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Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds

Xiang-Dong Li

Abstract

In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the ${\rm CD}(K, m)$-condition for $K\in \mathbb{R}$ and $m\in [n, \infty]$ and a family of Shannon and Rényi entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. {The rigidity models of the enhanced entropy differential inequalities are the $K$-Einstein manifolds and the $(K, m)$-Einstein manifolds}. Moreover, we prove the monotonicity and rigidity theorem of the $W$-entropy associated with the Shannon entropy and the Rényi entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD$(0, m)$-condition. Comparing with the characterization of the the CD$(K, m)$ curvature-dimension condition in the framework of the synthetic geometry developed by Lott, Sturm and Villani, we provide more simple equivalent characterizations for the CD$(K, m)$-condition, and we provide a characterization of the Einstein and quasi-Einstein manifolds by the enhanced entropy differential equality and the enhanced entropy power differential equality. These are new in the literature.

Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds

Abstract

In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the -condition for and and a family of Shannon and Rényi entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. {The rigidity models of the enhanced entropy differential inequalities are the -Einstein manifolds and the -Einstein manifolds}. Moreover, we prove the monotonicity and rigidity theorem of the -entropy associated with the Shannon entropy and the Rényi entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD-condition. Comparing with the characterization of the the CD curvature-dimension condition in the framework of the synthetic geometry developed by Lott, Sturm and Villani, we provide more simple equivalent characterizations for the CD-condition, and we provide a characterization of the Einstein and quasi-Einstein manifolds by the enhanced entropy differential equality and the enhanced entropy power differential equality. These are new in the literature.
Paper Structure (17 sections, 24 theorems, 316 equations)

This paper contains 17 sections, 24 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Differential inequalities for Shannon entropy
  4. Differential inequalities for Shannon entropy
  5. The $W$-entropy and NIW formula for Shannon entropy
  6. Rigidity of Shannon entropy and entropy power
  7. Differential inequalities for Rényi entropy
  8. Differential inequality for Rényi entropy
  9. The $W$-entropy and NIW formula for Rényi entropy
  10. Rigidity theorem of $W$-entropy for Rényi entropy
  11. Rigidity of Rényi entropy and entropy power
  12. The special case $p=1-1/N$ as in Sturm Sturm06b
  13. Limiting case $N\rightarrow \infty$ and $p\rightarrow 1$
  14. Comparison with Lott-Villani-Sturm's synthetic geometry
  15. Lott-Villani and Sturm
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $(M, g)$ be an $n$-dimensional complete Riemannian manifold, and $K\in \mathbb{R}$. Let $(\rho, \phi)$ be a smooth solution to the geodesic flow on $TP_2(M, v)$ equipped with Otto's infinite dimensional Riemannian metric, i.e., $(\rho, \phi)$ satisfies the continuity equation $(TA)$ and the Hami (iii) the Shannon entropy power differential inequality holds (iv) the Rényi entropy differential

Theorems & Definitions (29)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 19 more