Geometric analysis
 Author/Creator
 Li, Peter, 1952
 Language
 English.
 Imprint
 Cambridge ; New York : Cambridge University Press, 2012.
 Physical description
 x, 406 pages ; 24 cm.
 Series
 Cambridge studies in advanced mathematics ; 134.
Access
Available online
 dx.doi.org Cambridge Books Online Access limited to one user.

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QA360 .L53 2012

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QA360 .L53 2012
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Contents/Summary
 Bibliography
 Includes bibliographical references (p. 399403) and index.
 Contents

 Introduction 1. First and second variational formulas for area 2. Volume comparison theorem 3. BochnerWeitzenbock formulas 4. Laplacian comparison theorem 5. Poincare inequality and the first eigenvalue 6. Gradient estimate and Harnack inequality 7. Mean value inequality 8. Reilly's formula and applications 9. Isoperimetric inequalities and Sobolev inequalities 10. The heat equation 11. Properties and estimates of the heat kernel 12. Gradient estimate and Harnack inequality for the heat equation 13. Upper and lower bounds for the heat kernel 14. Sobolev inequality, Poincare inequality and parabolic mean value inequality 15. Uniqueness and maximum principle for the heat equation 16. Large time behavior of the heat kernel 17. Green's function 18. Measured NeumannPoincare inequality and measured Sobolev inequality 19. Parabolic Harnack inequality and regularity theory 20. Parabolicity 21. Harmonic functions and ends 22. Manifolds with positive spectrum 23. Manifolds with Ricci curvature bounded from below 24. Manifolds with finite volume 25. Stability of minimal hypersurfaces in a 3manifold 26. Stability of minimal hypersurfaces in a higher dimensional manifold 27. Linear growth harmonic functions 28. Polynomial growth harmonic functions 29. Lq harmonic functions 30. Mean value constant, Liouville property, and minimal submanifolds 31. Massive sets 32. The structure of harmonic maps into a CartanHadamard manifold Appendix A. Computation of warped product metrics Appendix B. Polynomial growth harmonic functions on Euclidean space References Index.
 (source: Nielsen Book Data)
 Publisher's Summary
 The aim of this graduatelevel text is to equip the reader with the basic tools and techniques needed for research in various areas of geometric analysis. Throughout, the main theme is to present the interaction of partial differential equations and differential geometry. More specifically, emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the underlying manifold and vice versa. For efficiency the author mainly restricts himself to the linear theory and only a rudimentary background in Riemannian geometry and partial differential equations is assumed. Originating from the author's own lectures, this book is an ideal introduction for graduate students, as well as a useful reference for experts in the field.
(source: Nielsen Book Data)  Supplemental links
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Subjects
 Subject
 Geometric analysis.
Bibliographic information
 Publication date
 2012
 Responsibility
 Peter Li, University of California, Irvine.
 Series
 Cambridge studies in advanced mathematics ; 134
 ISBN
 9781107020641 (hardback)
 1107020646 (hardback)