a.e. for x' E 9j-1(s'°) with
As, = max{a,,,,A, : s° n s'° 96 0}, 9
whereby denotes a bi-Lip constant of 0-10 0' considered on s° n s'° for those finitely many s E S(n) (X, T) for which s° n s'° 0 0. If X is not necessarily dimensionally homogeneous, define g, as above for
each homogeneous subpolyhedron Xn, where Xn denotes the union of all maximal n-simplexes of X . Clearly, each Xn is closed in X and independent of the chosen triangulation. Furthermore, X nnXn is a null set when m 76 n. The covariantly defined map g : s i g, thus defined on the set S' (X, T ) of all maximal simplexes s of T, is called the Riemannian (polyhedral) structure, or metric, on X .
4. Riemannian polyhedra
49
For simplicity of statements we shall sometimes require that, relative to a fixed triangulation T of a Riemannian polyhedron X,
A: = sup{As : s E S*(X,T)} < oo.
(4.2)
This condition of uniform ellipticity and uniform boundedness will, however, mostly be used with (X, T) replaced by some finite subpolyhedron, such as
the closed star of a point of X, and here the finiteness of A is of course a consequence of that of each As. The smallest constant A in (4.2) will be called the ellipticity constant of X = (X, T, g). Not every admissible finite Riemannian polyhedron (X, g) can be isometrically embedded in a Euclidean space (cf. Remark 3.5, in which (X, g) is a non-differentiable triangulable Riemannian Lipschitz manifold). A continuous map w : X - Y between Riemannian polyhedra is called simplicial if it can be realized as a simplicial map between (Lip) triangulations of X and Y. Then cp E Lip1oc(X,Y). A Riemannian metric g on a polyhedron X is said to be continuous if, relative to some (hence any) triangulation, gs is continuous up to the boundary on each maximal simplex s, and for any two maximal simplexes s and s' sharing a face t, gs and gs' induce the same Riemannian metric on t. There is a similar notion of Lip continuous Riemannian metric. In Part III in particular we shall often require that the Riemannian metric g on a Riemannian polyhedron X be simplexwise smooth. This means that,
relative to some triangulation (K, 9) of X, the Riemannian metric gs on the interior s° of each maximal simplex s of K is the restriction to s° of a smooth Riemannian metric on a neighbourhood of s in the linear span of s. This property is preserved under subdivision. Returning to the case of a general (bounded measurable) Riemannian
metric g on X, we shall often consider, in addition to g, the Euclidean Riemannian metric g° on a Lip polyhedron X with a specified triangulation
T = (K, B). For each s E S' (X, T), g; is defined in terms of a Euclidean frame on 0-1(s°) as above by the unit matrix (bed). Thus g` is by no means covariantly defined and should be regarded as a mere reference metric on (X, T). The ellipticity constant A of (X, T, g) from (4.2) equals the bi-Lip constant of the identity map (X, T, g) - (X, T, ge) in terms of the associated intrinsic distance dx = dX and its analogue dX, both of which are defined in the next section. Every connected open subset U of a Riemannian polyhedron X is a Riemannian polyhedron with the induced Riemannian structure.
Part I. Domains, targets, examples
50
Example 4.3. The cyclic group Z,,, of the m-th roots of 1 acts on C isometrically by multiplication; it has fixed point 0 E C. Form the orbit map
ltm:C,C/Zm=X. Anticipating Examples 8.9 (ii) and 8.11 (iv), X is a simple example of a complex orbifold; and 7r,,, a branched covering map with branching locus
0 E X. With the quotient metric, X is isometric to a standard cone of revolution in 1R3 with the geodesic distance on the cone (which is a geodesic space).
Choose a locally finite Euclidean triangulation of C, invariant under Z,,,. It induces a triangulation of X, which thus becomes an admissible
2-dimensional Lip polyhedron without boundary (cf. the argument in Example 4.2). Topologically, X is a 2-manifold, cf. Example 4.1. The projection map 7r,,, induces a Riemannian metric g on X (as in the above definition of a Riemannian polyhedron) from the Euclidean Riemannian metric ge on C. In local (polar) coordinate representation, g has the form
g = dr2 + m-2r2d82.
In terms of Cartesian coordinates (x, y) = (r cos 6, r sin 9) we have rdr = xdx + ydy, r2dO = xdy - ydx, and g is therefore indeed a Riemannian metric with measurable components and elliptic bounds (with ellipticity constant m). Thus (X, g) becomes a Riemannian Lipschitz manifold with one conical singularity (cf. Examples 3.6, 6.1 and 8.8). More generally, replacing 1/m in the above representation by any real number A > 0, we still obtain a metric g on C with a single conical singularity.
It can be shown that the curvature is >, 0 if A < 1 (as above), and 5 0 if A > 1 (as in the next example).
Example 4.4. For an integer m > 1 the Riemann surface X of zl/m is conceived in classical terms as consisting of m copies of C ("sheets") glued together so that there is a homeomorphism fm : C - X for which fpm 0 fm)(re:s)
=rein',
r > 0, 9 E IR/(27rZ),
C denotes the natural projection. Denote by 0 E X the where p,,, : X single point of pm1(0). In view of Example 4.1, X becomes a 2-dimensional normal Lip circuit without boundary when we take as an (auxiliary) distance function on X the image under fm of the Euclidean distance e on C. Then
X \ {0} is a smooth manifold, and pm : X \ {0} -+ C \ {0} is a local diffeomorphism.
4. Riemannian polyhedra
51
The whole of X becomes an admissible Riemannian polyhedron (and hence a Riemannian normal circuit, cf. below) when equipped (off 0) with the Riemannian metric purge, where ge denotes the Euclidean Riemannian metric on C \ {0}. Then X is also a Riemannian Lipschitz manifold, with an atlas consisting of the single chart fn,l : X C. In this representation the Riemannian metric p,',,ge on X \ {0} corresponds to the Riemannian metric g on C \ {0} given in polar coordinates (r, 8) by g = Id(reime)I2 = dr2 + m2r2d92,
and g has indeed bounded measurable components in terms of Cartesian coordinates on C, cf. the end of the preceding example. Thus (X, p,'ge) is (isometric to) a Riemannian Lipschitz manifold with one conical singularity.
Globally, we may view pm : X -+ C as a branched covering map with branching locus 0 = purl (0) on X. Thus pm corresponds via the isometry fm to the map Trm C -+ C/Zm from Example 4.3. Example 4.5. Every polyhedron X is clearly path connected, locally path connected and locally simply connected (being locally contractible). It follows that a Riemannian polyhedron X has a theory of covering spaces; in particular, a universal covering space p : X -+ X with k a Riemannian polyhedron, and p a simplicial map which is a local isometry. (See [Sp 1966, pp. 144f.].)
A Riemannian circuit is of course a circuit which is also a Riemannian polyhedron. The polyhedra in Examples 4.2 - 4.4 above are all (Riemannian)
normal circuits; and the Riemannian normal circuits are among the most important admissible Riemannian polyhedra, cf. Chapter 8. Example 4.6. Attaching a flat n-ball B" along an equatorial (n - 1)-sphere of the standard Euclidean n-sphere S" produces an admissible Riemannian n-polyhedron X = S" U B" (n > 1), without boundary. Clearly X is not a circuit.
The intrinsic distance dX Relative to a given (Lip) triangulation T = (K, 9) of an n-dimensional Riemannian polyhedron (X, g) (not required to be admissible), we have on X the distance function e induced by the Euclidean distance on a Euclidean
Part I. Domains, targets, examples
52
space V in which IKI is affinely Lip embedded, cf. Lemma 4.1. This distance e on X is not intrinsic and will play an auxiliary role in defining an equivalent intrinsic distance dX = dX as follows, by a slight adaptation of the procedure used by De Cecco and Palmieri [CP 1988-,1990] for the case of a Riemannian Lipschitz manifold, cf. Example 3.6. Let Z denote the collection of all null sets Z C X (cf. the section Lip polyhedron). For a given triangulation T = (K, 0) consider in particular the set ZT E Z obtained from X by removing from each maximal simplex s in X those points of s° which are Lebesgue points for ge (that is, for every component g ). For x, y E X and any Z E Z such that Z D ZT we denote by Lipz (x, y; X) the family of all Lip continuous paths y : [a, b] - (X, e) with -y(a) = x, y(b) = y which are transversal to Z in the sense that y-1(Z) is a null subset of [a, b]. The length LT(y) of such a path -y is well defined by
LT(y) _
(g
o 9-1 o y`)1'`'yj (< oo),
1'-'(s0)
where ('y1,... , yn) = 0-1 o y in terms of Euclidean coordinates on the open Euclidean simplex 0-1 (s°), and the dot means differentiation. Write pz(x, y) = inf{LT(y) : y E Lipz(x, y: X)}. (pz(x, y) depends on T as well.) Clearly, Z1 D Z2 (D ZT) implies pz1(x, y) pz2 (x, y). Finally set
dx(x, y) = sup{pz(x, y) : Z E Z, Z J ZT}. z
Note at this point that Lipz(x,y;X) # 0 (when Z E Z, Z D ZT), as shown by application of Fubini's theorem, cf. [CP 1988, p. 160). (Because X is connected, it suffices to consider points x, y of the same simplex s. The Euclidean segment [x, y] C a can then be slightly deformed to a path ry = [x, z] U [z, y] C s such that y-1(Z) is null.) Clearly pz satisfies the triangle inequality, and so therefore does dX. For the Euclidean Riemannian metric ge on X (induced by the Euclidean distance e on V) we have similar notions LT (y), p2 (x, y), dex (x, y). In
view of (4.2), each of the quotients LT(y)/LT(-y), oz(x,y)/pi(x,y) and dx(x,y)/dX(x,y) lies between A'1 and A. It is easily verified that e(x, y) < dX (x, y)
for x, y E X.
(4.3)
4. Riemannian polyhedra
53
Here the sign of equality holds if x and y are in the same maximal simplex s. This is shown by deforming [x, yJ into ry = [x, z] U [z, y] with 7-1(Z) null and LT(y) < e(x, y) + e, whereby oz(x, y) and hence dX (x, y) are 5 e(x, y) + e. Since X is connected it follows that dX (x, y) < oo and hence dx (x, y)
0 and hence dx (x, y) > 0 when x 0 y. Altogether, dx and dX are equivalent metrics on X, depending a priori on the triangulation T. They are locally equivalent to the given metric on X as a Lip polyhedron according to Lemma 4.2 below (to be used more fully in Lemma 4.3), applied to the star of any point of X. In view of Proposition 4.1 below, dx is called the intrinsic distance on (X, g).
Example 4.7. Let X be the Riemann surface of z1/' with the Euclidean Riemannian metric (see Example 4.4). We shall identify points x E X with
pn(x) E C, except that argx shall be counted mod 2m7r. The intrinsic distance dx is then given by dx(x, 0) = Ixi, while for x, y # 0,
dx(x,y)= (Ix - y) Sl Ix1 + Iy1
ifIargx - argyl, Ad (a). Fix x, y E X, and consider any other triangulation T' = (K', 9'). For any maximal simplex sQ E S*(X,T) and any maximal simplex sQ E S*(X,T') such that U sQ fl sl O 0 0 we have the Lip homeomorphism 91-1
0 9le-'(U) : 0-1(U)
0`1 (U)
between the open subsets B-1(U) and 9'-1(U) of the open Euclidean simplexes 0-1(sQ) and 9'-1(s'p°). This Lip homeomorphism is differentiable off some Lebesgue null set ZQp C 0-1(U), mapped by 9 onto a null set 9(ZQ0) C U. Writing
ZO=ZTUZT,U U9(ZQQ) (EZ), Q.A
we have for any Z E 2 with Z D Zo and any path y E Lipz0 (x, y; X) LT(-Y) = LT'('Y)
because this holds by covariance for the contributions from y-1(sQ fl s'') to LT(y) and to LT'(y). For any Z E 2 with Z D ZT, we have the quantities o'z (x, y) and d'x (x, y) corresponding to ez (x, y) and dx (x, y), respectively, but relative to T' in place of T. For any path y E Lipzuzo (x, y; X) we obtain ez (x, y) 5 e'zuzo (x, y) < LT' (y) = LT(y)By varying y E LipzUZO (x, y; X) (C LipzO (x, y; X)) we conclude that r
ez(x, y) 5 ezuzO (x, y) S dx (x, y), and finally d'x (x, y) . dx (x, y). Similarly, dx (x, y) 5 d'x (x, y). Ad (b). With the intrinsic distance dx we associate in the standard way the intrinsic length L(y) of a Lip path y : [a, b] - (X, dx) (or equivalently
Part I. Domains, targets, examples
56
y : [a, b] -+ (X, e), by Lemma 4.2, which is applicable because only finitely many maximal simplexes of X meet the compact image y([a, b])): ck
L(y) = sup A
dx(xi-1, xi), i-1
where rr ranges over all subdivisions
a=xo <xl
0 (a.e.) of class LipC(X). Similarly for other spaces, e.g., Wo'2(X)+.
W10C (X) denotes the linear space of all u E L °(X) such that uju E W1,2(U) for every relatively compact subdomain U of X. It suffices that X can be covered by such subdomains U; this follows from (a) in the following
remark, applied to functions i from a Lip partition of unity on the metric space (X, dx), as in the beginning of Chapter 4.
Remark 5.1. (a) For any E Lip°(X) and u E WW (X),,bu E W01'2(X). (b) For such 0, the map u ' -b Ou of W1'2(X) into Wo'2(X) is norm continuous. (c) W.11 2(X) C Wo'2(X); hence Wo'2(X) is the closure of W,1"2(X).
Ad (a). There are constants a, b such that
I'(x)I < a, IVO(x)I < b
a.e. for x E X .
For U E Lip"2(X) it follows that ,bu E Lip°(X) and V(Vu) _ -OVu+u0i/i (a.e.), hence after elementary manipulations IIV,ulI 0 to a smooth function ape : R -+ [-e,1 + E] with 0 5 cp'' 5 1 such that W. (t) = t for t E [0, 1], cf. [Fuk 1980, Problem II -
1.2.1].
5. The Sobolev space W',' (X)
67
Ad (b). That Jul E W1"2(X) follows from (a) applied to Ts = Isi. And IIIuIII = (lull follows from lVIu(x)II = Ivu(x)I for a.e. x, which holds by Remark 5.2 because Jul - u = 0 in {u 0}, hence DIul - Vu = 0 there, and similarly for {u < 0}. Ad (c). Consider a convergent sequence (uj) C W1"2(X) with limit u. In view of (b), II Iui I II = Ilu3II -+ (lull = IIIuIII, and this implies that indeed IIIuII - IulII -+ 0 according to a general result [De 1970, p. 140].
Ad (d). If e.g., (cps) C Lip' 2(X) converges to V > 0 in W1"2(X) then D
I cPi I _ I'pI = V in W1,2(X)I, by (c).
For u,v E W1"2(X) we write for brevity
E(u, v) = 2
r (Vu, Vv) = 2 E x
f(VuVv)
(5.2)
sES(")(X)
E(u) = E(u, u) = 2 f Ioui2 x
(5.3)
(the energy of u). Clearly, E(u) > 0. And E(u) = 0 (if and) only if u is constant a.e., first in each s E S'2(X), next in the star of any vertex p of X (by considering the traces of u on the (n - 1)-simplexes of X containing p, and using the local (n - 1)-chainability of X, see Chapter 4, Polyhedron), and finally in all of X (connected).
Lemma 5.3. Let u E W1'2(X). If X is non-compact, suppose that ess lim u(x) = 0, X-COX
where oox denotes the Alexandroff point. Then u E Wo'2(X). Proof. If X is compact, u E WCI"2(X) = Wo'2(X ), by Remark 5.1 (c). For e > 0 let TE : It -+ lR be odd, of compact support [-2e, 2e], and defined for
s>0by
T,(S) =
(s
for 0 < s , 2e} we have Tu = 0, hence V(TEu) = 0.
A Poincare inequality For any ball B(x, r) in the admissible Riemannian polyhedron (X, g) and any
function u integrable over B(x, r) with respect to the Riemannian volume measure µg on X we denote by u the meanvalue of u: U_
µ9(B(x,r))
udug, B(x,r)
and similarly under analogous circumstances, e.g., when B(x,r), µg, IVul are replaced by Be(x, r), p f IVeul, referring to the Euclidean Riemannian structure ge on X, see Chapter 4, Riemannian polyhedron. As shown by Sturm [St 19961 the "weak" Poincar6 inequality in the follow-
ing theorem implies the corresponding "strong" inequality whereby the in-
tegral on the left is taken over the same ball B(x, r) as that on the right, and the factor c(X, Xo)A6n is replaced by some c(X, A, Xo). A much weaker version of the Poincare inequality was given in [Wh 1986).
Theorem 5.1. For any relatively compact subset Xo of X there is a constant c = c(X, X0) such that, for any function u E W1'2(X) and any ball B(x. r) C Xo,
f (u - u)2dµg < cA6nr2 J B(x,=) B(x,r) Proof. Note that B(x, 2
(Vul2dµg.
) C Be(x, ' ) and Be(x, TA_
C B(x,r) C Xo.
5. The Sobolev space W1"2(X)
69
Writing I AI = fA dµ9 and IAIe = fA dµe for measurable sets A C X, we get
J B(r,-
(u - u)2dµ9 )
1
21B(x, 1"01
f
J 2A
\
A2n
=
2A
J
f
2IB(x, 22)I
cce1A5a-2r2
(u(x1) - u(x2))2dµe(xl)dµe(x2)
B`(x,') B°(x,c)
I BC(x, 2A)Ie A2n
I B(x,') I
u(xl) - u(x2))2d1a9(xl) du'9(x2)
B`(x,3
f
(u - u)2dtle
J
)
IQeU12dpe
B`(x.T)
S cA6nr2 f
Ioui2d,L9
B(z,r)
with c = c c1. The first equation reduces to the easy case -u = 0, replacing u by u - U. The second inequality is derived from Lemma 4.4 (assuming to begin with zA r. Because X is connected there also exists z E X with dX(x,z) = 2r, contradicting B(x,r) = B(x, Zr). The proof of (5.4) is by iteration, exploiting fully the polyhedral structure of X. The same iterative scheme will be used later (in the proofs of Proposition 6.2 and Theorem 7.4). We may suppose that r < 2 diame Xo; otherwise
Part II. Potential theory on polyhedra
70
Be (x,1 r) = Xo C Be (x, diame Xo), and (5.4) follows from its validity for r = 2 diame Xo. Suppose there is no such constant c1. There are then sequences of balls Bj := Be(xj,rj) C Xo and of functions uj E W1,2 (X) (or just as well uj E W1"2(Bjo)) such that
J B`(xj,Jr,)
Iuj - ujl2dµe > jrj2
I
IVeujl2dµe.
(5.5)
Be(xj,rj)
The same inequality then holds a fortiori when uj is replaced by any number. We may assume that xj x E Xo, rj - r < 2 diame Xo (< oo). Step 1. The case r > 0. We may then further assume that dX (xj, x) < a r and Irj - rI < It follows that sr.
Be(xj, 1rj) C Be(x, 4r) C Be(xj,rj), and so (5.5) contradicts the following "Euclidean" strong Poincarg inequality
(applied to the radius 4r):
f iu - iI2dl2e < Cr2 f IVeul2dit B° (x,r)
,
(5.6)
B° (x,r)
valid for fixed Be(x,r) C Xo with C independent of u E W1,2(Be(x,r)°). The proof of (5.6) is similar to the corresponding part of the proof of [Chen 1995, Proposition 2.21: We may assume that u = 0 and, writing
Be(x,r)° = B, that there is a sequence (uj) C W1,2 (B) such that uj = 0 and
(5.7) fe uj dµe > jr21B IVeujl2dµe. 1= These Dirichlet integrals thus converge to 0, and the Sobolev norms I Iuj I I
over B remain bounded. In terms of a triangulation T of X let a,, , sk denote the n-simplexes of T meeting B (C Xo). By Rellich's compactness theorem, applied to each s, fl B, we may assume that uj converges in L2(B) to some u E L2(B); and since V uj 0 in L2(B) we have u E W1"2(B)
and Veu = 0. It follows that u is constant (a.e.) in each si fl B, and these constants must be equal. Indeed, the Euclidean open ball B in the length space (X, dex) is connected' and therefore an admissible Riemannian 1 For any y E B there is (since deex (x, y) < r) a path ry in X joining x to y and of Euclidean length < r + da (y, X \ B), whence -1 cannot meet X \ B.
