Geometric Analysis

Peter Li is Chancellor's Professor at the University of California, Irvine.

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.