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 40 years, and has significant applications throughout mathematics. This 2001 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 work sets out much material on harmonic maps between singular spaces and will hence serve as a concise source for all researchers working in related fields.
Rezensionen / Stimmen
'This book can be highly recommended, both to specialists in the field, who will find a direct interest, and to geometers and analysts, who will find a source containing a large amount of material, with precise references. The organization of the chapters is excellent.' Luc Lemaire, Bulletin of the London Mathematical Society
Reihe
Sprache
Verlagsort
Zielgruppe
Maße
Höhe: 235 mm
Breite: 157 mm
Dicke: 23 mm
Gewicht
ISBN-13
978-0-521-77311-9 (9780521773119)
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Schweitzer Klassifikation
Gromov's preface; Preface; 1. Introduction; Part I. Domains, Targets, Examples: 2. Harmonic spaces, Dirichlet spaces and geodesic spaces; 3. Examples of domains and targets; 4. Riemannian polyhedra; Part II. Potential Theory on Polyhedra: 5. The Sobolev space W1,2(X). Weakly harmonic functions; 6. Harnack inequality and Hoelder continuity for weakly harmonic functions; 7. Potential theory on Riemannian polyhedra; 8. Examples of Riemannian polyhedra and related spaces; Part III. Maps between Polyhedra: 9. Energy of maps; 10. Hoelder continuity of energy minimizers; 11. Existence of energy minimizers; 12. Harmonic maps - totally geodesic maps; 13. Harmonic morphisms; 14. Appendix A. Energy according to Korevaar-Schoen; 15. Appendix B. Minimizers with small energy decay; Bibliography; Special symbols; Index.