5. The Sobolev space W1'2(X)
71
subpolyhedron (B, T') with each n-Fimplex s' of (B, T') being contained in some n-simplex of T; in particular, u is constant (a.e.) in s'. Any two nsimplexes s', s" of (B, T') can therefore be joined by consecutive (n-1)- and n-simplexes of (B,T'), and the trace of u E W1"2(B) on any such (n - 1)simplex is the same from both sides. Consequently, u is constant (a.e.) on
B, and in fact u = 0 on B because u = limi ui = 0. Thus ui - 0 in L2(B), contradicting the equation in (5.7). This proves (5.6), and so rules out the possibility r > 0. Step 2. We denote by s the carrier of x (i.e., x is an inner point of the
simplex s). After embedding the star st(s°) = st(x) in a Euclidean space V with origin at x and norm I I, we denote by p: V -i Rs the orthogonal x projection on IRs. Choose p > 0 so that Be(x,4p) C= st(x). Because xi (= 0) and p(xi) p(x) = x, we may assume that xi E Be(x, p) (C st(x)) and p(xi) E s° fl Be (x, p). Denote by 1/ii : V -+ V the homothety of V with centre p(xi) and factor
ai = 1 max{Ixi - p(xi)I,ri} > 0.
(5.8)
P
Writing xi =1(i ' (xi) (E V), we have x'i
- p(xi) = a
1(xi
-
p(Xj)),
(5.9)
and by (5.8)
Ixi - p(xi)I = aj 1lxi - p(xi)I < p
(5.10)
Since I p(xi I < p it follows from (5.10) that I x'i I < 2p, which in turn implies
that x'i lies in K' :_ {z E V : IzI < 2p}, a compact subset of st(x) (by the choice of p), and so is L:= {z E V : IzI < 3p}. Since the centre p(xi) of tpi is in s and ai < 1 we have iPi (s) C s, Oi(st(x)) C st(x). Writing r,' := a, 1ri < p, by (5.8), and recalling the notation Bi = Be(xi,ri), we see that
BB:=0.7-1(B1)={zEV:Iz-x I dims. Iterating the above procedure at most n - dims times therefore leads to an absurdity. This proves (5.4), and thereby the stated Poincare inequality. 0
Weakly harmonic and weakly sub/superharmonic functions Let (X, g) be an admissible Riemannian polyhedron of dimension n.
Definition 5.1. A function u E WWor (X) is said to be weakly harmonic if VV) dµ9 = 0
for every V E Lipc(X).
Definition 5.2. A function u E W2(X) is said to be weakly subharmonic, resp. weakly superharmonic, if
1,
(Vu, Vv) dµ9 < 0, resp. > 0,
for every V E Lips (X).
In these three definitions, Lipc(X) and Lip' (X) can be replaced equivalently by W,1"2(X), resp. W,'"2(X)+, the latter by Proposition 5.1 (d). If u E WI.2(X) then Lipc(X) and LipC (X) can similarly be replaced by Wo.2(X),
reap. Wo'2(X)+. Every constant function is weakly harmonic. If X is compact then every weakly harmonic function u E W1"2(X) is constant (a.e.). Indeed, U E Wo'2(X), hence E(u,u) = 0, by Definition 5.1 and the above observation, and so u is constant, as noted in text following (5.3).
5. The Sobolev space Wl'2(X)
73
The concepts of weakly harmonic and weakly sub/superharmonic functions are local concepts. For example, the restriction u1u of a weakly harmonic function u on X to an open set U is clearly weakly harmonic (strictly speaking: on each component) in U. And if a function u is weakly harmonic in each member of a family of open sets then so it is in their union U; in particular, u is of class Wo1 (U). (This is obtained using a Lip partition of unity on U, subordinate to a covering as above, cf. the first section of Chapter 4.) A classical variational argument (cf. e.g., [KS 1993, p. 6221) leads to the following characterization of weakly harmonic functions of class W"2(X) as local E-minimizers:
Proposition 5.2. A function u E W1'2(X) is weakly harmonic if and only if u minimizes the energy E(v) among all functions v E W1.2(X) such that v - u E Wo'2(X).
Remark 5.3. Under suitable smoothness assumptions, weak harmonicity and subharmonicity can be characterized in more classical terms. Consider a domain U C X which meets exactly one (n - 1)-simplex t. Let s1i , sk denote the n-simplexes of X containing t, and suppose that the Riemannian metric g is continuous and has smooth restrictions to each s fl U up to the boundary t° fl U (i.e., gu is continuous and simplewise smooth, cf. Chapter 4, Riemannian polyhedron). Consider a continuous function u on U of class C2 in each a,* fl U and of class C1 in each (sj* U t°) fl U. Then u is weakly harmonic (resp. subharmonic) in U if and only if (i) u is weakly harmonic , k, and (ii) (resp. subharmonic) in each s fl U, j = 1, k
E Dju(x) = 0 (resp. > 0) j=1
at almost every point x of t fl U. Here Dju(x) denotes the inner normal derivative of ul,,nu at x. (Proof by Green's formula.) Cf. Example 8.1 for the 1-dimensional case. The following maximum principles, including Theorem 5.3 below, are due to R: M. Herv6 [Her 1964, §11 for the particular case where X is a bounded domain in R" carrying a Riemannian metric with measurable components and elliptic bounds.
74
Part II. Potential theory on polyhedra
Proposition 5.3. Let u E W ','(X) be weakly subharmonic on X. In either of these two cases it follows that u < 0 (a.e. in X): (i) u E WOI,2(X) and 1 V Wo'2(X), or (ii) X is non-compact and ess lim sup u(x) < 0. Proof. If u is in W01,2 (X), as in case (i), then in view of Proposition 5.1 (c) so is Jul, Lip,(X) being dense in W01'2(X). In case (ii), u+ is in W1"2(X) by Proposition 5.1 (a), and in Wo'2(X) by Lemma 5.3. Thus, in either case, u+ E W01'2(X) . Taking cp = u+ in Definition 5.2 we obtain E(u, u+) < 0,
and so E(u+, u+) = 0 because E(u_, u+) = 0 by (5.2) and Remark 5.2. Consequently u+ is constant (see text following (5.3)), and indeed u+ = 0 in either case (i) or (ii). It follows from this proposition that a weakly harmonic function u of class
W"2(X) is - 0 if ess limy-,x u(x) = 0 (assuming that X is noncompact). Remark 5.4. Proposition 5.3 extends, as in [Her 1964], to weakly subharmonic functions u of class Woo, (X). In case (ii) there is, for any e > 0, a
compact set K C X such that u s e a.e. in X \ K. With U D K open connected relatively compact in X and containing a given point xo E X, it follows that u - e < 0 (a.e.) in U \ K, hence in U because u - e is subharmonic and of class W1"2(U); and so, in particular, u(xo) < e. As to case (i), see Theorem 5.3 below, applied to v = u - e and f = 0.
Theorem 5.2. (The variational Dirichlet problem.) Suppose the following Poincare inequality holds: Ju2
cJ IVul2 x
for all u E W01'2(X)
(5.12)
(or equivalently for all u E Lip,(X)), with c depending only on the admissible Riemannian polyhedron X. Then 10 WOI,2(X) and so X is non-compact. For any f E W1"2(X) the class of competing maps
Wf'2(X) = {v E W1'2(X) : v - f E WD'2(X)}
contains a unique weakly harmonic function u. That function is the unique solution u of the equation E(u) = E0, where
Eo :=inf{E(v) : v E W1'2(X), v - f E Lipc(X)} =min{E(v) : v E Wf2'(X)}.
(5.13)
5. The Sobolev space W1.2(X)
75
If f >, 0 off some compact subset of X, then u > 0. If instead f is weakly subharmonic, then u > f.
Proof. First note that 1 V W01,2 (X), by (5.12). In particular, X is not compact (for then 1 E W1'2(X) = Wo'2(X)). Consider a sequence (ui) in W1'2(X) such that
ui - f E Lipc(X),
E(ui)
with Eo defined by the former equation (5.13). Then for Eo, and by the parallelogram identity
E(ui - ui) 5 2E(ui) + 2E(ui) - 4Eo - 0
Eo.
(ui+ui) also competes 2
as i, j
oo,
(Vui) is a Cauchy sequence in L2(X) and therefore convergent. Since ui - ui = (ui - f) - (ui - f) E Lipc(X),
the Poincare inequality (5.12) shows that (ui) is itself a Cauchy sequence in L2(X), hence also in W1"2(X), and so has a limit u E W1'2(X). Moreover,
u - f = limi(ui - f) E W01'2(X), and E(u) = lim E(ui) = Eo, so that u is minimizing for the variational problem (5.13), and so u is weakly harmonic, by Proposition 5.2. For any two weakly harmonic functions u, u' of class W j'2 (X) we have
E(u - u') = 0 by Definition 5.1 because u - u' is weakly harmonic and of class Wo'2(X). Consequently, u - u' is constant, as mentioned earlier (after (5.3)), and u - u' = 0 because 1 0 W01'2(X ). This establishes the uniqueness assertions. Next, suppose that supp f- is compact, and again let u denote the minimizing function. Then Jul E W1'2(X) and II Jul II = (lull, by Proposition 5.1 (a), (b) and hence E(Iul) = E(u) (= Eo). It remains to show that f -Iul E Wo'2(X ), from which it follows that u = Jul (> 0) by the latter uniqueness property above. Write f - u = V (E W01'2(X)), and choose functions (pi) C Lipc(X) so
that liVi - pll - 0. Then ui := f -,Pi -4 f - co = u
in W1,2(X),
and consequently luil -' Jul in W1,2 (X) by Proposition 5.1 (c). It follows
that
f -lul=lim(f -Iuil)E W01,2(X)
Part II. Potential theory on polyhedra
76
because f - Iu.i I = f - If - w,i I has compact support (C supp Wj U supp f-). Finally, if instead f is weakly subharmonic, then so is f - u E Wo'2(X), 0 and hence f - u < 0 by case (i) of Proposition 5.3.
Remark 5.5. Note that this type of Poincare inequality (5.12) is different from that of Theorem 5.1. After a suitable subdivision of a given triangulation of X, the inequality (5.12) holds for the (open) star st(a) of any prescribed point a E X. In fact, with S := st(a), consider the Lip homeomorphism 0 : S -+ P from Lemma 4.3. Every n-cell Q of P has an (n - 1)-face F not containing 0, that is, F C 8P = P \ P°, cf. Remark 4.2. This establishes the Poincare inequality (5.12) for P°, and hence for st(a) = 8-1(P°), because it is known that (5.12) holds with X replaced by the above n-cube Q (e.g., endowed with the Euclidean metric) for any u E W1'2(Q) vanishing near the above (n - 1)-face F; cf. the proof of the Poincare inequality for a strip in R'' in (DL 1954, p. 318].
By Theorem 5.2, writing L f = u, the operator L : W"2(X) -+ W"2(X) is linear and positive, hence increasing, with ker L = W01'2(X ); and we have
f < Lf (a.e.) if f E W1'2(X) is weakly subharmonic. Furthermore, L
is continuous in the Sobolev norm II . II because f E W,"2(X) and hence E(u) < E(f). Indeed, (5.12) implies IILf
IIZ
= 2E(u) + Jx u2
(1 + c)2E(u) , 0 a.e. in Sr (the interior
of9-1(rP)) we have esssupu < coessinf u.
S,.
S,.1a
) only depends on (X, g) and a, and more precisely on n = dim X, N = the number of n-simplexes H e r e c o (as w e l l as subsequent constants c1, c2,
containing a, and A = the bi-Lip constant of 9 : S -' P with the given Riemannian structure on S (C X) and the Euclidean Riemannian structure on P. The dependence of co, say, on A, and similarly on N, should be understood in analogy with the precise formulation given after Theorem 6.1. The uniformity not just in u, but also in r, and the confinement in (6.3) of the inequality u > 0 to Sr will be crucial in the proof of Theorem 6.2 below. Since S,./5 is a neighbourhood of a, Theorem 6.1 for locally essentially bounded u follows from the above proposition for fixed r (e.g., r = 1) by a standard compactness argument (not quite as in [Mos 1961, p. 582], which seems to be erroneous), cf. e.g., [Hel 1969, 2.16]. The comment relating to Theorem 6.1 about the way c depends on (X, g) likewise follows from Proposition 6.1 in
6. Harnack inequality and Holder continuity
83
view of (6.1) above. The choice of the denominator 5 (in 5) in the proposition will be explained later. ([Mos 19611 uses 4 in place of 5, but this reflects an insignificant error, viz. the interchange of 2 and 3 in (4.5), p. 586 of the quoted paper.) Actually, an easy modification of the proof of Proposition 6.1 will show that Sr15 could be replaced by SQ,. for an arbitrary constant e with 0 < e < 1; with this change, co of course also depends on p. We begin by reducing Proposition 6.1 to the case r = 1. Using 0: S P we transform the problem to that of reducing the inequality ess sup v 0 is locally essentially bounded in X.
Completion of the proof of Theorem 6.1 We complete the proof by showing that every weakly subharmonic function
u on X is locally essentially bounded from above. We may suppose that u E W1'2(X), and also that u > 0 (otherwise replace u by Jul). It will follow that every weakly harmonic function u on X is locally essentially bounded (since Jul is weakly subharmonic).
Let 2 < p < oo be given. Inspired by [Mos 1960] we approximate as follows the convex even function f (t) = Jtip/2, t E R, from below by an increasing family of convex even functions fn (t) > 0, m > 0, having bounded continuous derivative f ,,,(t): tp/2
F n (t) = l
` MP /2 + 2mP/2-1 (t - m)
for 0
t
for t
m,
m
i.e., by replacing for t > m the graph of f by its tangent at m. Thus each fn, is a bounded Lip function. Given a weakly subharmonic function u > 0 of class W1"2(X) we write vm = ,fm o u
E W 1,2(X ),
Vm = [(fmfm) 0 u] 172
E W,1.2(X ),
for given ri E Lipc(X). Each of the functions v.. and cp,n is indeed of class W 1,2(X )+ because u is so and each of the functions F = fm and F = f,n f,' is uniformly Lipschitz with F(0) = 0, hence F(t) 0, c = max F'. For any function u E Lipl'2(X)+ it follows that F o u E Lip1"2(X)+ and IlFouli 5 c1lull; and in this statement we may clearly replace Lip1"2(X)+ by W1"2(X)+, in view of Proposition 5.1 (d). Write v = up/2, and for 0 < h < 1, cf. the paragraph containing (6.3), v2) 1/p
'(p, h) = (.f
= (f
uP)1/p,
h
(6.16)
h
h) = (
r v2),n1/p
\f/ $h
.
(6.17)
6. Harnack inequality and Holder continuity
89
For m / oo we have fm / f (pointwise), hence v,,, / v, and so
-tm(p, h) /
as m / oo.
h)
(6.18)
Proceeding as inr the (Vcproof of Lemma /a 6.1 we obtain, since P,,, > 0,
0>
om, Vu)
JX
ll.fmfm + (fm)2) o
X
u] ,n2IVtz 2
+ f ((fmfm) o u] 21(V . Vu). Inserting fm o u = v,,, and (f, o u)Vu = Vvm in the second integral gives
f Here
(f )2
+ 1) o uJ n210vmI2
-2 f 77vm(on, Vvm).
fm(t)fm(t) + 1 = f 2 - 2/p > 1 (f,', (t))2
1
(6.19)
t < m, for t > M. for
Applying Schwarz' inequality to the right hand side of (6.19) therefore leads to the following conclusion, similar to that of Lemma 6.1:
f
X
77 21VVm12 < 4 f
vmlV7/12,
, E Lip, (X).
(6.20)
X
For the given weakly subharmonic function u > 0 of class W1'2(X) we now proceed exactly as in the proof of (6.7) in Lemma 6.2, replacing Lemma 6.1 by (6.20), and using the notation in (6.17). As in (6.9) this yields
4'(np,h') < cioP(h'
-
for 0 < h' < h < 2h' and h < 1. As in (Mos 1961, p. 5851 it follows by iteration that *'m(Xv+1P, h"+1) < Cl1 Wm(p, 5)
, whereby h, = (1 + 2-"), hence ho = For m - oo s this leads to the same inequality with 4,, replaced by 4, c£ (6.17). Next, for v oo, it follows as in [Mos 1961), in view of (6.16), (6.17) and the former inequality (6.5), that f o r v = 0,1, 2,
5.
esssupu = lim 4b(IC"t1p,.1) < liminf A2n/(" "-ao " 00 S1,3
-
C114'(p,
5)
)4(K"+1p,h"+1) (6.21)
Part II. Potential theory on polyhedra
90
Because u E W1"2(X) C L2(X), 4(p, 5) is finite e.g., for p = 2, and we have thus proved that every weakly subhannonic function u : X -+ R is locally essentially bounded from above, viz. essentially bounded from above in a neighbourhood (denoted S115 = 9-1(S P)) of any point a E X. 0 Anticipating Definition 7.1 of harmonic functions as continuous weakly harmonic functions (with reference to Theorem 6.2 below) we derive in the following proposition a Harnack-style inequality for harmonic functions in
the complement of a variable point y. The result will be used in the study of the Green function on X in Chapter 7. As in Chapter 4 we denote by B° (y, r) = BX (y, r) the closed ball centred at y and of radius r for the distance function dX associated with the Euclidean Riemannian structure g° on X (induced by the metric on a Euclidean space in which X is afffinely embedded).
Proposition 6.2. Let (X, g) be an admissible Riemannian polyhedron of dimension n > -1 and ellipticity constant < A. For any compact proper subset K of X, any number B > 0 such that L :_ {x E X : dX (x, K) < 2g} is a proper subset of X, any y E K and any harmonic function u > 0 in X \ {y}, we have
max u
> 9 in, ul.
(6.22)
We may assume that yl -' y E K and rl - r E [0, e]. Step 1. The case r > 0. We may then further assume that d1X(y5, y) < r/4
and jrj - rl < r/4. It follows that 8B, C A := L \ B`(y, 2r)°, and u, is harmonic with respect to gj and > 0 in the connected open set Y obtained from X by removing y and all the y,,. (Since n > 1, a countable set does not divide an n-simplex, and therefore not X.) This leads to a contradiction with Theorem 6.1, applied with X replaced by the open subpolyhedron Y, and K by A. We are thus left with the case rj -+ 0.
6. Harnack inequality and Holder continuity
91
Step 2. Let s denote the carrier of y. After embedding st(s°) = st(y) in a Euclidean space V with origin at y and norm I I we consider the homothety ,Oj : V -+ V from Step 2 of the proof of Theorem 5.1 with x, xj, x, replaced
by y, yj, yj. We also consider the balls BJ := Be(y,, rj) C st(y) for which 1(ii(Bj) = Bi (= Be(yj,rj)). Writing uj o7ij = uj, we have from (6.22) its analogue
max u' > j min ul .
as;
aa,'
(6.23)
with respect to the Riemannian Here u' is harmonic and >, 0 in st(y) \ structure a 2(ali 1)*g with ellipticity constant 1 cannot be dropped, not even for fixed y, r, u (take e.g., X = R, y = 0 and u(x) = max{x, 0}).
Holder continuity Theorem 6.2. Every weakly harmonic function on X is Holder continuous (after correction on a null set).
Remark 6.4. In the particular case where X is a domain in lR' carrying a Riemannian metric with bounded measurable components, this is due to Morrey [Mor 1938] for n = 2 and to De Giorgi [Gi 1957] for arbitrary dimension n. The proof of Holder continuity of harmonic maps of an admissible Riemannian polyhedron X into a compact subset of a nonpositively curved Riemannian polyhedron Y, given in [Chen 1995], is faulty, even in the present
case Y = R, in particular because the conditions needed to apply Morrey's Dirichlet growth lemma at the end of the proof are not fully established. In our proof of Theorem 6.2 we continue to follow Moser [Mos 1961], who combined the Harnack inequality with an iterative procedure of De Giorgi. In this way one also obtains local essential equicontinuity of any locally uniformly bounded family of weakly harmonic functions, see Theorem 6.3 below.
Proof of Theorem 6.2. It suffices to consider a weakly harmonic function u of class W1"2(X). For a given point a E X and a suitable triangulation of
Part II. Potential theory on polyhedra
92
X having a as a vertex we continue to use the notation Sh = 0- 1(hP) for 0 < h < 1, cf. (4.2) and Remark 4.2. Because u is locally essentially bounded (see the preceding section), Proposition 6.1 asserts that u
0 a.e. in Sh implies ess sup u < co ess inf u,
(6.24)
Sh/5
Sh/S
where co is independent of u and of h < 1; in fact, co depends only on n, N, A. Write M(h) = esssup u, Sh
µ(h) = essinf u, Sh
so that w(h) = M(h) -µ(h) is the essential oscillation of u over Sh. Then M(h) - u and u -,u(h) are of class W12(S1) and weakly harmonic there, and > 0 a.e. in Sh, whence by (6.24)
M(h) - µ(h/5) < co (M(h) - M(h/5)), M(h/5) - µ(h) < co (µ(h/5) - µ(h)) Adding these inequalities gives, after simplification, w(h15) < qw(h),
q = Goo
+
By iteration we obtain
w(hl5') < q'w(h),
j = 1, 2, ... ,
(6.25)
which leads to
w(h) < (5h/h')°w(h')
for q=5-0, 0 < 5h < h' < 1.
