Geometric analysis (Q5891556)

The monograph is devoted to the important and interesting branch lying at the intersection of the up-to-date theory of partial differential equations and differential geometry, namely, geometrical analysis, or, in other words, subtle properties of solutions of partial differential equations and manifolds. The book is based on lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley and during the author's visit to the Global Analysis Research Institute at Seoul National University, also at the University of California, Irvine, and, at last, at the XIV Escola de Geometria Diferencial in Brazil during the summer of 2006. This book is not a systematical account of the theory; the author writes that the results presented in it are ``sometimes for their fundamental usefulness and sometimes for the purpose of demonstrating various techniques''. It is not possible in this review to describe the substance of this book in detail, it covers an immense volume of numerous and diverse results. NEWLINENEWLINENEWLINE NEWLINEHere is the general list of chapters of the book: 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 estimates 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 behaviour 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. \(L^q\) 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.NEWLINENEWLINEChapters 1-9 contain basic results in the field, and, in particular, demonstrate the power of the maximal principle method in obtaining estimates on a manifold. Chapters 10-19 give an outline of the theory of the heat equation and also establish various estimates for nonnegative solutions of this equation. Chapters 20-32 deal with harmonic functions, minimal surface and the geometric structure of certain manifolds. At the end of the book there are two applications: Computation of warped product metrics and polynomial growth harmonic functions on Euclidean space.NEWLINENEWLINEThe book is destined for specialists. However it will be useful for all mathematicians going to study the modern theory of partial differential equations and differential geometry.