Harmonic Maps Between Riemannian Polyhedra

by J. Eells , B. Fuglede , Preface by M. Gromov

• ISBN13: 9780521773119
• ISBN10: 0521773113
• Format: Hardcover
• Copyright: 2001-07-30
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

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