(6.26)
In fact, choose j so that h'/5'+' < h < h'/5'. Then, by (6.25), w(h) < w(h'/5j) < q'w(h') < (5h/h')°w(h'). Taking h' = 1 leads to w(h) < (5h)°w(1)
for 0 < h
0 so that B(a, o) C st(a) (= S°) and 5Ao < 1. For r < o we obtain x E B(a, r)
dx (a, x) < r
dp(0, O(x)) < Ar
x E ArP,
and so from (6.27)
for r < o.
essoscu < w(Ar) < (5Ar)°w(1) B(a,r)
Recall that a, from (6.26), only depends on n, N. A (like co and q). Finally we show that u is equal a.e. to a continuous function u first that ess lim sup u(x) = ess lira inf u(x)
x-a for every a E X. In fact, for r < o,
x-.a
(6.28)
Note (6.29)
esssupu - essinf it = essosc u < (5Ar)°w(1) B(a,r)
B(a,r)
B(a.r)
by (6.28), and each member of (6.29) is squeezed between the above esssup and ess inf. We may therefore define u* (a) = ess lim u(x), x-+a
a E X.
Select u within its equivalence class in L2(X). For any compact K C X there exists, by Lusin's theorem, a closed set F C K such that uIF is continuous
and the volume measure 1K \ Fl < E. For any point a E F such that IFnB(a, r) I > 0 for every r > 0 we have u* (a) = esslim u(x) = u(a).
x-.a, xEF
The remaining points of F form a null set, and we have thus found that if = u a.e. in F, hence a.e. in K, and therefore in X. For any ball B(a, r),
aEK,andanyxEB(a,r)°, ess inf u < u' (x) < ess sup u. B(a,r)
B(a,r)
This implies osc u' < essoscu < (5Ar)°w(1) B(ax)
B(a.r)o
for r < o,
94
Part II. Potential theory on polyhedra
by (6.28), and so u equals a.e. the Holder continuous function u'.
0
Example 6.1. (Cf. [Chen 1995].) The function z 1--+ z'1' on its Rlemann surface X (containing 0), with the Euclidean Riemannian metric ge (see Examples 4.4 and 4.7) is (complex) harmonic on all of X because zlfm is bounded near 0 and holomorphic on X \ {0}, and because {0} is a polar set by Corollary 7.2 below, hence a removable singularity for locally bounded harmonic functions (cf. Chapter 2 with reference to [Br 1959)). (Explicitly, the continuous function s defined on X by s(z) = - log )z[ for z E X \ {0} and s(0) = oo is superharmonic on X, being harmonic off 0, and equal to
ooat0.) In view of Theorem 6.2 above we shall establish (with reference to Definition 7.1) the following result on local uniform Holder equicontinuity of suitable families of harmonic (i.e., continuous and weakly harmonic) functions on an admissible Riemannian polyhedron X, cf. [Gi 19571 for the case where X is a domain in R" carrying a Riemannain metric with bounded measurable components.
Theorem 6.3. Every locally uniformly bounded family. of harmonic functions on X is locally uniformly Holder equicontinuous, that is, for every compact set K C X there are constants A, a, b, depending on (X, g) and K only, such that
osc u 0 by 1/51 = (co - 1)/(co + 1), which depends on n, N, A only. Following De Giorgi
and Moser as in the proof of Theorem 6.2, which was based on (6.24), we similarly obtain from (6.33):
w(t, h) < (5h/h')aw(t, h'),
0 < 5h < h' < b(t).
(6.34)
From this we shall derive the crucial inequality:
w(t, h) < 5'Mh",
(6.35)
valid for any u E F,
E !P and 0 < h < min{m, b(t) }, where we define a = a(n, N, A) and 7 _ 77(n, N,.\) by a = ,3/2",
q = 1/5. 2'x
(6.36)
6. Harnack inequality and Holder continuity
97
In the case l: = 0 this is clear from (6.34) with h' = b(l;) (= 1), for then w(l:, h') < M and hp < h°. Suppose therefore that r := m(1:) > 0, and define recursively (with to := £)
j=
hj = 6(Cj-1),
j=1,2, .,r.
By (6.31), hj_1 < hj (with ho := h), and m(lj) = r - j > 0 for j = , r - 1, while m(lr) = 0, i.e., Sr = 0. By (6.32),
1,
P(fj-1,hj-1)CP(Cj,hj),
j=1,...,r.
Consider first the case where, for some j = 1,
, r, we have
hj-1 < h and hj-1 0/2" = a, by (6.36).
In the remaining case, where (6.38) fails for every j = 1, , r, we see by induction that the latter inequality (6.38) does hold (trivially for j = 1),
and hence hj_1 > h, that is, h > h2 > h2 > ... > h2r.
(6.39)
Now applying (6.34) with £, h, h' replaced by 1;,. = 0, hr, b(£,-) = 1 we therefore obtain (6.35) from (6.37) and (6.39): 5#Mhp/2r
w(t, h) < w (Sr, hr) < (5h,.)'w(0,1)
0 of p being of class La 2(X) and weakly harmonic (because u and -u are majorized by the functions p and 0 of that class). Consequently, E(u, gyp) = 0 for every V E Lipc(X), or just O E L0' 2(X), in particular for W = u, and so u = 0. In the rest of the present chapter we require that (X, g) is an admissible Riemannian polyhedron such that the inequality (7.3) in Proposition 7.8 holds. In particular, X is noncompact. See, however, Remark 7.8.
Remark 7.5. See [HeKo 1998] for a more general framework for potential theory. There (X, dx,,u) is a metric measure space, regular with respect to Hausdorff dimension, and satisfying a Poincare inequality.
The Green kernel Using Theorem 7.2 we shall first deduce from results of Hervr [Her 1962], in
combination with [FP 1978], that X has a symmetric Green function, and so (X, 71) is a selfadjoint harmonic space in the sense of Maeda [Mae 1980]. Theorem 7.3. (X, g) has a unique symmetric, lower semicontinuous Green kernel G : X X X -+]0,00],
finite and continuous off the diagonal in X x X, such that (1) For every positive measure p on X the function Gµ : X -+ [0, oo] defined by
Gp(x) = fX G (x, y) dp(y),
x E X,
is a potential on X (in the harmonic space sense, Definition 2.3), unless G,u =- oo.
(2) Every potential p on X has the form p = Gp for a unique positive measure p on X. The support of µ is the complement of the biggest open set in which p is harmonic.
7. Potential theory on Riemannian polyhedra
109
(3) The E-potentials on X are those functions on X which equal a.e. a (unique) potential p = Gµ such that the energy 1 f Gµ dp is finite, hence equal to E(p). For any p E L0' 2(X) and any measure µ > 0 on X of finite energy we have p = Gµ a.e. iff E(p, cp) =
2
x
cpdu
for V E L0' 2(X) (or W E Lip°(X)).
Proof. According to [FP 1978, 37°], any two potentials on (X, it), harmonic off the same point of X, are proportional. Alternatively, this follows from Proposition 6.2 as in [Her 1965]. According to [Her 1962], this proportion-
ality property implies that (X, 7.1) has a Green kernel G with the properties (1), (2) above (see also (CC 1972, Theorem 11.5.2]). According to [FP 1978, 37°], G also has the property (3) and is thereby uniquely determined. Finally, the symmetry of G derives from that of the Dirichlet form E(u, v) = 1 fx (Vu, Vv) as follows. For any cp, Eli E C, (X), viewed as densities of positive measures 1A, v of finite energy on X,
ff
G(x, y)p(x)I(y) dx dy =
X X
Jx
Gudv = 2E(Gµ, Gv)
is symmetric in cp, t1i, and so we have G(x, y) = G(y, x) by the continuity of 0 G(x, y) for x 0 y.
We next deduce the following three further properties, well known in classical potential theory: (4) A set e C X has zero capacity if e is polar, that is, there should exist a superharmonic function u on X such that u =_ oc on e. In fact, if cap e = 0 then there even exists a potential u of finite energy on X such that u oc on e, cf. e.g., [FP 1978, 21°]. Conversely, if this relation holds for some superharmonic function u on X, then likewise for some potential u of finite energy, by [FP 1978, 32°], and hence cap e = 0 by
a classical argument: For any number a > 0, the set {u > a} is open and contains e, and so, as in [De 1970, p. 163], [Fuk 1980, §3],
cape ( cap{u/a > 1} 0 such that
IG(xi,yi)-G(x2,yi)I 0 and Guwy is a potential on U, it follows from F. Riesz' theorem (cf. e.g., [CC 1972, p. 38]) that by 0 in U. In view of Lemma 7.1 (with a replaced by y E V) we have y) = y) + by first in U \ V, then in U \ {y} by varying V, and finally in all of U. (The last assertion is trivial if GU (y, y) = oo, and follows otherwise by the continuity principle, stated in (6) above, and applied to the potentials Ge on X and Guey on U.) This establishes the decomposition G = Gu + Hu as stated, with Hu(-, y) = hy. Given a compact set K C U, choose a relatively compact open neighbourhood L of K in U, and write c = max{G(x, y) : x E 8L, y E K} (< oo). y) < For y E K we have y) < con 8L, hence y) < c in L, by the boundary maximum principle [Br 1958b] (see Chapter 2 above). In
7. Potential theory on Riemannian polyhedra
113
particular, HU < c on K x K, and so H is locally bounded in U x U. This leads to the Holder continuity of HU by application of Theorem 6.3, much as in Proposition 7.4. Ad (b). The function p := Gey - GwU is superharmonic in U because GwU is harmonic there by Lemma 7.1, which also shows that p -' 0 at
8U. It follows that p > 0 in U, and that p is an f-potential on U, both by classical minimum principles for superharmonic functions, valid in any Brelot harmonic space, [Br 1959, p. 1.04]. Clearly, p is harmonic in U \ {y}, and hence p = Gey - GwU = aGuey
for some constant a = a(y) > 0, as noted in the beginning of the proof of Theorem 7.3. Thus Gey = GUEY +
y) = aGuey + GwU.
Here H(., y) and GwU are harmonic in all of U, and Guey and aGuEy are 7{-potentials on U. By the uniqueness in Riesz' theorem we conclude that 0 a = 1, y) = GwU. Hence GUEy = p 0 at 8U.
It may be added that HU(x, y) > 0 for all x, y E U except if Gu = G, that is, if CU is polar.
Example 7.1. Given an admissible n-polyhedron X and a point a E X (which we may assume is a vertex), consider the closed star S = st(a), embedded in a Euclidean space V with origin 0 corresponding to a. We may
assume that S° = st(a) (cf. Remark 4.2). Fix e > 0 so that B" (a, p) C S°, and write S2 := B'(a, p)° = {x E S° : 1x1 < Q},
is the norm on V, and ge indicates the Euclidean Riemannian structure on X, so that 1x1 = dX(a,x) when x E S. The corresponding where 1
Green function Gn(x, a) for Il with pole at a = 0 is given outside the pole, up to a constant factor (depending only on X, a; and > 0 when n # 1), by
G'
a) .
J x12 n - gZ-n,
n#2
l log(B/1x1),
n=2.
In fact, the function x p(x) > 0 given by the right hand side of (7.7) for x E S° \ {a} is clearly a classical harmonic function in the interior of each n-simplex of S and has normal derivative 0 along the interior of each
114
Part II. Potential theory on polyhedra
(n - 1)-simplex of S containing a. Since p E W,1, (SZ \ {a}), it follows from Remark 5.3 (or directly by application of Green's formula) that p is weakly harmonic in 52 \ {a}, and indeed harmonic there, being continuous. If n > 2, p clearly becomes superharmonic in H when we define p(a) = oo. Because
p(x) - 0 as IxI - p, p is a potential on 1, for the only harmonic minorant 0 of p is 0, by the boundary minimum principle (see Chapter 2). If n = 1, p(x) is already defined at a by the right hand side of (7.7), and -p is then a potential on 1, cf. Example 8.1. We proceed to estimate the Green kernel G on an admissible Riemannian
n-polyhedron (X, g). Inspired by the study by Littman, Stampacchia and Weinberger [LSW 1963, §7] for the particular case where X is a Euclidean domain endowed with a Riemannian metric with bounded measurable coefficients, we begin by comparing the Green kernel Cu on suitable domains U C X with the corresponding Green kernel Gr with respect to the Euclidean Riemannian metric ge (Lemma 7.2 below). The proof uses, among other
things, that the sets Ja = {x E U : Gu(x,y) > a}, for suitable y E U, have capacity cap JQ = 1/a. Lemma 7.2. For every point a E X there are constants p, c > 0 depending only on X as a polyhedron, on a, and on the ellipticity constant A of (X, g), such that
c-1G`u(x,y) 0 will be that Be(a, 5p) Cc S := st(a). Set U = Be(a, 5p)°. Fix for a while y E Be(a, p), and write for brevity
u(x) = Gu(x, y),
x E U.
We begin by estimating u in terms of the capacity of the open sets
Ja={xEU:u(x)>a},
0 cap JJ = 1//3 = 1/ max u, OB` (y,r)
and, invoking (7.9),
min u
1
capBe(y,r)°
aB"(y,r)
max u,
r < 2p.
8Be(y,r)
(7.10)
By Proposition 6.2 (with p replaced by the present 2p, and K by Be(a, g)),
max u < cl min u,
aB"(y,r)
r < 2e,
OB"(y,r)
where cl depends only on X, A, and a. Inserting (7.9) and (7.10) here we obtain for any x, y E Be(a, e), writing dX(x, y) = r (< 2g) so that x E OBI (y, r),
u(x) - Gu(x, y) < i Considering instead of GU the Green kernel Ge and the capacity cape on cap B e (y, r) °
c1
(7.11)
cap Be (y, r)°
U with respect to the Euclidean Riemannian structure ge, we obtain as in (7.11) for the same x, y, r 1
1
cl cape Be (y, r)°
e < GU (x, y) < cl
1
.
cape Be (y, r)°
(7.12)
For any v E L01'2(U) we have A-n-2Ee(v) < E(v) < A"+2Ee(v), where Ee is the energy on (U, ge) and n = dim X. It follows by the definition of capacity in Chapter 2, Dirichlet structures on a space, that A-n-2 cape A < cap A < An+2 cape A (7.13) for any relatively compact open set A in U. Combining (7.11), (7.12), (7.13) 0 leads to the inequalities of the lemma with c = c2,A"+2
In the following lemma let V denote a Euclidean space into which the star S = st(a) of a point a E X is affinely embedded (cf. Chapter 4). Taking a as origin in V we consider a homothety 1' : V - V with centre a = 0 and factor 0 < a < 1; thus 7,1(x) = ax, x E V.
7. Potential theory on Riemannian polyhedra
117
Lemma 7.3. For any open set U C= S, GQU (ax, ay) = a2-"G, (x, y)
for x, y E U, 0 < a < 1.
Proof. For fixed y E U the function x '-- GQU (ax, ay) is a potential on U and harmonic in U \ {y}, being a Riemannian isometry up to the scaling
factor a. (If u is a harmonic minorant > 0 of x -- GQU (ax, ay) on U then u o -1 is a harmonic minorant > 0 of GQU( , ay) on aU, and so u o o-' = 0, u = 0.) By the uniqueness property (see the beginning of the proof of Theorem 7.3), GQU (ax, ay) = cGU(x, y)
(7.14)
for all x E U, whereby c might depend on y, but actually does not, by the symmetry of G. It remains to show that c = Consider a measure A > 0 on U of finite energy fu G' A dA (e.g., the equilibrium measure for some open set C= U), and a function W E Lipc(U). The measure tp*A on 1'(U) = aU likewise has finite energy, and W o 0-1 E Lip,(aU). By Theorem 7.3 (3) a2-n.
f(VeGiA,Ve)diZe = jcOdA
(7.15)
whereVe and µe are the gradient and the volume measure on X with respect to the Euclidean Riemannian structure ge. Via ii, (7.15) transforms into a2-n c
f
f
oU
QU
(gypo
A)+
on account of (7.14), and so indeed c = a2-n, by (7.15) applied with aU, i p * A, cp o ip-1 in place of U, A, gyp.
0
The following theorem can also be derived from [BM 1995, Theorem 1.3] by application of Lemma 4.4 and Proposition 7.5 (a) above.
Theorem 7.4. For every compact subset K of X there is a constant C = C(X, g, K) > 0 such that, when dim X = n > 2, C-1dx (x, y)2-n < G(x, y) 0 because u+ and u_ inherit the properties required for u. This time consider a point a E X(n-2) and its open star
S := stX(a), and write U = S \ Xin-2>. By Step 1, ulu E W1' (U), and even u E W1"2(U) by Lemma 5.1 because the Sobolev norm of uju is finite by hypothesis. It follows therefore from Theorem 5.2 and Remark 5.5 that there is a (unique) weakly harmonic function h > 0 a.e. on U such that uju - h E W01'2(U). We may assume that h is continuous and hence harmonic and > 0 on U. By Corollary 7.3, h extends (uniquely) to a superharmonic function s > 0 on all of S. Because ups - s is an extension of uju-h E W01,2 (U), and S\U C X(n-2) is polar, ups-s is quasicontinuous and of class W01'2(S). We may therefore replace u (or ups) by s (quasicontinuous)
in Step 2 (with X replaced by S), and our task is to prove that s E W,1,2 (S), hence actually s E W1'2(S). We first show, for given A > 0, that sa := min{s, A} E W1'2(S).
For any domain V of compact closure V C S, sa coincides in V with the potential p = k v on S obtained by sweeping the bounded superharmonic function sA onto V (see e.g., [CC 1972, Proposition 5.3.5]); p is harmonic in S \ V (see [Br 1959, p. 1.09]). Let G denote the Green kernel on S x S (cf. Remark 7.2). Then p = Gp for a measure p >, 0 of compact support
7. Potential theory on Riemannian polyhedra
125
(C V), and so f Gµ dµ < oo. By Theorem 7.3 (3), p E Lo2(S) = Wa'2(S) (invoking the Poincare inequality (5.12), cf. Remark 5.5). Thus (sa)Iv = ply E W1"2(V), whence sa E WW1, (S), and so indeed s.\ E W1,2 (S) by Lemma 5.1 because sA has finite Sobolev norm (no bigger than that of s). As \ -i oo (through the integers, say), sa s in L2(S, µ9) because s, like ups, has finite Sobolev norm, and in particular fS s2dp9 < oo. We complete the proof by showing that s E W1,2 (S), and for that it suffices that (s,\) is Cauchy in energy. For A' > A, sa - s,\ is constant off the set {A < s < A'}, and so, by Remark 5.2, Vs = 0 a.e. off that set. Thus IVs,, - Vsal2dµg = JS
f
a<e, 0 of finite energy E(p) = 1 fX p dµ = fX Gp du (cf. Theorem 7.3 (3) above). Recall that each such potential isz quasicontinuous, and that the axiom of domination holds (property (6) stated after Theorem 7.3). Ad (d). Choose a finite strict potential p > 0 on X (see e.g., [CC 1972, §7.2]), and write u = p - Rpu > 0, where RPM is obtained by sweeping p
on CU. Then u > 0 in U, and u = 0 q.e. in CU; hence u = 0 a.e. in X. It follows that u = 0 even q.e. according to [De 1970, Theoreme 5], and so
7. Potential theory on Riemannian polyhedra
127
u = 0 everywhere, by fine continuity, polar sets being finely discrete, by (a) above.
Proposition 7.8 clearly extends to any hypoelliptic Dirichlet space in the sense of Feyel and de La Pradelle [FP 19781.
Remark 7.8. The above quasitopological concepts have a local character and can therefore be defined on any admissible Riemannian polyhedron (X, g) without requiring that (7.3) hold. Say that a set U C X is quasiopen
if X can be covered by open sets X' for which (7.3) holds and such that U fl X' is quasiopen in X'. Recall that (7.3) always holds in the (open) star S = stX(a) of any vertex a of X in a triangulation fine enough so that S is the interior of its closure in X (Remark 5.5). This local concept of quasiopen sets is equivalent with the global concept
from Definition 7.2 in the case where X does satisfy (7.3). That follows immediately from Proposition 7.8 above, simply because the fine topology is a local concept (the subharmonic functions forming a sheaf, cf. Chapter 2), and because the union of countably many polar sets is polar. Similarly for other quasitopological concepts, such as quasi continuity (Definition 7.3 above). And now Proposition 7.8 holds without assuming (7.3). In the rest of this book we shall accordingly drop the hypothesis (7.3) without further ado when speaking of quasitopological or other local concepts. (The same can be done in the preceding section, beginning with Corollary 7.1. )
Sobolev functions on quasiopen sets In our subsequent study of (possibly discontinuous) maps of finite energy from an admissible Riemannian polyhedron (X, g) to a smooth Riemannian manifold or a Riemannian polyhedron enters the class W1"2(U) of Sobolev functions on a set U C X which need not be open, but only quasiopen. The following concept was introduced by Kilpeliiinen and Maly for the case X = R". Both the definition and the properties quoted below from [Ki1M 1992, § 2) extend immediately (in the absence of other indications) to the present case of an admissible Riemannian polyhedron (X, g), with the Riemannian measure µg replacing Lebesgue measure.
