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

 Machine generated contents note: Introduction; 1. First and second variational formulas for area; 2. Volume comparison theorem; 3. BochnerWeitzenböck 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.
 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" Provided by publisher.
 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
 1107020646