Geometric Analysis

This graduate-level text demonstrates the basic techniques for researchers interested in the field of geometric analysis.

Peter Li (Author)

9781107020641, Cambridge University Press

Hardback, published 3 May 2012

418 pages
22.9 x 15.2 x 2.8 cm, 0.73 kg

"This monograph is a beautiful introduction to geometric analysis."
Frederic Robert, Mathematical Reviews

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.

Introduction
1. First and second variational formulas for area
2. Volume comparison theorem
3. Bochner–Weitzenböck formulas
4. Laplacian comparison theorem
5. Poincaré 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, Poincaré 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–Poincaré 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.

Subject Areas: Geometry [PBM]