Definition 7.4. A function u : U -+ R defined on a quasiopen set U C X is of class W1"2(U) if: (a) U is the union of a sequence of quasiopen sets U3 such that each
restriction uju, can be extended to a function uj E W1"2(U3) on
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some usual open set Cl., D U,; and fv(u2 + IVuI2)di g is finite. (b) IIuII2 In (b), the gradient Vu E L2(U) is clearly independent of the choice of cover (Uj) and of extension u`j : U3 - R of each uju,, in view of Remark 5.2.
Every function u E W1,2(U) equals a.e. a quasicontinuous function on U, uniquely determined q.e. in U. Futhermore, (W1,2(U), II ' II) is a Banach space [KilM 1992, Theorem 2.6]. The restriction to U of the functions of class W"2(U) on usual open sets U D U are norm dense in (W1"2(U), II ' II). In particular, for a usual open set U C X, (W1"2(U), II ' II) as defined in Definition 7.4 is the same as in the usual sense. The chain rule (Lemma 5.2) extends immediately to Sobolev functions on a quasiopen set U C X. For example, if V C R' is open, if ce E C'(V) has bounded gradient, and if u = (u1, up) : U ---, V is of class [W1"2(U)]p, then ce o u E W 1,2 (U) and
V(
, 0, i-1 i-1
(8.1)
where Dju(p) = du(p)/dcr,j; here oj = dx(p, ) denotes Riemannian path length from p in s . As an application it is easily verified that the symmetric Green kernel GB
for a ball B = B(p, e)° C= st(p) is given (up to a constant factor > 0) in terms of lxi = dx (p, x), jyI = dx (p, y) by
G'
y) = I (P- IxI)(P- lyl) l (P - I xi)(,o + (k -
1)Jyl)
for x, y E B, the former expression applying when x and y lie on different 1-simplexes of st(p), the latter when x and y lie on the same 1-simplex and lxj > jyj. It follows that GB E Lip(B x B). The fact that the above map U - fl(U) is a Brelot harmonic sheaf (see Chapters 2 and 7) can alternatively be verified directly; and it can be shown that this remains true if equation (8.1) is replaced by EajDju(p) = 0 for given constants ay > 0 (j = 1, , k), allowed to depend on p. If, however, at some vertex p, aj = 0 is allowed for some, but not all j, then N becomes a Bauer harmonic sheaf, but not a Brelot sheaf. It is elementary to verify that every complete 1-dimensional Riemannian polyhedron (X, dx) has nonpositive curvature: curv(X, dx) < 0, in the sense of Definition 2.8.
Example 8.2. The need for dimensional homogeneity Let si be a Euclidean j-simplex, and form the polyhedron X = s1 Us2 C R2,
where sl n s2 is a common vertex p = so. X is not dimensionally homogeneous, and therefore not admissible. Each of the homogeneous parts s3, j = 1, 2, is an admissible Riemannian polyhedron with the induced Euclidean metric. Let 7{,2 denote the Brelot harmonic sheaf on s2 as defined in Chapter 7. We show below that there does not exist any Brelot harmonic sheaf H on X compatible with H,2 in the sense that 7{(U) = 7l,2 (U) for any
set U C s2 which is open in X (and hence in s2). Briefly, this is because p is a polar point for H,2, according to Corollary 7.2, hence a removable singularity for bounded 7{,2-functions.
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Suppose there does exist a harmonic sheaf 11 on X which is compatible with 71,3. Consider a regular domain U C X (with respect to W) containing
p and such that U C stx(p). Let u E H(U) extend continuously to U so that u = 1 on sl n 8U and u = 0 on the rest of W. By the local maximum principle for harmonic functions on a Brelot harmonic space [Br 1959] (cf.
Chapter 2) it follows that 0 < u(p) < 1; but on the other hand u(p) = 0, as we shall now see. Being 71,5-harmonic in U n s2 \ {p} (an open set in X contained in s2) by the required compatibility, and continuous in U n s2, u is in 71,5 (U n s2) by a well known result on removable singularities [Br 1959] (cf. Chapter 2) valid in any Brelot harmonic space (such as the Riemannian polyhedron s2). Because u = 0 on i9,2 (U n s2) = (8U) n s2, we conclude from the boundary minimum principle (cf. Chapter 2) that u - 0 in U n s2, in particular at p, a contradiction.
Example 8.3. The need for local chainability (Cf. [GMacP 1980, Fig. 5].) Let X be a Riemannian 2-torus with one meridian circle pinched to a point p.
As a polyhedron (indeed a circuit), X is not locally 1-chainable at p, and therefore not admissible. Again, there is no harmonic structure on X extending the usual harmonic structure on X \ {p}. To see this, take two meridian circles Co, Cl, one on either side of p, forming the boundary of a double cone Xo in X which we may assume is regular in X. Assign constant
values f = 0 to Co and f = 1 to Cl. Then f is continuous, but has no harmonic extension to X0, for such an extension would have to be both 0 and 1 at p, this point being polar (Corollary 7.2) in either half-cone (an admissible Riemannian polyhedron, cf. Examples 4.3 and 8.8), so p would be a removable singularity for bounded harmonic functions there according to [Br 19591 (as in Example 8.2 above).
Example 8.4. Manifolds as polyhedra Relative to a smooth triangulation, every connected smooth Riemannian manifold (possibly with boundary) is a Riemannian polyhedron, indeed a Riemannian normal circuit, and so is every connected triangulable Riemannian Lipschitz manifold M (cf. Example 4.2 and Subexample 4.2). That includes the case where M has a triangulable Riemannian Lipschitz boundary (e.g., a smooth Riemannian M whose boundary has corners).
Example 8.5. A kind of connected sum of polyhedra Let M0, Ml be admissible n-dimensional polyhedra, and fi : U , Mi proper
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133
embeddings (i = 0, 1) of the open unit ball U in R" onto the interiors of two n-dimensional simplexes of MO and M1, respectively. Use those to attach the cylinder (OU) x I to the disjoint union MO U M1, via the identification
fi c f6-1 : fo(aU) - f1(OU). The resulting space Mo#M1 is an admissible n-polyhedron. (That is like the connected sum Mo#M1, except that its construction starts with the disjoint union (Mo \ int fo(U)) U (M1 \ int f, (U)). But even when Mo and M1 are manifolds, Mo#M1 is not a circuit.) See [Mil 1959]. _ If Mo, M1 are Riemannian with metric structures go, gl, then Mo#Jvfl is an admissible Riemannian polyhedron with metric g given by gi on Mi and dt2 + (1
- t)fogo1fo(aU) + tfi 911 f,(OU)
on (OU) x {t} for each t E I.
Example 8.6. Riemannian joins of Riemannian manifolds If X and Y are subsets of Euclidean spaces V and W, respectively, with V and W embedded as orthogonal subspaces of a larger Euclidean space, the join X * Y is the set of all points of all segments xy joining x E X to y E Y; i.e.,
X*Y={z=(x,y,t)EV xW xl:z=(1-t)x+ty}.
If (X, dX) and (Y, dy) are metric subspaces of the above spaces V and W, respectively, a metric dX.y on X * Y is defined by dX.y (z, z') = dX (x, x') + dr (y, y') + It - t'I2.
If (X, dX) and (Y, dy) are length spaces, then so is (X * Y, dX.y ): Given z, z', let a be a path of X joining x, x' and of length L(a) close to dX (x, x'), and let 0 be a path of Y joining y, y' with L(,0) close to dy(y, y'). Then the following path y in X * Y joins z to z': -y(s) = (1
- ((1- s)t + st'))a(s) + ((1- s)t + st')Q(s),
and L(ry) is close to dx.y (z, z'). The join of two simplicial complexes P, Q is the simplicial complex P * Q whose set of vertices is the disjoint union of those of P and Q. A simplex of P * Q is the span of a simplex of P and one of Q. Thus, if s is an i-simplex
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of P and t a j-simplex of Q, then their span s * t is an (i + j + 1)-simplex of P * Q; its boundary is (b(s) * t) U (s * b(t)). Let R denote the set of all midpoints of all segments from points of P to those of Q. Then R = P x Q, and P * Q - P - Q = R* ]0,1 [. For the above, cf. [EiSt 1952, p. 74]. Now suppose M and N are smooth Riemannian submanifolds of V and W as above. Their join M * N is a length space with day, dN the indicated intrinsic distance of the Riemannian structure (and M*N is a geodesic space if M and N are complete). Relative to smooth triangulations of M and N, M * N is a Riemannian normal (m + n + 1)-circuit, in particular an admissible Riemannian polyhedron. (Presumably, this is still true if M, N are themselves smooth Riemannian normal circuits.) In [CD 1993, Appendix] a detailed description is given of the Riemannian join of two finite Riemannian polyhedra whose simplexes are spherical (i.e., of constant positive curvature). Here is a simple model to serve for most of the following examples:
Example 8.7. Riemannian orbit spaces In [Sc 1983, §§1-2] Scott discusses in detail the Riemannian orbit spaces X = M/t of discrete groups r' acting isometrically (hence ± holomorphically) on the 2-dimensional space forms M. In that special situation, X is a topological manifold, perhaps with boundary. It has a naturally induced Riemannian metric, possibly with singularities, of three types (cone points, reflector lines and corner reflections). Thus X is an admissible Riemannian polyhedron, and even a Riemannian normal circuit. The following is a closely related situation (see also Subexample 8.13 (i)): With all structures smooth, let (M,F, g) be a compact oriented 3-manifold, and: (i) F a foliation by oriented circles; (ii) g a Riemannian metric on M relative to which F is a horizontally conformal foliation by geodesics. Then the leaf space X = M/F' is a Riemannian 2-orbifold with a cone point corresponding to each critical leaf of F; X has a unique Riemann surface structure, and the leaf map 7r : M -' X is a harmonic morphism [BW 1992]. (For harmonic morphisms, see Definition 13.1 below.)
Example 8.8. Conical singular Riemannian spaces (Cf. [Chee 1980;1983].) Let X be a compact n-circuit with triangulation
8. Examples of Riemannian polyhedra
135
T, and singular set S(X), a subcomplex of the (n - 2)-skeleton X (n-2) of X. Cheeger studied the (generally incomplete) Riemannian metrics g on the complement X \ S(X) which are conical around S(X). or more universally: around X(n-2). For any j-simplex a of T the dual linking (n - j - 1)-cell Q of the first barycentric subdivision of T determines a local product structure on small neighbourhoods of interior points of a, homeomorphic to P = Co.1(Q)x ]0,1[',
where Co 1(Q) is a linear truncated cone over Q with vertex a. We endow P with the metric i hp = r2hQ + dr2 + L dr? (0 < r, ri < 1). i=1
A Riemannian metric g on X \S(X) is said to be conical if locally it has such a form near each simplex of S(X ), in the strict sense - not just locally quasiisometric to a conical metric. X is then a Riemannian normal n-circuit, and so an admissible Riemannian polyhedron. The geometry of Riemannian spaces X has been studied in a more general framework by Liu and Shen [LS 1998], with special attention to convergence of the Riemannian metric of X \ S(X) near S(X) in metric conical charts.
Subexample 8.8. Let it : V - Al be an oriented vector bundle of Euclidean spaces over a compact manifold M. Its Thom space is the one-point compactification X = V U {oe}, expressed as the space X = D U cone OD formed as the union of the closed ball bundle D of V with the cone over its boundary. X is a normal circuit whose singular set S(X) = oo, the vertex of the cone. For instance, the quadric cone X given in homogeneous coordinates [zo, z1, z2, z3] of CP3 (the complex projective 3-space) by zo +zi +zz = 0 is a normal 4-circuit with isolated singular point [0, 0, 0, 1]. X is homeomorphic to the Thom space of the tangent bundle of the 2-sphere. [McC 1977].
Example 8.9. Normal analytic spaces with singularities (Cf. [Lo 1991].) Let X be a complex analytic space in the sense that X is a topological space with a maximal open cover (9i, Ui) such that: (a) 9i : Ui V, C Cn- is a homeomorphism onto an analytic subset [ (i.e., given locally by the zero locus of a finite number of holomorphic functions), and (b) every 9j o 0, 1 is holomorphic.
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Part II. Potential theory on polyhedra
Let X° denote the set of regular points of X (i.e., points having neighbourhoods which are manifolds). Then X° is a disjoint union of open closed submanifolds, and is open and dense in X. Its complement S = X \ X° of singular points is an analytic nowhere dense subspace of X. The space X is irreducible if X° is a connected manifold; thus dim X = dim X°. Say X is normal if, for every open U C X and every nowhere dense analytic subspace A C U, every holomorphic function on U \ A which is locally bounded near A has a holomorphic extension to U. Every analytic space X has a normalization 7r : X X, unique up to isomorphism; thus 7r is a finite (i.e., proper and with finite fibres) holomorphic map of a normal analytic space k onto X whose restriction to the regular set of k is biholomorphic, with dense image [Lo 1991, Chapter VIA]. Now let X be a complex analytic space whose irreducible components have complex dimension n. Then there is a semi-analytic triangulation of X into a locally finite simplicial complex with its singular set triangulated as a subcomplex. X is an orientable 2n-circuit without boundary (Giesecke [Gie 1964] and Lojasiewicz [Lo 1964]). And normality as a complex analytic space is topologically equivalent to normality as a circuit [Hae 1984, §1.6]. As an explicit example of a non-normal complex analytic space, take the pinched torus X in CP2 given (in homogeneous coordinates) by x3 + y3 = xyz; its singular set S is the point [0, 0, 1]. As a circuit, X is not normal (Example 8.3). Finally, suppose that X is a normal complex analytic subset of a Kiihler
manifold M. There is a triangulation of X with the interior of each top dimensional simplex s holomorphically embedded in M [Gie 1964, (6.1)]. That induces a Riemannian metric on each simplex, and so X becomes a Riemannian normal circuit. Furthermore, g restricted to the interior of s is a Kahler metric.
Subexample 8.9 (i). If X is a normal complex analytic space of complex dimension 1, then X is a Riemann surface (by Example 4.1).
Subexample 8.9 (ii). Suppose that G denotes a group of holomorphic automorphisms of a compact analytic space X; and that G acts properly discontinuously, i.e.,
(a) each pair of points x', x" in X in different G-orbits have neighbour-
hoods U', U" such that sU' fl sU" = 0 for all s E G (that insures that the quotient space G\X is Hausdorff), and (b) for any a E X, its isotropy group G. is finite; and there is an open neighbourhood U of a which is G0-stable and for which the following
8. Examples of Riemannian polyhedra
137
implication holds: s E G, x E U, sx E U implies s E Ga. (That ensures that the image of U in G\X is isomorphic to GaIU.) A theorem of H. Cartan [Ca 1957] asserts that G\X is a complex analytic space, and normal if X is. The orbit map 7r: X -+ G\X is holomorphic.
Now assume that X is a complex manifold of dims X = n. Then Y = G\X is also a complex orbifold; i.e., each point y E Y has a neighbourhood biholomorphic to a neighbourhood of the origin in Gy\C", the isotropy group G. of y being a finite group of linear automorphisms of C".
Subexample 8.9 (iii). Fix a set of positive integers Q = (qo,
, qr),
and form the finite group GQ whose elements are (go, , gr), where each gj = exp(27ribj/qj) (for integers bj with 0 < bj < qj). Define the isometric action of GQ on Cr+l \ {0} by ((g0, ... , gr), (ZO, ... , Zr)) '-+ (BOZO, ... , grzr)
Its orbit space P(Q) = GQ \ (Cr+1 \ {0}) is a weighted projective space [Dol 1982]. It is a normal, irreducible, projective algebraic variety. All of its singularities are cyclic quotients, so in particular, IP(Q) is a Riemannian orbifold, as in Example 8.13 below.
Subexample 8.9 (iv). The Grassmannian Gk(C") is the space of k-dimensional subspaces of C". It is a compact homogeneous projective (hence Kiihlerian) manifold. (G1(C") is the complex projective space CPn-1 ) Gk(C") has a natural decomposition into open complex cells, called Schubert varieties, whose closures are complex analytic spaces with explicit and simply described singularities [GrHa 1978, Chapter 1, §5]. There are analogous complex cellular decompositions of compact homogeneous Kiihlerian manifolds; the closures of these cells are geometrically significant (usually singular) complex analytic subspaces [Bor 1954].
Subexample 8.9 (v). Suppose that X is a compact projective irreducible complex analytic space. (Thus X is a connected algebraic variety.) Then an application of Hironaka's theorem [Hir 1964] on the resolution of singularities gives a compact connected Miler manifold (M, g) with dim M = dim X, and a surjective holomorphic map 7r : Al ---+ X. Such resolutions can often be used to recognize harmonic maps in a holomorphic framework.
Subexample 8.9 (vi). Assume that X has a Riemannian metric g as in Example 7.2, and that glx\s is Kiihler. Then any bounded holomorphic
Part II. Potential theory on polyhedra
138
function u : U C is harmonic. Indeed, U \ S is a Kiihler manifold, so the restriction of u to U is harmonic (see e.g., [ES 1964, §2C]). The assertion now follows upon application of Corollary 7.3.
Remark 8.1. Similarly, a real (semi-)analytic or (semi-)algebraic set is triangulable [Har 1975], [Hir 1975]. The key ingredients are
that such sets admit suitable stratifications (in the sense of Thom and Whitney), that these are triangulable; see [Ver 1984, Theorem 7.8].
That monograph provides a thorough treatment and fine perspective on stratifications and triangulations.
Example 8.10. The Kobayashi distance (Cf. [Kob 1970;1976;1998].) Let U denote the open unit disc in C, and Lou its Poincare metric. Given two points p, q of a complex analytic space X, join them by a chain of holomorphic discs, i.e., points a1, bi, , ak, bk in U, ,pk = q in X, and holomorphic maps fl, , fk : U -+ X with p = P O,** f. (a,) = pi-1, fi(bi) = pi (1 < i , 2).
Subexample 8.11 (ii). Complex conjugation in C3 induces an involution a : CP2 - CP2 with real projective space RP2 ti CP2 as fixed point set. The quotient map 7r : CP2 -' CP2/a endows the quotient with a smooth manifold structure; in fact CP2/a is explicitly diffeomorphic to S4 (Hitchin, unpublished). Then it : CP2 S4 is a Riemannian branched covering, with A = RP2.
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141
Subexample 8.11 (iii). Take a configuration L of k lines (k 3 3) in GP2 not all passing through the same point, and any branching order r > 2. This determines Kummer's branched (over L) cover 7r : X,. (L) -. GP2 [BHH 1987, §§1.5, 3], with Xr(L) represented as the complete intersection of k - 3
hypersurfaces of degree r, and thus a normal complex analytic space. The space Xr(L) is simply connected.
Subexample 8.11 (iv). Suppose cp : X
Y is a finite, surjective, continuous open closed map between polyhedra. Its local degree (if finite) at a point x E X is deg(cp; x) = inf sup{card (cp-1(cp(z)) n u) : z E U},
where U ranges over the neighbourhoods of x. Define the degree (if finite) of cc:
deg(cp) = sup{cardcp-1(y) : y E Y}. Say that cp : X y E Y,
Y is a finite branched cover if deg(cp) < oo and for each deg(cp) =
{deg(cp;x) : x E V-1(y)j.
Let A,, denote the branch set of cp (i.e., the set of points in X at which cp fails to be a local homeomorphism). If cp : X , Y is a finite, surjective, continuous open closed map and X a normal circuit, then deg(cp) < oo, and cp(A,p) does not locally separate Y. Furthermore, Y is also a normal circuit, and cp a finite branched cover [Ed 1976, Theorem 3.5].
A surjective proper holomorphic map cp : X -* Y with finite fibres between connected normal analytic spaces is an analytic branched covering. There is a proper analytic subspace B C Y of pure complex codimension one such that Wax\,p-'(B) --* Y \ B is unbranched, [BPV 1984, I.16]. X and Y can be triangulated as normal circuits (Example 8.9), relative to which the map becomes simplicial [Gie 1964]. If Y = (N, h) is a Kahler manifold,
then h induces a metric dx on X relative to which (X, dx) becomes an admissible Rieman nian polyhedron, by Example 8.9.
Let X be a connected normal analytic subspace of dim X = n, in GP'. GP" [Sh Then there is a finite holomorphic branched covering map X 1974, p. 52]. If cp : GP" --+ Y is a nonconstant holomorphic map onto a normal analytic space, then cp is an analytic branched cover, [RV 1961] and [RS 1960].
