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; 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 Hölder 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. Hölder 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.
Reviews for Harmonic Maps between Riemannian Polyhedra
'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
