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Harmonic Maps between Riemannian Polyhedra

Harmonic Maps between Riemannian Polyhedra

Part of Cambridge Tracts in Mathematics

  • Date Published: August 2001
  • availability: Available
  • format: Hardback
  • isbn: 9780521773119


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About the Authors
  • 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.

    • Written by leading researchers
    • Presents new material which has never before been brought together in book form
    • Unique treatment - there are no directly comparable books on the subject
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: August 2001
    • format: Hardback
    • isbn: 9780521773119
    • length: 312 pages
    • dimensions: 229 x 152 x 21 mm
    • weight: 0.63kg
    • availability: Available
  • Table of Contents

    Gromov's 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
    Special symbols

  • Authors

    J. Eells, University of Cambridge

    B. Fuglede, University of Copenhagen

    Preface by

    M. Gromov

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