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Example 8.12. The quotient M/K Let K be a compact group of isometries of a complete smooth Riemannian manifold (Al, g), and let it : M M/K be the projection onto the orbit space M/K. There is a smooth triangulation of M for which it induces an admissible Riemannian polyhedral structure on M/K such that the associated intrinsic distance dAt/K(y1, y2) between elements yl and y2 of M/K equals the (constant) intrinsic distance in M between the corresponding compact orbits 7r'' (yl) and 7r'' (y2), cf. Subexample 8.12 (ii); in particular, M/K admits a triangulation such that every open simplex consists of points of the same orbit type [Ver 1984, Corollary 7.10]. (As in Example 8.9, M/K has a suitable stratified structure.) Furthermore, with the distance dnt/K the polyhedron M/K is a geodesic space whose curvature is bounded below by the infimum of the sectional curvatures of Al. The polyhedral structure determines a Brelot harmonic sheaf lM/K on M/K as described in Theorem 7.1. On the other hand, it carries the Brelot harmonic sheaf Wm (the local solutions to the Laplace-Beltrami equation on M) onto a Brelot harmonic sheaf 1M1K = on M/K. It is obtained by assigning to each open set V C M/K the linear space MIK(V) of all functions u : V - IR such that u o it E NM (7r-'(V)). Accordingly, 7r : (M, 7{M) - (M/K,11''y/K) is a harmonic morphism, by the very definition of such morphisms in [CC 1965, §3], cf. Definition 13.1 below; and this characterizes the Brelot harmonic sheaf 7{'' IK. The proof of these assertions will be given after Subexample 8.12 (iii). The two harmonic structures HAl/K and 7{'MIK on M/K are generally distinct, cf. Subexample 8.12 (iii).
Subexample 8.12 (i). [Ed 1976], [Ja 1966]. In the case where K is a finite group of isometries acting on (M, g), conditions are known to ensure that the orbit map 7r : Al N = M/K is a smooth branched cover of a smooth manifold; and of a geodesic space (N,dN). Subexample 8.12 (ii). Here is a simple example of a fibration with singular fibres (a general picture of which can be found in [CT 1968]). If K = S' is the circle group acting on M as above, then the orbit space M/K is a Riemannian normal (n - 1)-circuit (n = dim M) [EdFi 1976, Lemma 2.6], and thus an admissible Riemannian polyhedron. Therefore, for K = S' we have the Brelot harmonic structure 1M/K from Theorem 7.1. A concrete example follows:
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143
Subexample 8.12 (iii). Take M = S2 with its standard metric, K = S' = the group of orientation preserving isometries of M leaving a given point p E M (and -p) fixed. If 9 denotes the geodesic distance from p, M/K can be identified as a metric space with the interval 0 < 9 < 7r, which we shall view as a Riemannian 1-circuit with just one 1-simplex, embedded in R and equipped with the Riemannian metric ds = d9.
The corresponding Brelot harmonic sheaf 7IM/K from Theorem 7.1 consists of the linear combinations of 1 and 9 - 9, i.e., the affine functions of 9, as in Example 8.1. However, the constants are the only functions harmonic on
an interval containing 0 or it (cf. Remark 5.3); but the function 0 u 9 is subharmonic at 0 and superharmonic at 7r. Every point of M/K is non-polar (for the sheaf 7IM/K) by Corollary 7.2. In addition to the above harmonic sheaf 7IM/K we have, as mentioned above, the harmonic sheaf 7I'M/K = 7r.7IM on M/K = [0, 7r] induced by 7IM so that the projection M M/K becomes a harmonic morphism, i.e., pulls harmonic functions back to harmonic functions. The harmonic functions on M/K = [0, 7r] are here the solutions to sin 9 d9 (sin 9d9
) = 0,
and these are the linear combinations of the constant 1 and the function log cot 9/2, the latter being (finite and) harmonic only for 0 < 9 < it. Thus the constants are the only functions harmonic on an interval containing 0 or 7r; but the function log cot(9/2) is superharmonic at 0 and subharmonic at in (when given the value +oo at 0 and -oo at 7r), so 0 and in are polar points for the sheaf 7I'M/K, while the remaining points of M/K = [0, 7r] are nonpolar. The above two harmonic sheaves on M/K are obviously different, and neither contains the other, in agreement with Corollary 13.1 (ii).
Example 8.12 (continued). We now prove the assertions about the sheaf xM/K = 7r.Nm on M/K. For simplicity we now write 7IM/K in place of First a more general setup: MM/K.
Let (X, ?ix) be a Brelot harmonic space for which the constant functions are harmonic. Let Y be a Hausdorff space, and in : X -+ Y a surjective,
144
Part II. Potential theory on polyhedra
continuous, open map. Then Y is connected, locally connected and locally compact, because X is so. For open V C Y define:
vEfy(V)
v o7r E %x(7r-1(V)).
(8.2)
Then Hy (V) is a linear subspace of C(V), and fly (= 7r.7-lx) is a sheaf containing the constants on Y. We show that this sheaf has the Harnack monotone convergence property. Consider a domain V C= Y and an increasing sequence of nonnegative functions v E fly (V), and write v = supn vn. Then
vo7r=Sup(vno7r), n
0, a} = x({u > a}) is compact because {u > a} is a closed subset of the compact set U, and similarly {v < a} is compact. Furthermore,
viev = f because vlav o 7r = (v o lr) lau = U OU = fox, by (8.3). Finally, f > 0 implies
v > 0 because f o 7r >, 0 implies u > 0, by regularity of U. This shows that V = By(yo,r)° is indeed regular in (Y,fy) for small enough r, thus completing the proof that Y = M/K, endowed with the sheaf fly = zr.lim (= irdix) from (8.2) above (originally denoted 7{MIK), is a Brelot harmonic space.
Assume now that the group K is also connected Let M° be the union of the K-orbits in M of maximal rank. As discussed in [ER 1993, Chapter IV, §3], M° is an open dense submanifold of M. The harmonic morphism 7r M M/K contains an open dense harmonic Riemannian submersion 7r° M° -+ M°/K induced from that on M.
Example 8.13. Riemannian orbifolds The construction in Example 8.12 can be adjusted to apply to Riemannian orbifolds (roughly speaking, spaces modelled on quotients of domains in R" by finite groups of isometries) [Mol 1988, §3.6]: A Riemannian orbifold has a natural geodesic structure, if complete, and natural harmonic space structures. Especially important are those Riemannian orbifolds with curvature bounded above [CD 1993, §5].
Subexample 8.13 (i). If F is a Riemannian foliation (i.e., any two leaves are locally equidistant) of a smooth Riemannian manifold (M, g) with closed
leaves, then the leaf space M/.F is a Riemannian orbifold [Rei 1961]. For instance, let K be a closed subgroup of isometries of a Riemannian manifold (M, g), all of whose orbits have the same dimension. Then their components form such a Riemannian foliation [Rei 1983, IV, Example 4.10]. More generally, any closed, connected subgroup of isometries of (M, g) determines a model singular Riemannian foliation, in the sense of [Mol 1988, Chapter 6].
Query. Let (M, g) be a Riemannian manifold, and F a Riemannian foliation with closed leaves, all of which are minimal submanifolds. Is the leaf-quotient map 7r : M -+ M/.F harmonic [a harmonic morphism] in the sense of Defini-
tion 12.1 [Definition 13.1]? That is so over the regular points of M/.F (i.e., where the leaves have no.F-holonomy), by [EV 1998, §5].
8. Examples of Riemannian polyhedra
147
Subexample 8.13 (ii). Let &M denote the r-fold symmetric power of the Riemannian manifold (M,g): 6rM = (M x ... x M)/er, where 67r denotes the symmetry group on r factors. It acts isometrically, so 6rM is a Riemannian orbifold (nonsingular if dim M = 2; see [DuL 1971] and [MacD 1962]). In particular, with Al the Euclidean n-sphere S" (n > 0) and r large, 6rS" serves as a classifying space for n-dimensional integral cohomology, by a theorem of Dold and Thom [DT 1958]. Thus for any compact polyhedron X, the homotopy classes of maps X -+ 67,S" are in natural bijective correspondence with the cohomology classes in H" (X, Z) [Sp 1966, Chapter 8, §1].
Example 8.14. Buildings of Bruhat-Tits Here we describe a construction which associates Riemannian polyhedra X of nonpositive curvature to certain algebraic groups. In fact, X are Euclidean buildings associated to Tits systems. For motivation, background, and relevant literature, see [Ti 1974], [Bro 1989], [BruTi 1972] - especially, the formal analogy between Euclidean buildings and Riemannian symmetric spaces of noncompact type. Step 1. [Bou 1974, Chapter IX, §3J, [Bou 1981, Chapter IV-VIII]. The valuation vp : Q -+ R associated to a prime p is defined by letting vp(x) be the exponent of p in the prime factorization of x; associated to vp is the ultrametric dp(x, y) = exp(-vp(x - y)).
Thus dp(x,z)
sup(dp(x, y), dp(y, z)) < dp(x,y) + dp(y,z).
The completion of Q with respect to dp is the field Qp of p-adic numbers. Qp is locally compact and totally disconnected. A subgroup G c GL(n, Q) is algebraic over Qp if it is the set of invertible matrices whose entries annihilate some collection of polynomials in n2 indeterminates with entries in Qp. Let Gp be the group of invertible n x n matrices with coefficients in Qp and which satisfy the polynomial equations defining G. We refer to Gp as a p-adic Lie group. Step 2. As a special case, Gp = SL(n,Qp), the group of invertible n x n matrices of determinant 1, with coefficients in Q,. Let B denote the sub-
group of upper triangular matrices in Gp; and N the normalizer of the subgroup D of diagonal matrices. Set W = N/D, which is the permutation
148
Part II. Potential theory on polyhedra
group on n elements; and S = {s1, , sn_ 1 }, where sj denotes the permutation of the lines in (Qp)n spanned by ej and ej+1, leaving the other lines fixed. Then (W, S) is a Coxeter system, in the sense of Step 3 below. Associating the complex (W, S) to the group Gp is a model example of a general method of associating a Euclidean building to a suitable algebraic group over a field. Step 3. A Coxeter matrix on a finite set S of n elements is a generating set S and the relations
m93'=1 ifs=s', m93'>, 2 ifs0s'. See [Bro 1989]. Associated to (m8s') is the group W (the Coxeter group) defined by the presentation W = {S :
1}.
Call (W, S) a Coxeter system of rank n. The natural map S --+ W is injective.
The Coxeter complex of (W, S) is defined as follows: Set m = n - 1; let As be an open m-simplex whose vertices are the elements of S; and for any subset T of S let oT be the open face of As spanned by the vertices
s E S \ T. Define X = (W x Os)/ -, where (w1ix1) - (W2,X2) if (i) X1 = x2 and (ii) if OT is the open simplex containing x1i then w1 1w2 E WT,
the subgroup generated by T. Then X is a simplicial complex on which W acts simplicially. Step 4. Consider now a simplicial complex X all of whose maximal (open) simplexes (called chambers here) have the same dimension m. Suppose X is expressible as a union of subcomplexes A (called apartments) such that, (a) every apartment is a Coxeter complex of dimension m;;
(b) any two simplexes of X are contained in an apartment; (c) given any two apartments A, A' with a common chamber, there is an isomorphism A -# A' fixing A n A' pointwise. Such a complex is called a (Bruhat-Tits) building. The group W acts on X simplicially and preserves the rank (i.e., the isomorphism type of the apartments). Furthermore, (i) every simplex of a building is contained in one of dimension m; (ii) any two chambers are (globally) (m - 1)-chainable by chambers. Step 5. Each apartment A carries a canonical Euclidean metric 9A such that the isomorphisms in (c) above are isometries. Such an X will be called a Euclidean building. Here it is appropriate to relax the hypothesis of local
8. Examples of Riemannian polyhedra
149
finiteness on X, requiring instead that X have only finitely many isometry types of simplexes. The required properties of X as a complete geodesic space are retained [Bri 1991]. These 9A together extend to a complete geodesic structure on X. Furthermore, X is contractible and has nonpositive curvature. (See [Bro 1989, §VI.3], [BruTi 1972].) Verifying nonpositivity of curvature is equivalent to this: Let d denote the metric of the geodesic structure of X. For any points w, x, y E X (with x and y sufficiently close) and t E [0,1] let z = (1 - t)x + ty be the unique geodesic segment joining x to y. It is required (compare the reference to Reshetnyak towards the end of Chapter 2) that d2(w, z) 0 is of class LoC(X,µ9), cf. [KS 1993, §1.2], which applies because W equals µ9-a.e. a Borel measurable map X -p Y. Definition 9.1. The energy E(cp) of a map cp of class L C(X,Y) is E(W) =
sup fECc(X,1O,11)
(limsupJ fee(co)dµs) ( 0; it follows that ec(cp P) 5 A2m+2e'Af(,pIP) I
In the rest of the proof we therefore refer tacitly to the Euclidean Riemannian structure on P (omitting the dashes), and the associated distance dp, relative to which P is a geodesic space (viz. a compact length space, see Chapter 4, The intrinsic distance). By a chain we shall understand the union of a finite sequence Qo, , Qp of distinct m-cells Qi of P (i.e., cubes of the above form Qj) such that, for
each i = I,... p, dim(Q; fl Qi-1) = m- 1 and (Qi \ Qi-1) fl Q, = 0 for j < i - 1. There is only a finite number N of such chains.
9. Energy of maps
157
Any two m-cells Q, Q' of P can be joined by a chain A = Qo U U Qp, Qo = Q, Qp = Q'. Since (P, dp) is admissible this is easily shown by joining
a point x of Q to a point x' of Q' by a (minimizing) geodesic segment xx' in P, and noting that the intersection of xx' with each m-cell of P is either a single straight line segment or a single point, if not empty. Every chain A = Qo U U Qp is Lip homeomorphic to a cube in R. For the construction of a Lip homeomorphism a : A - [-1,1]' it suffices, by recursion with respect to p, to construct (if p > 1) a Lip homeomorphism of Qp U Qp_1 onto Qp_1 which (if p >, 2) leaves the (m - 1)-dimensional face Qp_i fl Qp_2 of Qp_1 pointwise fixed. Here we may replace Qp_1 by [0,1]"` = [0,1]2 x [0,1]m-2, QP U QP_1 by [-1, 1] x [0,1] x [0,1]m-2, and the face Q,_1 f1Qp-2 by [0, 1] x {0} x (0,11--2. We only need to consider the case
m = 2 (for if c : E1 -+ E0 is a bi-Lip bijection between sets E1, E0 C 1R2, then k x id : E1 x [0,1]si-2 --p E0 x [0,1]re-2 is likewise a bi-Lip bijection). When m = 2, we thus consider the rectangle E1 = [-1, 1] x [0, 1] and its right half Eo = [0,1]2. Now, E1 is composed of three triangles t1, t2, t3, all with 0 as a vertex, the opposite sides being {1} x [0,1], [-1,1] x {1} and {-1} x 10, 1]. We therefore divide Eo into 3 triangles $1, $2, $3, again with 0 as a common vertex, now with opposite sides (1) x [0, 1] (as before), [2, 11 x {1} and [0, x {1}. The obvious affine maps ti si (i = 1,2,3) are bi-Lip bijections2]which combine to produce the desired bi-Lip bijection k : E1 -* E0 extending the identity map of t1 = s1, in particular that of the side [0, 1] x {0} thereof. From (a) A (b') follows by p applications of [KS 1993, Theorem 1.12.3]
that V o a-1 is a finite energy map [0,1]r` - Y. Every cube is a Lip ball. (For example the map x '- (IIxII2/I1xII.)x is a Lip homeomorphism of the closed unit ball U in R' onto [-1,1]', as seen by replacing II 11, by II 112 in the proof of Sublemma 4.3.) Therefore we also have a Lip homeomorphism 8 : A -+ U, and V o,3-1 has finite energy. We may assume that V does not contain the point /3-1(0), and so O(A fl V) does not meet pU for some radius p > 0. By reflection in the unit sphere Y extends to a finite energy map OU the finite energy map cyo,3_1 : U p* of X* := (2 - y)U° to Y, again by [KS 1993, Theorem 1.12.3]. By the same reflection, 3(A fl V) extends to an open set V* C= X*; and -
-
the functions f o,8-1 and 77o,(3-1 extend to functions f * and,*, both of class C(X*, [0,1]). The above extension of the unit ball U to the larger, open ball X* (a Riemannian domain) was designed to achieve that supp(1- 77*) C V* (C= X*). (The quantities $,X*,V*, f*,r7*,cp* all depend on the chain A.)
Fix temporarily an m-cell Q of P. For any point x E Q, any chain X*), A starting with Q, and any e > 0 such that I/3IE
1R' have quasicontinuous versions. For each
chart 77 : V - RJ from V we have, by (ii), 171 ° Vi,p-,(v) E
[W1,2(Sp-1(V))]n
in the sense of Definition 7.4 (extended to maps into Rn). For any open set By the extended chain Z C= V the map i o i7-' E C' (r7 (V )) is Lip on rule (Chapter 7, Sobolev functions on quasiopen sets) it follows that 'plw-'(z) = (t o 77J-1) o (y ° PI,P-,(Z)) E
[1V1.2(,P_1(Z))]9.
, q, of = t o is of class W1,2(X) in the sense of Definition 7.4 (hence also in the usual sense), because
This means that each component ep', i = 1,
the quasiopen sets W-1(Z) form a countable cover of X. Conversely, suppose that cp = t o V = (cp1, , cp') is of class [W1"2(X)]Q
in the usual sense. By Proposition 7.7, (p and hence O = t"1 o (p` have quasicontinuous versions, and for each pair (V. 77j) E V 77J °Wl ,-1(v) = (,/.'1,...
)n) E [w1,2(V_1lV))]n.
This shows that V satifies (i) and (ii) from Definition 9.2 relative to the above atlas V. Because t is a Riemannian isometry, the covariant components h,O
of h relative to the chart i7 : V
1R' are (in the case j,, = a, say)
8yi 8y i
h«li = 6.a + E 8y° 8y n cmE(cp); and if E'(ep) < oo then e'(cp) > cme(cp) a.e. in X. (b) Suppose either that W(X) has compact closure in N, or that N admits a C'-embedding into some Rq, whose inverse is globally Lipschitz:
dN(y,y) 0. If
(10.4)
oo then
(10.4) holds, more generally, for every A E W0"2 (X) fl L°° (X), A > 0.
Proof. We adapt and complete the proof from [J 1997a), in which V is compared with maps obtained by pulling W(x) towards a given point q of Y, an
Part III. Maps between polyhedra
182
uq = v. idea used in [JK 19791. For brevity write Uq = u, In proving (a) we may suppose that X is compact. Consider first the case that A E Lip(X), 0 < A < 1, and that supp A C U for some domain U as in Definition 10.1. For x E X denote yx : [0,1] Y the constant-speed geodesic in Y from 'y.,,(0) = yo(x) to 'yx(l) = q. Define a map cpa : X Y by
,p,\(x) = yx(A(x)),
x E X,
hence Va = off U. We shall compare the energy of W with that of spa. To see that VA E L2(X,Y) (cf. Chapter 9), note that
0 such that r < Q/6 write U = Be(a,5r)0,
µr = (µ9)IB"(a.r).
Sublemma 10.1. There are constants -yl > 'ry2 > 0 and o > 0, depending only on X, Ax, a, such that GUµr 5.y1r2
GUµr >72r2
on U = Be(a, 5r)°, on Be(a,r)
for any0 c2 > 0 depending only on X, AX, a (after fixing p) such that on Be (a, p)
C2 < GV L p < cl
Applying Lemma 7.3 to a = r/p (< 1) leads to on Be(a,r)
c2r2/p2 < Guµr < cir2/p2
when r < p; and hence by Lemma 7.2 c-1c2r2/p2 < GUpr < cc1r2/p2
on Be(a,r).
The latter inequality actually holds in all of U by Frostman's maximum principle (see property (5) in the text following Theorem 7.3).
Henceforth, p = p(X, Ax, a) will remain as in Sublemma 10.1. Because f GUpr dpr < oo for r < p it follows from Theorem 7.3 (3) and Remark 7.2 that Gup,. E L'2(U) = W0'2(U) andjdr,
J(VGuizrVtI)dg =
i E Wp'2(U).
(10.7)
For the following inequality recall the notation U+,r and Ur preceding Lemma 10.1, to be applied now to the weakly subharmonic function u = Uq =
q E Y, see (10.2).
