Li P. Geometric Analysis

Cambridge University Press, 2012. — 418 p. — (Cambridge Studies in Advanced Mathematics 134) — ISBN: 052137331X.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.First and second variational formulas for area
Volume comparison theorem
Bochner–Weitzenb¨ock formulas
Laplacian comparison theorem
Poincar´e inequality and the first eigenvalue
Gradient estimate and Harnack inequality
Mean value inequality
Reilly’s formula and applications
Isoperimetric inequalities and Sobolev inequalities
The heat equation
Properties and estimates of the heat kernel
Gradient estimate and Harnack inequality for the heat equation
Upper and lower bounds for the heat kernel
Sobolev inequality, Poincar´e inequality and parabolic mean value inequality
Uniqueness and the maximum principle for the heat equation
Large time behavior of the heat kernel
Green’s function
Measured Neumann Poincar´e inequality and measured Sobolev inequality
Parabolic Harnack inequality and regularity theory
Parabolicity
Harmonic functions and ends
Manifolds with positive spectrum
Manifolds with Ricci curvature bounded from below
Manifolds with finite
Stability of minimal hypersurfaces in a 3-manifold
Stability of minimal hypersurfaces in a higher dimensional manifold
Linear growth harmonic functions
Polynomial growth harmonic functions
Lq harmonic functions
Mean value constant, Liouville property, and minimal submanifolds
Massive sets
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