Geometric Analysis

Description

This graduate-level text demonstrates the basic techniques and how to apply them to various areas of research in geometric analysis. The author focuses mainly on the interaction of partial differential equations with differential geometry and only a rudimentary knowledge of Riemannian geometry and partial differential equations is required. Cover CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 134 CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Title Copyright Dedication Contents Geometric Analysis 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 hypersurface 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