Lemma 10.3. Suppose that X is compact, (Y, dy) is a simply connected complete geodesic space of nonpositive curvature, and that w : X --+ Y is locally E-minimizing. For given a E X there is a constant c depending only on X, AX, a such that, for r < p, q E Y, and corresponding uq from (10.2), r2
p9(Be(a, r))
du,
f° a,r
c(z19,+,6r - u9,+,r/5)
Proof. With u = uq from (10.2) and 0 < r < p from Sublemma 10.1 set z = U+,6r - u,
a superharmonic function of class W1'2(X) f1 L°° (X) satisfying z > 0 in Be(a,6r)°. With U = Be(a,5r)° as in Sublemma 10.1, gr
Gu pr
10. Holder continuity of energy minimizers
187
is bounded and of class W0'2(U) = Lo 2(U), hence so are gT and zgr. (Indeed,
gr can be approximated in W01'2(U) by a uniformly bounded sequence of functions of class Lip°(U), as seen by truncation.) Because E(W) < oo (X being compact), this further implies by the last assertion of Lemma 10.2 and by polarization (cf. (2.1) and (10.3)),
2f e(,P)grdµg 0 in U C Be(a,6r)°. From (10.8) and from (10.7) applied to 11i = zgr E
W01, 2 (U)
we obtain, by Sublemma 10.1, (recall the abbreviation
Pr = (µ9)B°(a,r))' 1
fB(a.r)
e(w) dµg
24
Y2r
U
2r4 f ( (zgr), V9r)d1t9 = 2rr4 L(a,r) zgdµ'g
Iy2T2
Je(ar CoC
zdµ9=
`72µg(Be(a,r))zr
1
C
TAg (BC (a, r)) (u+,e,r - U+,r/5)
For the last inequality we have applied Lemma 10.1 (a) with r replaced by r/5, noting that z = u+,ar - U E W1"2(X)f1L°O(X) is weakly superharmonic in X by Lemma 10.2 (since u = uq from (10.2)). O For the following lemma see [KS 1993, Proposition 2.5.4]; the easy proof is included for completeness.
Lemma 10.4. Suppose that X is compact and that Y is a simply connected complete geodesic space of nonpositive curvature. For every measurable set A C X with µ9(A) > 0, the meanvalue coA over A of a map (P E W1'2(X,Y) lies in the closed convex hull of the essential image y (A). The essential image co(A) is defined as the closed set of all points q E Y such that A fl W-r (V) has positive measure for any neighbourhood V of q
Part III. Maps between polyhedra
188
in Y. The closed convex hull of a set B C Y is defined as the intersection of all closed convex subsets of Y containing B. By the meanvalue 1PA of cP over A is meant the unique pointy E Y for which JA
d..(co('),y)dit9=miYJ dy('P('),y)dp9.
(10.9)
A
See [KS 1993, Lemma 2.5.1] for the existence and uniqueness of V. (By triangle comparison it is shown that every sequence which is minimizing for fA dY (co(-), y) d' g, is a Cauchy sequence in (Y, dy).) Proof of Lemma 10.4. Let C denote any closed convex set containing cp(A). We shall rule out the possibility that a point y E Y \ C can be the meanvalue
of cP over A. Let y' denote the point of C nearest to y. (Any sequence (yy) C C, minimizing for on C, is Cauchy in (Y,dy), by triangle comparison, and hence has a unique limit y', lying in C.) It suffices to show a.e. in A, and that will follow from that
dy(z,y) > dy(z,y*)
for every z E C.
Consider any point to E zy' (the geodesic segment in C with endpoints z, y* ). In the Euclidean comparison triangle z` y y' let w correspond to to, i.e. Jw - y I = dy(w, y'). Because Y has globally nonpositive curvature and w E C, jw - yi >, dr (w, y) > dr (y*, y) = Iy - y'l
For w near y', hence io near y', this shows that the Euclidean angle at is > 7r/2, and so indeed
dy(z,y)=jz-yI>Iz-yI =dr(z,y') 0 Corollary 10.2. Under the hypotheses of Lemma 10.4 we have for every q E Y and every measurable set B D A esssupdy(co('),r ) < 2esssupdy(ip('),q) B
B
In fact, this reduces, by the triangle inequality, to
dY(c 5A, q) < ess sup dy (,p(.), q) (< esssupdrq)), A
B
10. Holder continuity of energy minimizers
189
which in turn follows from Lemma 10.4 because W(A) C By (q, R), where R = ess SUPA dy I q); and every ball By (q, R) is closed and convex.
Proof of Theorem 10.1 The proof will be given in the more general setting where cp maps into a simply connected complete geodesic space Y of nonpositive curvature and such that Y is Lip homeomorphic to a closed subset of a Euclidean space (cf.
Lemma 4.1), this being sufficient for the Poincare inequality (Proposition 9.1 and Remark 9.3) in view of Corollary 10.1 above. The assertion of the theorem being local, we may suppose that X is compact. Fix a point a E X. As in Lemma 10.3, suppose r < e (cf. Sublemma 10.1). For brevity, write 5r/6 for the meanvahie of cp over Be(a,5r/6), and consider on X the function w = dY(i
Returning to (10.1) and the text preceding it, we now fix e = 1/8 and p = p(X, Ax, a) so that 6p < 62. By Corollary 4.1 (ball doubling, now with "Euclidean" balls) there is a constant c3 = c3(X, AX, a) such that, for
i = 1,... ,p,
e ((a, a, 5r/6))
µ9 w5rW < µ9 (Be
/
5r/6P))
war/6 < C3w5r/6
(10.10)
By (10.9) we have, w5r/6 = J
dy (W('), cp5r/6) 0.9
f
= min B"(a,5r/6) dy(V('), y) du,. According to Corollary 10.1 and Remark 9.3 we may apply the weak Poincare
inequality (Proposition 9.1, again with "Euclidean" balls). For a suitable constant K = ic(X, A X, a) > 1, we have from (10.10) and (10.11) when 5r.r < 6p c4r2 max w5r/6+ < C3W5r/6 < {Lg(Be (a, 5rl6))
JB(a,5r/6)
C5 (tLq,+,5,cr - tLq,+,, r/6)
e(w) dµ9
(10.12)
Part III. Maps between polyhedra
190
by Lemma 10.3 for any q E Y, with uq from (10.2). Here c4 and hence c5
depend on X, Ax, a and A. Combine that with (10.1) applied to the function w = which is weakly subharmonic by Lemma 10.2. Taking q in the essential image W(Be(a, epr)), cf. Lemma 10.4, we have for any 77 > 0
µ9({x E Be(a,-Pr) : dY(cv(x),q) 0, and hence
ess sup dy(W('), 0), its closed convex hull containing the meanvalue ipr of cp over Be(a, r), by Lemma 10.4. In the rest of the proof we accordingly fix a point
q E n co(Be(a,r))
(
O).
r>O
Following De Giorgi [Gi 1957] and Moser [Mos 19611, cf. [GT 1998, Lemma
8.23], we iterate the Harnack type inequality (10.14) j - 1 times:
j = 1, 2, ....
uq,+,Q1 r < y2juq,+,r+
(10.15)
For r > 0 let w(r), resp. we(r), denote the essential oscillation of cp over B(a,r), resp. Be (a, r), i.e., the diameter of the (bounded) essential image cp(B(a, r)), reap. o(Be(a, r)). For any y E Y, uy,+,r is the square of the maximal distance between y and points of W(Be(a, r)), and hence, for r < ro, (10.16)
uq,+,r < we(r)2 < 4uq,+,r
because q E W(Be(a, r)). From (10.15) and (10.16) we get for r < ro j = 1, 2, -
we(Qjr)
0 defined by /i° = y this leads to
we(r) < 2(
-
r awe(ro)
for r 0. Let cp : X -' Y be a locally E-minimizing map, and suppose that the essential range cp(X) is contained in a convex compact set V C Y of diameter D < a/(2vrK-) such that geodesics in V are uniquely determined by their endpoints. Then V is Holder continuous. Some limitation on the size of cp(X) is necessary, see Example 12.3.
Proof. The proof of Theorem 10.2 will be given in the more general setting where V maps into a compact geodesic space V which is Lip homeomorphic
to a closed subset of a Euclidean space (cf. Remark 9.3) such that V has diameter D < a/(2VK-) and satisfies bi-point uniqueness. We may assume that X is compact and K = 1 (after resealing the metric dy). We first establish Lemma 10.2, now for q E V and with a suitable constant factor > 0 on the right. Every geodesic triangle in V is 1-dominated, cf. Chapter 2. Consider in V a triangle yy'q with sides dy (y, q) = v,
dy (y', q) = W,
dy (y, y') = d
(all < D). For given 0 .A, A' < 1 let ya E (y, q] and y' E (y', q] be given by dy(y,ya) = Av, dy(y',y,\) = ,Vv', and write dy(yA,y,) = da. We shall estimate da from above in terms of d. Triangle comparison for the triangle qyy', in V gives, by spherical trigonometry after elimination of the angle 9 between yaya and yaq, writing dy (y, ya) = a, sin v cos da > sin(v - .1v) cos a + sin(Av) cos(v' - A'v').
10. Holder continuity of energy minimizers
193
Similarly, for the triangle qy'y, sin v' cos a >, sin(v' - A'v') cos d + sin(.A'v') cos v.
Eliminating a between these two inequalities leads to sin v sin v' cos da > sin(v - AV) sin(v' - )'v') cos d + sin(v - Av) sin(A'v') cos v + sin v' sin(Av) cos(v' - A'v').
Inserting cos d = 1 - 2 sin2( d) and similarly for cos da, and dividing by
sin v sin v', gives the following2 basic estimate (after some manipulations serving to render Theorem 9.1 and Corollary 9.2 applicable): sin D o n (da - d2)
2 sine (2 d,\) - 2 sine (2 d) < R =
4
R; .
(10.17)
i=1
Here,
R1: = -2sin2 (2d) 1-
sin(v - AV) sin(v' - \'v') l
1
sin v
sin v'
= -2sin2(1d)(Avcoty+A'v'cotv')(I+0(A2+A'2))
- sin22 D) 2d2D cot D(2A - (A - A'))(1 + O(A2 + A'2)), the functions t-1 sin(t/2) and t cot t being decreassing for 0 < t < D (< a/2); R2: = cos(Av) cos(A'v')(cos v -cos v')
(tan(Av) - tan(AV) )
sin V sin v' tan(Av) _ tan(AV) =(cosy-cosy')( (1+O(az +. ,z)); sin v' sine
R3: = (cosy -
cosv')2sin(av) sin(A'v')
sin v
sin v'
= (cosy - cosv')2O(A2 + A'2); A'y' 2 v - v' sin(Av) sin(A'v') R4 : = 2 sin 2 AV -2 - 2 sin 2
(Av - A'v')2. 2
sin v
sin v'
Part III. Maps between polyhedra
194
With U as in Definition 10.1, consider a function A E Lip,, (U), 0
-
fu 'Oe(cp)
dpy,
E Lip+ (U),
(10.19)
where c = c'D-'sin D (cf. [JK 1979, Lemma 41). In particular, cosy is weakly superharmonic and hence superharmonic on U (after correction on a null set, cf. Remark 7.4), and therefore actually on all of X. The last step in the proof of the present version of Lemma 10.2 is to pass from cosy in (10.19) to v itself and to uq = v2 = d'
q) = F o (1 - cos v),
where
F(s):=arccos2(1-s),
0<s 0 on the right. In particular, Uq = q) is subharmonic for any q E V, and so Lemma 10.3 remains in force with the same proof (now with Y replaced by V). Lemma 10.4 and hence Corollary 10.2 likewise carry over when the previous meanvalue irpA of cp over A C X is replaced by the unique point 47A of V minimizing
I(y) = J sin 2[2dy(co(x),y))dµs(x),
y E V.
A
In order to show that any I-minimizing sequence (yj) C V is Cauchy, let z E V denote the midpoint of y;yj, and write d = dy(y1, yj) and ui = dy (po(x), yi ),
U3 = dy (so(x), yj ),
u = dy (W(x), z)
for x E A. From spherical trigonometry we then have by triangle comparison,
noting that d < D < 7r/2 when we take K = 1: 2cosDsin2 Zd Y can be approximated uniformly by simplicial maps cp' relative to fine subdivisions of given triangulations of X and Y; and simplicial maps are Lipschitz. We show that cp' can be taken to be homotopic to V. Because Y is compact and has nonpositive curvature, there exists r > 0 such that every ball By (y, 2r) is convex and that triangles therein are 0, k, dominated (cf. Chapter 2). Finitely many balls By(yi, r), i = 1, cover Y. We may assume that IV* (x) - V(x)I < r
for x E X.
(Henceforth, by abuse of notation, we write Iy - y'9 for dy(y, y'), y, y' E Y.)
For given i = 1, , k and x E Ui := V- '(By (yi, r)) there is a unique constant-speed geodesic yy : I --> By(yi,2r) joining V(x) to V`(x) within By (yi, 2r). The map cp : [0,11 x Ui - By (yi, 2r) given by V(t, x) = -yz (t) is continuous because geodesic segments in the convex ball By(yi,2r) vary continuously with their endpoints, Y being compact. Step 2. Take an E-minimizing sequence of continuous W1,2-maps in H, and an L2-convergent subsequence (cpi), convergent also pointwise a.e. (This is possible by Lemma 9.2 because dy (Wi (), q) < diam Y and f X dµ9 < oo.) Each of these neaps lifts to a continuous map yoi E i.c (X,Y) between the indicated Riemannian universal covers:
X
Y
(Pi(xo) 1-Y
7tXI
xoEX
°-Y
11. Existence of energy minimizers
201
Each of the maps Bpi is equivariant with respect to the covering groups FX, ry; i.e., letting (cpi).: rx -> I'y denote the homomorphism induced by the map Bpi,
cpioy=(cpi)*(ry)o,pi
for all yEl'X.
(11.3)
Normalize these lifted maps by fixing a point xo E X and choosing an image point called cpi(xo) belonging to Try 1( I'y, independently of the choice of api(io) in the case of the minimizing sequence ((pi); so choose all (ij. = t01s. In view of (11.3) the pointwise limit cp = limi-,,. cpi satisfies
00-Y=01.('Y)0'
a.e. in k for ally E rx.
(11.4)
Now rX acts isometrically and simpliciallyon X. We shall call a compact set_F C X a fundamental domain of rX if 8F has measure 0 and each point of X is Fx-equivalent either to exactly one point of the interior of F or to at least one point of F. Because X is compact, such a fundamental domain F can clearly be assembled as a suitable union of top-dimensional simplexes of k, possibly after subdivision. Furthermore, F is contained in the interior U of a suitable union G of finitely many (say N_) Fx-translates of F. Denote by A the class of all maps in WJ., (X, Y) which are equivariant, as in (11.4). Then the above map cp = lim 0 so that every ball By (y, 3r) is convex and its triangles are 0-dominated. Again, let By (yi, r), i = 1, , k, cover Y, and write Ui = W-' (BY (yi, r)),
i = 1, ... , k.
(11.5)
In the beginning of the proof of Theorem 11.1 we found a map gyp' of class Lip(X, Y) such that
IW*(x) - s(x)I < r,
x E X.
(11.6)
It remains to replace gyp" by a map u)` E C,,(X,Y) such that IV,* (x) - s(x)I < 2r,
x E X;
(11.7)
11. Existence of energy minimizers
207
indeed, such a map 7P* will be homotopic to W relative to CO (X, Y) by the argument in Step 1 of the proof of Theorem 11.1. For x E X write
l;(x) = min dX(x,x'). x'EbX
Choose 6 > 0 so that, for x, x' E X, IV)(x') - O(x)I,
I
"(x') - W*(x)I < r when dX(x,x') < 36,
(11.8)
and, for x E X, Iv(x) - O(x) I < r when fi(x) < 36.
(11.9)
Now define a continuous map 1b* : X -+ Y by z/,*(x) = (1 - 6-11;(x)),O(x) +
for £(x) 26 we have, by (9.2), ee(l/i*)(x) = ee(W*)(x) because 26 - 6 on BX(x,e) and hence Vi* = cp* there. Because 'p* E Lip (X, Y) has finite energy, to show that also 7/i* has finite energy amounts, by Definition 9.1, to proving that
limsupJ e-.o
e,,(i/5*)dµy < oo.
(11.11)
And to verify this, fix x E X with e(x) Y is the same as a continuous locally E-minimizing map.
Proof. Suppose that cp is harmonic. For given a E X let U D a and V be as in Definition 12.1. We may assume that a is a vertex of X, that the open star stX (a) has compact closure S in U, and that stX (a) = S°, i.e., bS = OS, cf. Remark 4.2. By Theorem 11.3, WSP1'2(S, Y) has an E-minimizer cp*. Let cpj : S -+ Y be the geodesic homotopy joining Wo = cp to cpl = cp*, cf. Proposition 11.1. Because cp(S) and cp*(S) are compact, and geodesics in Y vary continuously with their endpoints, it follows from w(S) C w(U) C V that cp* (S) C V for small t, and hence
E(cp) < E(tpt) < (1 - t)E(cp) + tE(cp*), by Definition 12.1 and Proposition 11.2. This implies E(W) < E(cp*) < E(zP) for any map ?P E W,,''2(S,Y), and so cp is indeed locally E-minimizing. 0
Remark 12.1. If instead curv Y < K for some constant K > 0, a similar result is established in the same way in the setting of Theorem 11.4 for maps cp of X into a closed ball of radius < ir/(2VK-) satisfying radial uniqueness. Say that a geodesic space Y has locally upper bounded curvature if Y can be covered by convex subdomains of upper bounded curvature. For harmonicity of continuous maps into such a geodesic space Y, a certain infinitesimal type of bi-local E-minimization suffices:
12. Harmonic maps. Totally geodesic maps
219
Lemma 12.2. Let Y be a locally compact geodesic space with locally upper bounded curvature. A continuous map cp E W"2 (X, Y) is harmonic if X can be covered by relatively compact subdomains U for each of which there
is a compact convex set V C Y satisfying bi-point uniqueness such that W(U) C V and, when E(cp) < oc, lim sup t-.o+ t
E(co)) >, 0
(12.1)
for any geodesic homotopy (cpt)tE J Within W,Oc (X, V) with cp1 = cp in X \ U and cpo = W.
If E(W) = oo, replace co and cot in (12.1) by their restrictions to U.
Proof. By continuity of cp we may suppose that cure Y < K and, after rescaling, that K = 1. Fix P E X, and write W(p) = q. Choose a compact convex ball B = By (q, R) of radius R < 7r/2 so that B satisfies bi-point uniqueness (see Chapter 2). Then B is simply connected (cf. the text preceding Theorem 11.4).
Suppose first that cp is harmonic, and choose U and V as in Definition 12.1. By diminishing R and U we arrange that B C V and co(U) C B. Clearly, (12.1) holds for any geodesic homotopy (cf. Proposition 11.1) as stated in the lemma (now with V replaced by B). Conversely, with U and V as in the lemma and with a continuous map E W11.1,2 (X, Y) such that i(U) C V and z[' = cp off a compact subset of U, there is a geodesic homotopy (cpt)tE J within Wio2 (U, V) joining coo = VIU to col = ?GNU, cf. the proof of Proposition 11.1, applicable to B in place of Y in
view of [AB 1990, Theorems 3 and 6]. Then E(cot) < (1 - t)E(coo) + tE(coi),
0 < t < 1,
first for t = 1 by Serbinowski's inequality (11.15). By repeated application the same follows for any dyadic number t E [0, 1]; and finally it holds for any t E [0,1] by lower semicontinuity of the energy functional (Lemma 9.1 and Remark 9.6) because the distance function dy(cpt(), co()) converges locally uniformly, and hence in L 102C(X) to 0 on X as t - 0. Hence,
0 < limsup1(E(cot) - E((po)) < E(cpl) - E(co), t-,o+ t and so indeed E(cpIU) < E(ii)iu).
0
220
Part III. Maps between polyhedra
Remark 12.2. There is equality in (12.1) when Y is a Riemannian manifold (see Theorem 12.1 below), but not always for polyhedral Y, as shown by the following example. Take X = R, Y = the union of the x-axis and the positive y-axis in the xy-plane (Cornea's tripod, cf. Remark 5.6). The inclusion map cp : X -+ Y given by cp(x) = (x, 0) is clearly locally E-minimizing and hence harmonic (this also follows from Lemma 12.5, cp being a speed 1 geodesic). For any a > 0 consider the geodesic homotopy (Vt)t3o within WI.. (X, Y), given by wt (-x) = -Wt(x) and by wt (x) = (x,0) (= W(x)) for lxi >, a, while
for0<x E(cpjU), by Definition 12.1. Inserting this in (12.5) and dividing by t leads for t - 0 to 6E(cp) = 0 in view of (12.7) and the linearity of 6E(cp). By (12.6) this implies (12.3) and therefore (12.2).
12. Harmonic maps. Totally geodesic maps
227
Conversely, suppose cp is weakly harmonic. This time choose U and V as in Definition 12.2, again so that p E U, V C= W, and rl(V) is convex. Next, choose a ball B centred at cp(p) and having bi-point uniqueness; and a domain U' with p E U' C= cp 1(B). For any geodesic homotopy (Vt)tEl within Woe (U', B) with WO = cpIU, it follows from (12.3) that 6E(W) = 0, and hence from Lemma 12.3 that
lim 1(E(ct) - E(cp)) = 0 to+ t on account of (12.5) and (12.7) applied to A = At := cpt - c'o, which leads to
R(cp, At) = O(t2). Indeed, At(x) = O(t) and IVeAt(x)Ie = O(t), uniformly for x E U', by smoothness of the exponential map exp : T(N) -4 N and continuity of cp and ,0. (b) e==> (c). For any continuous map cp E W11, (X, N), for any open sets
V C N and U C W-1(V) with U of compact closure in X; and for any functions v E C2(V) and A E W,1,2 (U) n L°°(U), we have by Remark 9.7,
- f (VA, V(v o cp)) f([(Ov) o SOJVA, VV-) - j(V(A [(a. V) o cP]), VP-) + j A [(a°apv) o c01(VW', Vi)
(12.9)
Inserting (12.8) in the last integral gives
-f (VA,V(vo,p))
_ - j(V(A [(Okv) o S])V) + fUA(v0 o S)(VSO°,V)
(12.10)
+ f A [(Okv) o w1(I'a0 o cp)(Ocp°, vi). U
Suppose first that the continuous map cp is weakly harmonic. For any chart 77 : V -+ R' on N and any function A E Lip°(,p 1(V)), the first and
the third integral on the right of (12.10) are then equal according to (12.2) applied with A [(Okv) o 9] E W, ','(U) n L°°(U), k = 1, , n, in place of A E W,:1, 2(U) n L- (U).
