Book contents
- Frontmatter
- Contents
- Preface
- 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 the 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
28 - Polynomial growth harmonic functions
Published online by Cambridge University Press: 05 June 2012
Book contents
- Frontmatter
- Contents
- Preface
- 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 the 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
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- Geometric Analysis , pp. 340 - 348Publisher: Cambridge University PressPrint publication year: 2012