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Geometric Analysis

Part of Cambridge Studies in Advanced Mathematics

  • Author: Peter Li, University of California, Irvine
  • Date Published: May 2012
  • availability: Available
  • format: Hardback
  • isbn: 9781107020641


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About the Authors
  • 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.

    • Originates from the author's lectures to graduate students
    • A short treatment of the heat equation provides background for research in geometric flows
    • The tools presented here can also be used in nonlinear theory
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    Product details

    • Date Published: May 2012
    • format: Hardback
    • isbn: 9781107020641
    • length: 418 pages
    • dimensions: 229 x 152 x 28 mm
    • weight: 0.73kg
    • availability: Available
  • Table of Contents

    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

  • Instructors have used or reviewed this title for the following courses

    • Topics in Geometric Analysis
    • advance mathematics for computer vision and robotics
  • Author

    Peter Li, University of California, Irvine
    Peter Li is Chancellor's Professor at the University of California, Irvine.

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