If v E C2(V) is convex, (12.10) therefore shows that f (VA, V(v o cp)) 2t2. Dividing by t2 and letting t - 0 leads to (v o -t)-(0) >, 2. By (12.8) and the differential equation for the geodesic y of speed 1: V r12 = (hQp o y)yQ Yp = 1, we have (vap °7)1QTp % 2(hap o'y)''yQryp
at t = 0.
Inserting -y(O) = y this establishes (12.15) because we may take y'Q(0) = Q (a = 1, ... , n) when (hQp o 'y)'S,c5 = 1. As in the proof of (b) (c) in Theorem 12.1, we have from (12.2) and (12.10) (cf. Remark 12.6) - /x (VA, V(v ° (G)) _ Lip,(X)
for A E
x
A(vQp °
Inserting (12.15) (evaluated at y = cp(x), x E X) gives
- f x(VA, V (v O 1P)) > 2 fx \(hQp o'p)(V, V
)=2
Jx
\e(W)
(12.16)
by Definition 9.2, using the representation of a positive matrix (in this case
vap - 2h0p) in the paragraph following (12.10). And (12.16) extends to A E W0'2(X) f1 L°°(X) because such A can be approximated in W01, 2 (X) by a uniformly bounded sequence of functions of class Lip,(X), by truncation. From Lemma 10.2 (b) thus established in the present setting follows that
v o W = A ((p(.), q) is weakly subharmonic, and the rest of the proof of Theorem 10.1 carries over immediately.
0
Now pass to the case where N is a smooth Riemannian manifold with sectional curvature < K for some 0 < K < oo. Proposition 12.2. Let W : X -p N be a weakly harmonic map, and suppose that the essential range W(X) is contained in an open convex set V C N of diameter D < 7r/(2v'rK-) such that V lies within normal range of each of its points. Then ,p is Holder continuous.
Proof. We may suppose that X is compact and K = 1. Because V lies within normal range of its points, geodesics in V are uniquely determined by
Part III. Maps between polyhedra
232
their endpoints, and vary continuously with them, V being locally compact. Every triangle in V has perimeter < 27r and is consequently K-dominated with K = 1 [A 1951; 19571, cf. Chapter 2. For each q E V take normal coordinates in V centred at q, and write u(y) = dy(y, q),
v(y)=u(y)2,
y E V.
Consider a geodesic segment y : [-s, e] -i V with y(0) = y 36 q. Triangle comparison of y(t)y(-t)q with median 'y(0)q gives, writing u(t) for (uo-y)(t),
cosu(t) +cosu(-t) - 2cosu(0)cost S 0. Subtracting 2 cos u(0) (1-cost) on both sides, and dividing by t2, we obtain
for t - 0
d Zt
cos
v(t) It_o < -cosu(0),
and so v(0)
2u(o) + 1 tan u(0) 2u(0)
1
(0)
- tan u(0) )
x,(0)2
> 2c
with c = D/ tan D, noting that D < sr/2. Thus (v o y)"(0) > 2c, and hence Hv - 2ch is positive semidefinite, as in the proof of Proposition 12.1 (also if y = q). Again, this is found in [HKW 1977, (2.2)] with the same constant c. The proof is completed in quite the same way as that of Proposition 12.1, cf. also the last three paragraphs of the proof of Theorem 10.2. 0 Corollary 12.2. Every harmonic map W of (X, g) into an arbitrary smooth Riemannian manifold (N, h) without boundary is Holder continuous.
Remark 12.7. If N has even dimension n and is complete with sectional curvature satisfying 0 < curv N < K, then every ball in N of radius R < it/(4v) is convex and within normal range of its points; similarly for odd n provided N is simply connected and K/4 < curv Y 2 this is optimal, as shown by the following example. When n = 2, no restriction on the size of V(X) is needed, see [HW 1975, p. 67].
Example 12.3. [HKW 1977, §6]. Let S"-1, resp. S", be the unit sphere in R", resp. R'+1, n > 2. Let p : Ill" \ {0} -+ S"-1 denote central projection from 0 onto S"-1; and i : S"-1 -+ S" the inclusion map, which is clearly a totally geodesic map (Definition 12.3 below). It is known that p is a usual harmonic map (even a harmonic morphism, cf. Definition 13.1); this follows from [BE 1981, §5] because p is horizontally homothetic and has minimal fibres (straight lines in this case). The composite map i o p : R" \ {0} -+ S" is therefore harmonic, by Theorem 12.2 and Theorem 12.1, (a) e=> (c).
For n > 3, i o p is clearly of class WIo,, (R", S"), and the singularity at 0 is therefore removable, so that i o p extends to a weakly harmonic map E Woo, (R°, S"), according to [EP 1984]. The essential range cp(R") equals S"-1, which is contained in a closed hemisphere in S", i.e., in a closed ball in S" of radius it/2 = it/(2vlK-); but does not agree a.e. with a continuous map because of the essential discontinuity at 0.
Query. Can the assumption on V(X) in Proposition 12.2 be weakened to yp(X) being contained in an open ball V in N of radius < 7r/(2VK-) such that V lies within normal range of its points (cf. Remark 12.8)? What about the particular case when N is the standard n-sphere of radius 1/v/'K--? Similarly for Theorem 10.2?
Totally geodesic maps We extend to suitable geodesic spaces X, Y the pull-back characterization of totally geodesic maps (p : X -+ Y, due to Ishihara [Ish 1979, Theorem 3.2] for the case of maps between smooth Riemannian manifolds (without boundary). In the rest of this chapter, geodesics ry : I -+ X (with I an interval on R or a standard circle) are understood to have constant speed. In the present section we do not require that geodesic spaces be locally compact. A geodesic space Y of locally upper bounded curvature (as defined in the text preceding Lemma 12.2) can be covered by convex subdomains V with curv V , cosm cos i s > 1(cos a+ cosb),
and this implies indeed m 5 (a + b) because cosine is concave on 10, 7r/21. (Cf. [AB 1996, Theorem 4.3],zwhere y is fixed.)
Definition 12.3. A continuous map (p : X - Y between geodesic spaces (X, dX) and (Y, dy) is said to be totally geodesic if (p o y : I -+ Y is a (constant-speed) geodesic in Y for every geodesic ry : I -p X in X.
Remark 12.9. (a) In this definition it suffices to consider (minimizing) geodesic segments ry : [0,1] -+ X, even with images contained in prescribed open sets with the union X. (b) The composite of two totally geodesic maps V : X -> Y and i,b : Y --+ Z is a totally geodesic map ip o (p : X --' Z.
(c) If X and Y have locally upper bounded curvature, the inverse of a totally geodesic homeomorphism (p : X -+ Y is a totally geodesic homeomorphism (P-1 : Y -+ X ([ES 1964, §5] for the case of manifolds). To
see this, cover X and Y by relatively compact convex domains U C X and V C Y such that (p(U) C V and that geodesics in U, resp. V, are uniquely determined by their endpoints. For a geodesic r : [0,1] -+ V(U) write (p'1(i(0)) = Co, (p' 1(,i(1)) = l;l; and let l; : [0,1] -' U be the unique geodesic in U such that l;(0) = to, g(1) = t;l. Then (P o C : [0,1] -+ V is the unique geodesic in V with endpoints 77(0),77(l); and so (p o C =17.
Remark 12.10. In the case of smooth Riemannian manifolds Mm, N" without boundary it is known that a C2-map (p : M - N is totally geodesic
iff,fori,j=1, ((pk)ij+(r';
o(p)at(p`YajYo
=o,
in terms of local coordinates x' in M and y° in N, whereby cpa = y* o (p, and ((pk),j denote the covariant second partials of (pk.
In order to obtain a characterization of totally geodesic maps we first consider maps ry : I - Y of an interval I C R into a geodesic space (Y, dy) of locally upper bounded curvature.
12. Harmonic maps. Totally geodesic maps
235
Lemma 12.4. A continuous map ry of an interval I into a geodesic space Y of locally upper bounded curvature is a constant-speed geodesic if y pulls convex functions v : V -+ JR on open sets V C Y back to convex functions v o y on each component of y-1(V) C I. The proof will show that it suffices to verify this pull-back condition for convex functions v : V -+ ]R with V convex and v of the form v = dy (., y), y E V.
Proof. If y : I Y is a (constant-speed) geodesic then v o y : J -+ ]R is convex for any convex function v : V --+ R, J being any component of y-1(V) C I; this expresses the definition of v being convex. In the opposite direction we may assume. by continuity of cp, that the entire space Y has the properties of a convex subdomain V stated in the
second paragraph of the present section. To prove that y : I --+ 1R is a geodesic, let t1, t2 E I, t1 < t2, and write yi = y(ti), i = 1, 2, and further, on Y,
vi = dr
y, ),
v = v1 + v2.
Then vi is convex, and so are therefore vi o y and v o y, by hypothesis. The
triangle inequality shows, for any y E Y, that v(y) > b := dy(yi,y2), and that v(y) = d 4=* y E (y1, y2],
the unique geodesic segment joining yi and y2. In particular, (voy)(ti) = S, hence v o y = 6 in [t1, t2], by convexity of v o y > 6. This shows that y maps (t1, t2] into (even onto) the geodesic segment (yi, y2] in Y. The map 71 [t, t2l is affine-linear (in terms of Riemannian path length on (yi, y2]) because the convex functions vi o y = y(ti)) on [t1, t2] are affine linear, their sum v o y being constant = 6 on R1, t2]. Thus y is indeed a (constant-speed) geodesic. 0
Remark 12.11. The curvature condition in the above lemma is clearly not needed for the "only if part", but cannot be dropped in the "if part". As an example, let Y be a standard cone of revolution such that Y, when cut along a generator, unfolds as the upper halfplane {(yi, y2} E R2 : y2 > 01, say. To the line t F-+ (t, e) (e > 0) in J2 there corresponds on the cone Y a geodesic ye: 1R -+ Y, converging as e --+ 0 to the "limit geodesic" y : R -+ Y formed by the generator g (directed towards the vertex of Y) followed by -g;
and this path y is clearly not a geodesic. But y does pull germs of convex functions in Y back to germs of convex functions in R. This shows that Y
Part III. Maps between polyhedra
236
does not have locally upper bounded curvature (this is also clear from the existence of two geodesic segments joining points of any standard cone of revolution Y on opposite generators). Every standard cone of revolution has nonnegative curvature (as mentioned in Example 4.3).
Theorem 12.2. Let (X, dX) and (Y, dy) be geodesic spaces, and suppose that Y has locally upper bounded curvature. A continuous map cp : X -' Y is then totally geodesic if and only if V pulls convex functions on open sets V C Y back to convex functions on cp-1(V). This theorem is proved by a straightforward application of Lemma 12.4. Again it suffices to take v = (y E V) on convex domains V, as in the beginning of the present section.
Corollary 12.3. Let (X, g) be a complete admissible Riemannian polyhedron without boundary and with continuous and simplex-Wise smooth Riemannian metric g. Let (N, h) be a complete Riemannian manifold without 1,2 boundary. Then every totally geodesic map W E W10C (X, N) is harmonic. Indeed, X and N are geodesic spaces, and N has locally upper bounded curvature. By Theorem 12.2 and Proposition 7.9,
(q - 26)e,
and so from (12.18), for any f E CC(I°, [0,1]) with f (t) = 1 for t E
l
(y) f dt >
(y) dt > q - 2b. a lE
By (12.21) this implies for n -+ oo, hence e = (/3 - a)/n - 0: L(y) >, limsup
r0
Ja
1,(-y) dt > q - 2b.
By varying q we infer that indeed
L(y) > C(yll.,01) - 26.
(12.25)
For any q* < ,C(-y), (12.24) holds with q replaced by q*, now for any < tm = b of I = [a, bJ. We may
sufficiently fine subdivision a = to < t1
b - b; and by (12.23) that dy(a,ti) < S, we obtain dy(tm-1i b) < b. Applying the above to a = ti, 6 = m-1
L('i11.>RI) > E dy('y(ti-i),7(ti)) > q* - 25, i=2
and so, by (12.25), L(7) > q` - 45. Making first q' -+ £(-y) and next 5 -+ 0, we conclude that indeed L(7) > G(7). In proving the opposite inequality C(7) > L(,y), we may suppose that G(7) < oo, i.e., 7 is rectifiable. Consider the increasing bounded continuous function
tEI=[a,b],
A(t) = C(-Yl[..,tl) ,
with the approximate length density lE(A)(t) _
\(t - E)
.1(t + 6) 2e
> 1-(7)(t),
(12.26)
cf. (12.18). By the consistency result [KS 1993, p. 582], the 1-energy L(A) of the function A equals the usual 1-energy Var A = A(b) - A(a) = C(7).
Applying (12.21) to both maps 7 and A, and inserting (12.26), leads to G(7) > L(-y).
Having thus proved that L(7) from (12.21) is the usual length of -y, the remaining former inequality (12.22) is obvious. Replacing [a, b] by a subinterval [t, t'] we derive from (12.22) the uniform Holder continuity of 7: d2y (7(t),7(t')) E(7) according to Definition 12.1. Suppose therefore that 7'(t') V for some t' E U'; then dy(7'(to ± 5'),7'(t')) > 0/2 and e.g., t' > to. Applying (12.22) to the interval [to - 6', t'] of length b' gives
E(7') %
(P/2)2/S' > E(7)
for sufficiently small 6'. (c)
(a). From (c) it follows that every point to E ]a, b[ has a neigh-
bourhood U = ]to - 6, to + b[ Cc ]a, b[ such that 7[U is E-minimizing for fixed endpoints, and 7u is therefore a constant-speed geodesic segment. When to varies, this shows that 7 is itself a geodesic, and has constant speed (because the speed must be the same in overlapping time intervals).
Corollary 12.4. In a geodesic space (Y, dy ), a continuous path 7 : I -+ Y from an open interval or a standard circle I is a constant-speed geodesic if 7 is harmonic, or equivalently if y is locally E-minimizing.
Note that the "only if part" breaks down if I has a boundary, e.g., I = [a, b], the only harmonic (or continuous locally E-minimizing) maps I -+ Y being then the constant maps; this should be compared with Lemma 12.5
12. Harmonic maps. Totally geodesic maps
241
(b), in which `harmonic' is replaced by `E-minimizing for fixed endpoints', which is weaker in the presence of a boundary.
Jensen's inequality for maps This inequality, included here primarily on account of its use in Proposition 12.4, is stated without proof in [J 1994, Lemma 3.1]. Local compactness of geodesic spaces is not needed in this section. We first characterize the (constant-speed) geodesics on the product of two geodesic spaces, invoking Lemma 12.5:
Lemma 12.6. Let (X, dx) and (Y, dy) be geodesic spaces, and define a metric dz on Z = X x Y by d2
2 ((x, y), (x', y)) = dX (x, x') + dY (y, y)
Then (a) (Z, dz) is a geodesic space.
(b) A path I E) t H z(t) = (x(t),y(t)) E Z is a constant-speed geodesic i5 t ' x(t) and t --+ y(t) are such. (c) If X and Y have globally nonpositive curvature then so has Z. Proof. With I = (0, 1] let 19 t H X(t) E X and I -3 t H y(t) E Y be continuous paths of finite energy. Then the path 13) t '-+ z(t) := (x(t),y(t)) E Z has (12.28)
eE(z(.)) = e, (x(.)) + eE(y(.)),
cf. (12.17), and hence E(z(.)) < E(x(.)) + E(y(.)) < oo, cf. (12.20). By [KS 1993, Theorem 1.5.1] applied to the Riemannian domain ]0, 1( it therefore follows from (12.28) that actually
E(z(.)) = E(x(.)) + E(y(.)).
(12.29)
Ad (a). Given two points z; = (x2, y2) E Z, i = 0,1, let I D t - x(t) E X be the (constant-speed) geodesic segment in X with endpoints x(0) = xo, x(1) = x1, and similarly for 19 t H y(t) E Y. By Lemma 12.5 the continuous path 13) t H z(t) = (x(t), y(t)) E Z of finite energy is rectifiable, and its length L(z(.)) satisfies, by (12.29), E(z(.)) = E(x(.)) + E(y(.)) = 2
= dx (xo, xi) + d2 (yo, yi) = dz (zo, zi ),
L(y(.))2
Part III. Maps between polyhedra
242
so that (Z, dz) is indeed a geodesic space. And z(.) is a geodesic in X, by Lemma 12.5, (e)
:
(a).
Ad (b). The "if part" was shown in the proof of (a). Conversely, if z(.) = is a geodesic in Z then the continuous paths and in X and Y, respectively, clearly have finite energy, and by (12.22), (12.29),
dz(xo,xi)+d2 (yo,yi) < E(x(.))+E(y(.)) = E(z(.)) =d2z(zo,zi), (e) in Lemma 12.5. This shows that the equalthe last equation by (a) ity sign prevails in the four inequalities d2x(xo,xi) < are geodesics, again by and so x(.) and d2Y(yo, yi) < Lemma 12.5.
Ad (c). Given triples zi = (xi, yi) E Z, i = 1, 2, 3, and comparison triples 1, and yi in R2, the triple z, _ (ii, yi) in R4 lies in a 2-dimensional Euclidean (affine) subspace V of R4 and is clearly a comparison triple for (zi) satisfying the triangle comparison inequality; thus Z has globally nonpositive curvature.
Remark 12.12. There is a well known, quite elementary analogue of (a) and (b) in Lemma 12.6, in which the exponent 2 is replaced by 1 in the definition of the distance on Z, and energy is replaced by length in (12.29). However, Lemma 12.6 as above, depending on Lemma 12.5, is indispensable for the proof of Jensen's inequality below.
Proposition 12.3. (Jensen's inequality.) Let (Y,dy) be a simply connected, complete geodesic space of nonpositive curvature, v a lower semicontinuous convex function on Y, and µ a probability measure on Y. Then
v(y,,) < fvdp when y,, denotes the centre of mass of µ, i.e., the unique point of Y minimizing
I(y)
f d2y(y,0)d/I(rl),
y E Y.
Proof. The centre of mass y, is the same as the meanvalue of the identity map id : Y --a Y, and so y, lies in the closed convex hull of supp p, cf. Lemma 10.4.
12. Harmonic maps. Totally geodesic maps
243
Recall that Y has globally nonpositive curvature (cf. Chapter 2). By Lemma 12.6 the product space Z = ]l( x Y is therefore a complete geodesic space of globally nonpositive curvature (hence simply connected), with metric dZ defined by dz ((t, t'), (y, y')) = (t - t')2 + dr(y,?! ) The convexity and lower semicontinuity of v : Y
R amounts to its epigraph
G={(t,y):yEY,t>, v(y)} being a convex closed subset of Z. The image v of a under y " (v(y), y) is a probability measure on G. The centre of mass of v is (f v dµ, y,,), as shown presently, and it belongs to G, so that indeed v(y,,) < f v dµ. For any (t, y) E R x Y we have in fact
fZd2Z
=f
((t, y), (T, i)) dv(T, n)
dz ((t, y), (V(77),77)) dgrl)
= f (t -
v(,q))2dµ(7])
_ (t- f vdµ)2+ Y
+ j dY(y, n)da(i) v2da
f vdµ)2+ f dY(y,r1)d/,(r!), Y
Y
which is smallest for t = f v du and y = y,,.
0
Harmonic maps from a 1-dimensional Riemannian polyhedron We bring a version of Ishihara's characterization of harmonic maps, cf. Theorem 12.1, (a) (c), for the case of a 1-dimensional Riemannian polyhedral domain (X, g). Every map p E (X, Y) of X into a geodesic space (Y, dY) has then a continuous version (Remark 9.2), and even a locally uniformly Holder continuous version with exponent an easy consequence of Lemma 12.5 applied to the 1-simplexes of X containing a given vertex of X.
244
Part III. Maps between polyhedra
Proposition 12.4. Let (X, g) be a 1-dimensional Riemannian polyhedron and (Y, dy) a locally compact geodesic space of nonpositive curvature. For a continuous map cp E Y) the following are equivalent: (a) cp is harmonic (Definition 12.1), (b) W is locally E-minimizing (Definition 10.1), (c) cp pulls convex functions v on open sets V C Y back to subharmonic functions v o cp on o-1(V).
