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Geometric analysis

Author: Peter Li
Publisher: Cambridge : Cambridge University Press, 2012.
Series: Cambridge studies in advanced mathematics, 134.
Edition/Format:   Print book : EnglishView all editions and formats
"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  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Peter Li
ISBN: 9781107020641 1107020646
OCLC Number: 774016638
Description: x, 406 pages ; 24 cm.
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.
Series Title: Cambridge studies in advanced mathematics, 134.
Responsibility: Peter Li, University of California, Irvine.


This graduate-level text demonstrates the basic techniques for researchers interested in the field of geometric analysis.  Read more...
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"This monograph is a beautiful introduction to geometric analysis." Frederic Robert, Mathematical Reviews

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