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Entropy power concavity inequality on Riemannian manifolds and Ricci flow

  • Songzi Li and Xiang-Dong Li
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Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs
This chapter is in the book Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Abstract

In our recent works [17, 21, 22], we prove the concavity of the Shannon and Renyi entropy powers for the heat equation and the nonlinear diffusion equation associated with the usual Laplacian or the Witten Laplacian on Riemannian manifolds with CD(K, m)-condition and (K, m)-super-Ricci flows, m ∈ [n,∞) and K ∈ ℝ. The rigidity models are the Einstein and quasi-Einstein manifolds. Inspired by Perelman’s work, we prove the convexity of the Shannon entropy power for the conjugate heat equation on Ricci flow. The corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with nonnegative (Bakry-Emery) Ricci curvature and maximal volume growth condition. The purpose of this paper is to give a survey on our results obtained in [17, 21, 22].

Abstract

In our recent works [17, 21, 22], we prove the concavity of the Shannon and Renyi entropy powers for the heat equation and the nonlinear diffusion equation associated with the usual Laplacian or the Witten Laplacian on Riemannian manifolds with CD(K, m)-condition and (K, m)-super-Ricci flows, m ∈ [n,∞) and K ∈ ℝ. The rigidity models are the Einstein and quasi-Einstein manifolds. Inspired by Perelman’s work, we prove the convexity of the Shannon entropy power for the conjugate heat equation on Ricci flow. The corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with nonnegative (Bakry-Emery) Ricci curvature and maximal volume growth condition. The purpose of this paper is to give a survey on our results obtained in [17, 21, 22].

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Preface VII
  4. Part I: Fractals and graphs
  5. Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes 1
  6. The pointwise existence and properties of heat kernel 27
  7. Resistance estimates and critical exponents of Dirichlet forms on fractals 71
  8. A survey on unbounded Laplacians and intrinsic metrics on graphs 103
  9. Energy measures for diffusions on fractals: a survey 119
  10. Hyperbolic graphs induced by iterations and applications in fractals 143
  11. Geometric implications of fast volume growth and capacity estimates 183
  12. Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set 201
  13. Metrics and uniform Harnack inequality on the Strichartz hexacarpet 275
  14. Part II: Euclidean spaces and manifolds
  15. Analysis on manifolds and volume growth 299
  16. Geometric analysis on manifolds with ends 325
  17. A matrix Harnack estimate for a Kolmogorov type equation 345
  18. Entropy power concavity inequality on Riemannian manifolds and Ricci flow 359
  19. Fractional differential operators and divergence equations 385
  20. Interior gradient estimates for mean curvature type equations and related flows 421
  21. An alternate induction argument in Simons’ proof of holonomy theorem 443
  22. Higher integrability for nonlinear nonlocal equations with irregular kernel 459
  23. On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds 493
  24. Index 515
https://doi.org/10.1515/9783110700763-013

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Preface VII
  4. Part I: Fractals and graphs
  5. Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes 1
  6. The pointwise existence and properties of heat kernel 27
  7. Resistance estimates and critical exponents of Dirichlet forms on fractals 71
  8. A survey on unbounded Laplacians and intrinsic metrics on graphs 103
  9. Energy measures for diffusions on fractals: a survey 119
  10. Hyperbolic graphs induced by iterations and applications in fractals 143
  11. Geometric implications of fast volume growth and capacity estimates 183
  12. Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set 201
  13. Metrics and uniform Harnack inequality on the Strichartz hexacarpet 275
  14. Part II: Euclidean spaces and manifolds
  15. Analysis on manifolds and volume growth 299
  16. Geometric analysis on manifolds with ends 325
  17. A matrix Harnack estimate for a Kolmogorov type equation 345
  18. Entropy power concavity inequality on Riemannian manifolds and Ricci flow 359
  19. Fractional differential operators and divergence equations 385
  20. Interior gradient estimates for mean curvature type equations and related flows 421
  21. An alternate induction argument in Simons’ proof of holonomy theorem 443
  22. Higher integrability for nonlinear nonlocal equations with irregular kernel 459
  23. On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds 493
  24. Index 515
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