Book contents
- Frontmatter
- Contents
- Preface
- 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 the 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
- Frontmatter
- Contents
- Preface
- 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 the 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
Summary
The main goal of this book is to present the basic tools that are necessary for research in geometric analysis. Though the main theme centers around linear theory, i.e., the Laplace equation, the heat equation, and eigenvalues for the Laplacian, the methods of dealing with these problems are quite often useful in the study of nonlinear partial differential equations that arise in geometry.
A small portion of this book originated from a series of lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. The lecture notes were revised and expanded when the author taught a regular course in geometric analysis. During the author's visit to the Global Analysis Research Institute at Seoul National University, he was encouraged to submit these notes, though still in a rather crude form, for publication in their lecture notes series [L6].
The part of this book that concerns harmonic functions originated from the author's lecture notes for a series of courses he gave on the subject at the University of California, Irvine. A part of this material was also used in a series of lectures the author gave at the XIV Escola de Geometria Diferencial in Brazil during the summer of 2006. These notes [L9] were printed for distribution to the participants of the program.
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- Information
- Geometric Analysis , pp. ix - xPublisher: Cambridge University PressPrint publication year: 2012