Description 
x, 406 pages ; 24 cm. 
Content Type 
text 
Format 
volume 
Series 
Cambridge studies in advanced mathematics ; 134.

Bibliography 
Includes bibliographical references and index. 
Contents 
First and second variational formulas for area  Volume comparison theorem  BochnerWeitzenböck 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 NeumannPoincaré 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 3manifold  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 CartanHadamard manifold Appendix A. Computation of warped product metrics  Appendix B. Polynomial growth harmonic functions on Euclidean space. 
Summary 
"The aim of this graduatelevel 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 
