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Informationen zum Autor Peter Li is Chancellor's Professor at the University of California, Irvine. Klappentext This graduate-level text demonstrates the basic techniques for researchers interested in the field of geometric analysis. Zusammenfassung 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. Inhaltsverzeichnis Introduction; 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 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 hypersurfaces 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....
