Harmonic Maps Between Riemannian Polyhedra

Harmonic maps between smooth Riemannian manifolds play a ubiquitous role in differential geometry. Examples include geodesics viewed as maps, minimal surfaces, holomorphic maps and Abelian integrals viewed as maps to a circle. The theory of such maps has been extensively developed over the last 30 years, and has significant applications throughout mathematics. This book extends that theory in full detail to harmonic maps between broad classes of singular Riemannian polyhedra, with many examples being given. The analytical foundation is based on existence and regularity results which use the potential theory of Riemannian polyhedral domains viewed as Brelot harmonic spaces and geodesic space targets in the sense of Alexandrov and Busemann. The authors set out much new material on harmonic maps between singular spaces for the first time in book form. The work will hence serve as a concise source for all researchers working in related fields.

Gromov's Preface

ix

Authors' Preface

xi

Introduction

1

(13)

The Smooth framework

1

(3)

Harmonic and Dirichlet spaces

4

(1)

Riemannian polyhedra

5

(1)

Harmonic functions on X

6

(1)

Geometric examples

7

(1)

Maps between polyhedra

8

(3)

Harmonic maps

11

(1)

Harmonic morphisms

12

(1)

Singular frameworks

12

(2)

Part I. Domains, targets, examples

14

(48)

Harmonic spaces, Dirichlet spaces, and geodesic spaces

15

(15)

Harmonic spaces

15

(5)

Dirichlet structures on a space

20

(4)

Geodesic spaces

24

(6)

Examples of domains and targets

30

(11)

Riemannian manifolds

30

(1)

Almost Riemannian spaces

31

(1)

Finsler structure on a manifold

31

(2)

Metric associated to a holomorphic quadratic differential

33

(1)

Lie algebras of vector fields on a manifold

33

(4)

Riemannian Lipsechitz manifolds

37

(2)

The infinite dimensional torus T∞

39

(2)

Riemannian polyhedra

41

(21)

Lip continuous map. Lip homeomorphism

41

(1)

Simplicial complex

42

(2)

Polyhedron

44

(1)

Circuit

45

(1)

Lip polyhedron

46

(1)

Riemannian polyhedron

47

(4)

The intrinsic distance dx

51

(6)

Local structure in terms of cubes

57

(3)

Uniform estimate of ball volumes

60

(2)

Part II. Potential theory on polyhedra

62

(88)

The Sobolev space W1,2(X). Weakly harmonic functions

63

(16)

The Sobolev space W1,2(X)

63

(5)

A Poincare inequality

68

(4)

Weakly harmonic and weakly sub/superharmonic functions

72

(5)

Unique continuation of harmonic functions

77

(2)

Harnack inequality and Holder continuity for weakly harmonic functions

79

(20)

Proof of Theorem 6.1 in the locally bounded case

79

(9)

Completion of the proof of Theorem 6.1

88

(3)

Holder continuity

91

(8)

Potential theory on Riemannian polyhedra

99

(31)

Harmonic space structure

99

(5)

The Dirichlet space L01,2(X)

104

(4)

The Green kernel

108

(17)

Quasitopology and fine topology

125

(2)

Sobolev functions on quasiopen sets

127

(2)

Subharmonicity of convex functions

129

(1)

Examples of Riemannian polyhedra and related spaces

130

(20)

1-dimensional Riemannian polyhedra

130

(1)

The need for dimensional homogeneity

131

(1)

The need for local chainability

132

(1)

Manifolds as polyhedra

132

(1)

A Kind of connected sum of polyhedra

132

(1)

Riemannian joins of Riemannian manifolds

133

(1)

Riemannian orbit spaces

134

(1)

Conical singular Riemannian spaces

134

(1)

Normal analytic spaces with singularities

135

(3)

The Kobayashi distance

138

(1)

Riemannian Branched coverings

139

(3)

The quotient M/K

142

(4)

Riemannian orbifolds

146

(1)

Buildings of Bruhat-Tits

147

(3)

Part III. Maps between polyhedra

150

(127)

Energy of maps

151

(27)

Energy density and energy

151

(11)

Energy of maps into Riemannian manifolds

162

(11)

Energy of maps into Riemannian polyhedra

173

(3)

The volume of a map

176

(2)

Holder continuity of energy minimizers

178

(20)

The case of a target of nonpositive curvature

179

(10)

Proof of Theorem 10.1

189

(3)

The case of a target of upper bounded curvature

192

(6)

Existence of energy minimizers

198

(19)

The case of free homotopy

200

(6)

The Dirichlet problem relative to a homotopy class

206

(2)

The ordinary Dirichlet problem

208

(3)

The case where the target is a Riemannian manifold

211

(1)

The case of 2-dimensional manifold domains

211

(2)

Questions and remarks

213

(4)

Harmonic maps. Totally geodesic maps

217

(30)

A concept of harmonic map

217

(4)

Weakly harmonic maps into a Riemannian manifold

221

(9)

Holder continuity revisited

230

(3)

Totally geodesic maps

233

(3)

Geodesics as harmonic maps

236

(5)

Jensen's inequality for maps

241

(2)

Harmonic maps from a 1-dimensional Riemannian polyhedron

243

(4)

Harmonic morphisms

247

(12)

Harmonic morphisms between harmonic spaces

247

(2)

Harmonic morphisms between Riemannian polyhedra

249

(2)

Harmonic morphisms into Riemannian manifolds

251

(8)

Appendix: Energy according to Korevaar-Schoen

259

(5)

Subpartitioning Lemma

259

(2)

Directional energies

261

(1)

Trace maps

262

(2)

Appendix: Minimizers with small energy decay

264

(13)

T. Serbinowski

Introduction and results

264

(1)

Embedding Y into an NPC cone

265

(3)

Holder continuity of the minimizer

268

(5)

Proof of Theorem 15.1

273

(2)

Lipschitz continuity of the minimizer

275

(2)

Bibliography

277

(14)

Special symbols

291

(3)

Index

294

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Harmonic Maps Between Riemannian Polyhedra

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