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By Peter Li
Publisher: Cambridge University Press
Print Publication Year:2012
Online Publication Date:June 2012
Online ISBN:9781139105798
Hardback ISBN:9781107020641
Paperback ISBN:9781107471719
Book DOI: http://dx.doi.org/10.1017/CBO9781139105798
Subjects: Geometry and topology , Differential and integral equations, dynamical systems and control theory
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.
Reviews:
pp. i-vi
pp. vii-viii
pp. ix-x
1 - First and second variational formulas for area: Read PDF
pp. 1-9
2 - Volume comparison theorem: Read PDF
pp. 10-18
3 - Bochner–Weitzenböck formulas: Read PDF
pp. 19-31
4 - Laplacian comparison theorem: Read PDF
pp. 32-39
5 - Poincaré inequality and the first eigenvalue: Read PDF
pp. 40-56
6 - Gradient estimate and Harnack inequality: Read PDF
pp. 57-67
7 - Mean value inequality: Read PDF
pp. 68-76
8 - Reilly's formula and applications: Read PDF
pp. 77-85
9 - Isoperimetric inequalities and Sobolev inequalities: Read PDF
pp. 86-95
10 - The heat equation: Read PDF
pp. 96-108
11 - Properties and estimates of the heat kernel: Read PDF
pp. 109-121
12 - Gradient estimate and Harnack inequality for the heat equation: Read PDF
pp. 122-133
13 - Upper and lower bounds for the heat kernel: Read PDF
pp. 134-148
14 - Sobolev inequality, Poincaré inequality and parabolic mean value inequality: Read PDF
pp. 149-168
15 - Uniqueness and the maximum principle for the heat equation: Read PDF
pp. 169-176
16 - Large time behavior of the heat kernel: Read PDF
pp. 177-188
17 - Green's function: Read PDF
pp. 189-202
18 - Measured Neumann Poincaré inequality and measured Sobolev inequality: Read PDF
pp. 203-215
19 - Parabolic Harnack inequality and regularity theory: Read PDF
pp. 216-240
pp. 241-255
21 - Harmonic functions and ends: Read PDF
pp. 256-266
22 - Manifolds with positive spectrum: Read PDF
pp. 267-283
23 - Manifolds with Ricci curvature bounded from below: Read PDF
pp. 284-298
24 - Manifolds with finite volume: Read PDF
pp. 299-305
25 - Stability of minimal hypersurfaces in a 3-manifold: Read PDF
pp. 306-314
26 - Stability of minimal hypersurfaces in a higher dimensional manifold: Read PDF
pp. 315-325
27 - Linear growth harmonic functions: Read PDF
pp. 326-339
28 - Polynomial growth harmonic functions: Read PDF
pp. 340-348
29 - Lq harmonic functions: Read PDF
pp. 349-360
30 - Mean value constant, Liouville property, and minimal submanifolds: Read PDF
pp. 361-369
pp. 370-380
32 - The structure of harmonic maps into a Cartan–Hadamard manifold: Read PDF
pp. 381-391
Appendix A - Computation of warped product metrics: Read PDF
pp. 392-394
Appendix B - Polynomial growth harmonic functions on Euclidean space: Read PDF
pp. 395-398
pp. 399-403
pp. 404-406