The proof will show that it suffices to consider convex functions v on Y of the form v(y) = dy(y,q), q E Y. Proof. Because cp is continuous we may assume that X and Y are compact and that Y has globally nonpositive curvature, so that Y is simply connected. (Cf. the text at the beginning of Totally geodesic maps; and Chapter 2, Geodesic spaces.) (a) (b) follows therefore from Lemma 12.1. (b) (c). Suppose that cp is locally E-minimizing, and let v be a
convex function on an open set V C Y. By Lemma 12.5, (c)' (a), and by Lemma 12.4, v o cp is convex and hence subharmonic in the part in V-1 M of each open 1-simplex of X. It remains to show that v o cp is subharmonic at each vertex xo E W'1(V) of X in a given triangulation; i.e., in terms of the 1-simplexes sl, , sk of X having xo as a vertex, k
(12.30)
Di(v o w) (xo) % 0, i=1
where Di(v o co)(xo) denotes the derivative along si at xo with respect to path length from xo, see Example 8.1. We may assume that the open ball Bx(xo,r)° C v-1(V)flstx(xo) can serve as U in Definition 10.1. Let xi E si have dx (xo, xi) = r, i = 1, , k. Writing yi = v(xi) for i = 0, 1, , k, we have from Lemma 12.5, (d) (e), applied to each s9,
E(vju) = r
(12.31)
dy(yo,yi), :=1
and we conclude by varying yo in Y that yo must be the centre of mass of + cy,, )). Y1, , yk (i.e., of the measure µ = k-1(ey, + The constant speed of the geodesic is dy(yo, yi)/r, and so Di(v o P)(xo) =
aidy(yo,yi)
r
= lim v(yi) - v(yo) r--o
r
(12.32)
12. Harmonic maps. Totally geodesic maps
245
where ai denotes the derivative of the convex function vl1y0,,,j at yo with respect to path length dy(yo, y) on [yo, yi]. Adding over i = 1, , k leads to (12.30) in view of Jensen's inequality (Proposition 12.3), according to which v(yo) , Ed2y(yo,yt). For any i = 1, , k let Bt denote the angle at yo = 0 in the Euclidean comparison triangle ?/oyi4 for yoyiq Then dy(yo, q) = 141, dy(yo, yi) = 1pi1; and
Edgy(q,yi)=EI4-yi12=E(14I2+IyiI2-21411& 1cos 9i) = kd4 (yo, q) + E d2y (yo, yi) - 2dy (yo, q) E dy (yo, yi) cos Bt. It suffices therefore to establish the inequality k
E dy (yo, yi) cos Oi -oo and v # oo. To see that v is lower semicontinuous, let Yo E V and a < v(yo). According to [CC 1965, Lemma 3.1] the set {v > a} = cp({v o w > a}) is indeed a neighbourhood of yo because {v o cp > a} contains the compact fibre cp-1(yo), hence also its components which are likewise compact.
Consider now a regular set W in Y such that yo E W, W C V. (If X is compact we further choose W small enough so that cp 1(W) X ; this is possible because W-1(yo) 54 X, cw being nonconstant.) For any f E C(OW) with f 5 vjew denote by h the continuous extension of Hf (Definition 2.1) to W by h = f on W. Then hoop is harmonic on W-1(W) (Definition 13.1) and continuous on w-1(W) (C cp-1(W)); and 0
on 19v-1(W) (C W- 1 (09w))
because h = f < v on OW. Since cp-1(W) 0 X (connected), OW-'(W) 36 0, and we conclude from the boundary minimum principle (cf. Chapter 2) that voce> h o V in V-1(W), and hence
v > h = Hx'
in W = cp((p 1(W)).
Because h(y) = f f dwx' for y E W (Definition 2.1), it follows by taking supremum over all continuous f
J v dwy
for every y E W,
and so v is indeed superharmonic on V. If v o
2) be the remaining 1-simplexes of Y containing yo. Define a harmonic function v on V (cf. Example 8.1) by (k - 1)dy(y,yo)
v(y) -
{ -dy(y,yo)
for x E t1 n V
forxEtjnV, 2<j 2 because b(Y) = 0. We therefore have on V = sty (yo) the harmonic function v from (13.2). As before, let U be a connected open neighbourhood of x0 such that U C cp-1(V). We shall prove that U C X \A. If W were constant in some neighbourhood U* of a point x' E U then v o cp would be constant in U* and hence in U, by Proposition 5.4. It would therefore follow that v o W = v(yo) = 0 in U because v-1(0) = {yo}. But this would contradict x0 E X \ A. We conclude that every point x' E U is indeed in X \ A. 0
Remark 13.1. In case (ii) with n = 1, Y cannot be allowed to have a boundary. (For example: W : IR - [0, oo] given by V(x) = max{x, 0}.) See Remark 5.6 as to the need for X to be circuit, and Remark 5.7 for g to be Lipschitz continuous if m > 2, even when Y = R.
13. Harmonic morphisms
251
Harmonic morphiams into Riemannian manifolds In this section we shall consider maps from an admissible Riemannian polyhedron (X, g) to a smooth Riemannian n-manifold (N, h) without boundary. We study the relation between harmonic morphisms and harmonic maps of X into N and extend known results from [Flag 1978] and [Ish 1979] for maps between smooth Riemannian manifolds. A continuous map w : X N is of class W, (X, N) if the components !p° = y° o cp of tw in terms of local coordinates (y°) in N are of class W (X) (Definition 9.2). For every v E C'(N) we then have v o W E Woe (X), by Remark 9.7.
o
Definition 13.2. A continuous map sp : X - N of class WW (X, N) is called horizontally weakly conformal if there exists a scalar A, defined a.e.
in X, such that (V(v o (p),V(w o vp)) = A[(VNV, VNW) o gyp]
a.e. in X
(13.3)
for every pair of functions v, w E C' (N).
Henceforth in this chapter the gradient operator V without a subscript always refers to the domain space (X, g). By polarization, it suffices in Definition 13.2 to consider pairs v, w with
v = w. The property of horizontal weak conformality is a local one, and (13.3) reads, in terms of local coordinates (y°) in N, (Vsp°, VvO) = A(h°p o,p)
a.e. in X
(13.4)
for a = Q this shows that A is uniquely determined and > 0 a.e., and that A E LI,C(X) because VWk E L10C (X). A is called the
dilation (or dilatation) of W. The term `horizontally weakly conformal' comes from the case where also X is a smooth Riemannian manifold without boundary and W E Cl (X, N), for then Definition 13.2 is equivalent to dsp(x) : T,, (X) - T,,(,) (N) having, at each point x E X where do(x) # 0, a conformal restriction to the horizontal space H. at x, H. being the orthogonal complement of the kernel of dcp(x) within Tx (X); Hx is spanned by Vcpl (x), , Vo"(x), cf. [Fug 1978, p. 120]. A continuous map cp E Wig (X, N) is harmonic (Definition 12.1) if W is weakly harmonic (Definition 12.2), i.e., in charts on N, fx(VO,V(pk)dµ9
= J (r, o w)(Vw°,V
(13.5)
Part in. Maps between polyhedra
252
k = 1,
, n, for every E LipC(X), and the same then also holds for any (X) fl L- (X). (See Theorem 12.1.) According to [GW 1975], N admits local coordinates (yk) which are harmonic functions. For local questions we may therefore assume that N admits a diffeomorphism y = (yl, , yn) onto an open set in R such that each yk is harmonic. zb E
W01,2
Lemma 13.1. Let cp : X -+ N be horizontally weakly conformal with dilation .1. Then cP is a harmonic map if the components cpk = yk oip in terms , n, are harmonic functions. of harmonic local coordinates yk, k = 1,
Proof. The identity map id : N -+ N is harmonic. This follows e.g., from Theorem 12.1 because every convex function on N is subharmonic, see [GW 1973]. For a direct proof, see [Fug 1978, p. 123]. Taking id : N - N for cp in (13.5) we obtain in terms of harmonic local coordinates, i.e., ANyk = 0: h013
=0,
Inserting (13.4) in (13.5) we find that cP is a harmonic map if
f
(V O, °Wk)
=
f o r every t E Lip°(X), k E {1,
[(h«ar«a)
f 0A ,
W]
o
=0
n}; that is, if each component cpk is
harmonic.
Theorem 13.2. For a continuous map cp : X - N of class WWa, (X, N) the following are equivalent:
(a) cp is a harmonic morphism (Definition 13.1). (b) cp is a horizontally weakly conformal, harmonic map (Definitions 13.2 and 12.1).
(c) There is a scalar )1 E L' 10_(X) such that fX(OW, V(v O 0)) =
f
wI\ [(ANV) o 0]
for every v E C2(N) and 0 E Lip, (X) (or
E
W01,2
(x) fl L- (X)).
13. Harmonic morphisms
253
In the affirmative case, A from (c) equals a.e. the dilation A of W (as a horizontally weakly conformal map).
In the case where also X is a smooth Riemannian manifold (without boundary) and o E C2(X, N), this theorem was obtained in [Fug 1978] and independently in [Ish 19791; then (c) reads as follows: AX (V 0 p) = A [(ANV) o (a],
v E C2 (N).
The target manifold N cannot be replaced in Theorem 13.2 by an admissible Riemannian polyhedron, not even by a smooth Riemannian manifold with boundary (see Remarks 13.2 and 13.3 below).
Proof of Theorem 13.2. A constant map satisfies (a), (b) and (c), the last with A = 0. Each of the properties (a), (b), (c) is a local one. (For (c), this is seen by multiplying z/i E LipjX) with the functions from a suitable partition of unity.) We may therefore assume that cp is nonconstant and that the whole of N is diffeomorphic with a domain in R1. In terms of the corresponding coordinates yl, , y" on N we have for any v E C2(N) (writing 8a for 8/8y", etc.)
ANV = h'R8a8/jv+ (ONy")8av,
(13.6)
and for any ' E Lip,(X), by partial integration (as in (12.9)),
-f (ov, v(V' W)) (13.7)
- fX (v
[(8av) o p]), pip") + fX zG [(aaOQv) o'P](V
,
0'p0)
In the rest of the proof we choose the above coordinates y" as harmonic functions, appealing to [GW 1975]. We may then assume that the components cp" = ya o cp are harmonic on X, for this holds under any one of the
hypotheses (a), (b) or (c). (Ad (a). Take v = ya in Definition 13.1. Ad (b). Apply Lemma 13.1. Ad (c). Take v = y" in (c); then Definition 5.1 is fulfilled with W replaced by V; and u by W".) For v E C2(N), the identities (13.6) and (13.7) now reduce to ANV = ha08a80v,
-J(Vb 0(voW)) =fX
[(8a80V)o'p](ve,V
(13.8)
)
(13.9)
254
Part III. Maps between polyhedra
for E Lipe. (X) since (8"v) o E W,o,c' (X) fl C(X) by Remark 9.7, and so Vi[(8"v) o cp] E W,'"2(X) by Remark 5.1. Consequently, (c) is equivalent to
a.e. in X,
[(8000 OP] (VV*, VV') = A [(ONV) o gyp]
(13.10)
because two functions gl, g2 E L ,,(X) are equal a.e. if fX z0igl = fx r1i92 for all E Lip. (X); indeed, Lipc(X) is positively rich in CJX), as noted in the text following inequality (12.12). (c). Multiplying (13.4) by (8"8pv)oip, and inserting (13.8) on the (b) right, leads to (13.10), and hence to (c). This also proves the final assertion of the theorem. (c)
(b). Inserting (13.8) in (13.10) yields
) = A [(h18"8pv) ° v]
[(8"apv) ° A (TAP V
a.e. inX
for every v E C2(N). Taking v =
2y"yO leads to (13.4), that is, V is horizontally weakly conformal with dilation A. By Lemma 13.1, rp is a harmonic map (its components being harmonic, as noted earlier). (c) (a). Clearly, (c) can be localized so as to apply to functions v defined merely in open subsets V of N. In this localized form, (c) obviously implies (a). (a) (c). If n = 1 we may assume that N = R. The concepts harmonic morphism and harmonic map then both reduce to that of harmonic function; and every continuous map W E WWoc (X, R) is horizontally weakly conformal
(with A = IV I2). Thus (a) 4=* (b) (c). In the remaining case n > 1, suppose W is a nonconstant harmonic morphism. Using (13.4) with a = Q = 1 as a definition of A, we put
-h"oV, Consider the linear subspace of and note that A > 0 a.e. and A E L C(X) P = {[(.NV) o co)/(hll o cp) : v E C2(N)} ,
and the linear map T : P -* L C(X) defined by
T (ANV) o
_
[(8680V) o'P]
(0'P,
O
cf. (13.10). This map is well defined. In fact, if (ANV) o
1). It follows that v o cp is harmonic on X: Ix (VV,, V(v o cp)) = 0
for every O E Lip,(X),
and so indeed [(8a(9pv) o ioj (V', V ) = 0 a.e. by (13.9), again because Lipe(X) is positively rich in CC(X). The same argument shows that T is local, that is, for any open set U C X and any p E P, p = 0 in U implies Tp = 0 a.e. in U. Similarly, T is positive:
p > 0 in U implies Tp > 0 a.e. in U,
(13.13)
invoking (13.12), (13.9) and Theorem 13.1 (i). Note at this point that the subharmonic function v o W is of class W, (U) by Remark 9.7, and hence weakly subharmonic in U by Remark 7.4.
For v =
2(yl)2
we obtain from (13.6): ANV = h", and so 1 E P.
Furthermore, by (13.11), (13.12), T1 = A,
(13.14)
and (13.10) will follow, thus completing the proof of the theorem, once we show that Tp = Ap for every p E P. (13.15) Suppose, for some p E P and e > 0, that ITp- Apl > 2e in some set A C X of volume JAI > 0. We may further arrange that A is contained in the interior
s° of a top-dimensional simplex s of X, and that A is bounded: A < a in A. Let xo denote a point of density for A with respect to the Riemannian volume measure on s°, or equivalently with respect to Lebesgue measure on s°. Because p is continuous, there is an open neighbourhood U of xo in X such that
p(xo) - e/a < p < p(xo) + e/a
in U.
(13.16)
Since T is linear and positive, cf. (13.13), it follows that
(p(xo) - e/a)T1 5 Tp < (p(xo) + e/a)Tl
in U,
i.e., by (13.14),
ITp - Ap(xo) I < Ae/a < e
a.e. in An U.
By (13.16), IAp - Ap(xo)I < e a.e. in A n U, and so altogether ITp - Apl < 2e a.e. in AnU. This leads to a contradiction because IAnUI > 0. We conclude that (13.15) and (13.10), and hence (c) hold true. 0
256
Part III. Maps between polyhedra
Corollary 13.2. If cp : X -+ N is a harmonic morphism and -0 : N --* P is a harmonic map between smooth Riemannian manifolds without boundary, then the composition ip o cp : X --+ P is a harmonic map.
This follows by invoking Theorem 12.1. Indeed, i/i pulls germs of convex functions on P back to germs of subharmonic functions on N; and cp pulls such germs back to germs of subharmonic functions on X, according to Theorem 13.1 (i).
Example 13.1. In the cases N = R or S' it was noted in the proof of The(c), that a function it then such a geodesic is unique and given by
Pa=SIr [P1, At1 - (1 - \)to] [Po,(1-A)to-At1]
for 0 0) 180 q) for q E Y 181 uq = 4 it,. restriction of p to ball 185 gr mollified Green function 186 V(A) essential image of A 187 ip(A) meanvalue of map cp over set A 188
Jul : M -+ [0, co) 267
Chapter 11 cp.:7rl(X) - ir,(Y) a homomorphism 198 Try : Y -+ Y universal cover 198 1-l a homotopy class of maps 200 W0 1,2
Y), C,j,(X, Y) 206
Chapter 12 6E(V) first variation of E 224 vap second covariant derivative 226 1E(y) approximate length density 237 yµ centre of mass of p 242
Chapter 13 A dilation (dilatation) 251
Chapter 14 f°(2) 259 de(W) energy measure of cp 260 ZeE(cp), ZE(w), dZe(cp) 261 Ze(cp) directional energy
density 262 trB(cp) trace of Won B 262
293
Index
admissible Dirichlet space 23
convex hull 188
- polyhedron 5, 45 - Riemannian polyhedron 5
-subset 25
almost Riemannian space 31 analytic branched cover 141 apartments, chambers 148 approximate energy density 152 axiom of domination 109
Coxeter matrix, system 148 curvature bounds 27
balayage ("sweeping") 111 ball doubling 23, 61 Bauer harmonic space 19 bi-locally E-minimizing 217 boundary of a polyhedron 45
core 20
De Giorgi-Nash theorem 38 diffusion type 20 dilation (=dilatation) 251 dimensionally homogeneous 45 directional energies 261 Dirichlet problem 2, 15, 74, 206, 208
- form 20
- minimum principle 18
- space 23, 104
branching locus 140 Brelot harmonic space 15 Bruhat-Tits building 148
K-dominated triangle 26
capacity 21 Caratheodory distance 22 Carnot-Caratheodory metric 35 Cartan's maximum principle 110 chain rule 66, 167 Chow's theorem 35 circuit (or pseudomanifold) 45 complex (locally finite simplicial complex) 42 conical singularity 135 continuity principle 110
cone 265 connected sum 133 convex function 4, 25, 226
Doob convergence property 20
E-critical map 220 elliptic bounds 48, 162 energy density 3, 153, 163, 174
- measure 21 - of a function 2, 6, 67 - of a map 3, 152, 163, 174 equilibrium potential 115 essential image 187 Euclidean Riemannian structure 49 Euler-Lagrange operator 2 Feyel-de La Pradelle theorem 106 fine topology 121, 126 finely. (sub)harmonic 184, 230 Finsler structure 31
Index
foliation 146 Frostman's maximum principle 110 fundamental domain 201
295
Jensen's inequality for maps 242 join 133 Kobayashi's distance 138 Kummer's branched cover 141
geodesic segment, geodesic 25
- space 25 globally nonpositive curvature 27 Green kernel 19, 108 Gromov's gluing lemma 29
f-potential 108 Hadamard-Cartan theorem 26 harmonic function 2, 15, 99
- map 11, 217 - morphism 12, 247 - sheaf/space/structure 15 Harnack's convergence theorem 2, 15, 103 -inequality 17, 79 Hessian (second covariant
differential) 4, 226 Hironaka's theorem 137 Holder continuity 6, 91 holomorphic quadratic differential 33 (free) homotopy 10, 200 Hopf-Rinow theorem 28 horizontally weakly conformal 12, 251
Hormander's condition on iterated brackets 35 hyperbolic analytic space 138 hypoelliptic 34, 107
infinite dimensional torus 39 intrinsic distance 51 Ishihara's characterization 4, 225, 233, 243, 246 iterated suspension 140
length space 25 Lipschitz continuity 1
- manifold 37 - triangulation 46 local form, local type 20 local E-minimizer (i.e., locally Eminimizing map) 24, 178, 218 locally (n - 1)-chainable 45
- equivalent metrics 35 - squared convex 27 lower semicontinuity of
energy 159 maximal simplex 44, 48 maximum/minimum principle 18, 73, 76 meanvalue of a function 68
- map 188 measure class 37 measurable contraction 22 Mokobodzki's theorem 17 Nash's embedding theorem 162 nonpositive curvature 27 normal analytic space 136
- circuit 45 - contraction 20 normalization 136
orbit space 142 p-adic Lie group 147 P-harmonic space 19 pluriharmonic map 204
296
Index
Poincare inequality for functions 6, 68, 69, 104
- maps 160, 161 polar set 18 potential 18 quasicontinuous function 21, 106 - map 9, 153, 163, 174 quasi-everywhere (q.e.) 21, 106 quasiopen set 127 quasisubharmonic function 229 quotient space 142
Rademacher's theorem 37 radial uniqueness 28 rectifiable path 24 reduced function 106 regular set 15
- Dirichlet form 20 Riemannian branched cover 140
- domain 8, 153, 259, 264 - Finsler structure 32
sweeping (balayage) 111 symmetric power 147 Teichmiiller space 139 tension field 3, 222 Thom space 135 totally geodesic map 234
- subspace 29 trace map 153, 262 transverse vector field 262 triangle comparison 26 triangulation 44, 46 unique continuation 39, 78 variational Dirichlet problem 74 volume element 37, 176, 177 weakly harmonic map 3, 222, 223 weakly (sub/super)harmonic function 72 weighted projective space 137
- foliation 146 -join 134
- Lipschitz manifold 37 - orbifold 146, 147 - polyhedron 47 Schubert varieties 137 separating core 22 simplicial map 43 skeleton 42 Sobolev space of functions 63
- maps 152, 163, 174 star of a simplex 44 stratified space 215 strongly harmonic 19 strongly polar 247 sub/superharmonic function 17, 18 subpartitioning lemma 259
X-Dirichlet integral 34 X-Laplacian 34
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