California State University, Northridge WordmarkOviatt Library WordmarkOviatt Library Catalog Wordmark
Author Li, Peter, 1952-
Title Geometric analysis / Peter Li, University of California, Irvine.
Published Cambridge : Cambridge University Press, 2012.
Book Cover

 Floor4  QA360 .L53 2012    IN LIBRARY
Description x, 406 pages ; 24 cm.
Content Type text
Format volume
Series Cambridge studies in advanced mathematics ; 134
Cambridge studies in advanced mathematics ; 134.
Bibliography Includes bibliographical references and index.
Contents First and second variational formulas for area -- Volume comparison theorem -- Bochner-Weitzenböck formulas -- Laplacian comparison theorem -- Poincaré 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, Poincaré inequality and parabolic mean value inequality -- Uniqueness and maximum principle for the heat equation -- Large time behavior of the heat kernel -- Green's function -- Measured Neumann-Poincaré 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 volume -- 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.
Summary "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"-- Provided by publisher.
Subject Geometric analysis.
ISBN 1107020646 (hardback)
9781107020641 (hardback)
OCLC number 774016638