Geometric Analysis

Table of Contents

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.

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