Geometric Analysis
Published on 3. May 2012
Book
Hardback
418 pages
978-1-107-02064-1 (ISBN)
Description
The aim of this graduate-level 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.
Reviews / Votes
"This monograph is a beautiful introduction to geometric analysis."Frederic Robert, Mathematical Reviews
More details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
College/higher education
Dimensions
Height: 235 mm
Width: 157 mm
Thickness: 27 mm
Weight
753 gr
ISBN-13
978-1-107-02064-1 (9781107020641)
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Schweitzer Classification
Other editions
Additional editions
Peter Li
Geometric Analysis
E-Book
05/2012
1st Edition
Cambridge University Press
€66.49
Available for download
Person
Peter Li is Chancellor's Professor at the University of California, Irvine.
Content
Introduction; 1. First and second variational formulas for area; 2. Volume comparison theorem; 3. Bochner-Weitzenboeck 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 Neumann-Poincare 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 3-manifold; 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 Cartan-Hadamard manifold; Appendix A. Computation of warped product metrics; Appendix B. Polynomial growth harmonic functions on Euclidean space; References; Index.