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On the $W$-entropy and Shannon entropy power on RCD$(K, N)$ and RCD$(K, n, N)$ spaces

Xiang-Dong Li, Enrui Zhang

TL;DR

This work generalizes core entropy-analytic concepts from smooth manifolds to non-smooth metric measure spaces with Ricci curvature lower bounds. By establishing entropy dissipation identities, a W-entropy formula and its monotonicity on RCD$(K,N)$ spaces, as well as the $K$-concavity of Shannon entropy power, the authors extend information-thermodynamic inequalities (entropy isoperimetric and Stam LSI) to the RCD framework, including a rigidity theorem for sharp Stam constants on non-collapsing spaces. They further broaden two classical entropy inequalities—Wang’s and Ye’s formulas—to RCD spaces, enhancing the toolkit for studying heat flow and entropy in singular geometric settings. The results unify geometric analysis and information theory in non-smooth spaces, with implications for isoperimetric-type inequalities, sharp LSI constants, and rigidity phenomena in metric-measure geometry.

Abstract

In this paper, we prove the $W$-entropy formula and the monotonicity and rigidity theorem of the $W$-entropy for the heat flow on RCD$(K, N)$ and RCD$(K, n, N)$ spaces $(X, d, μ)$, where $K\in \mathbb{R}$, $n\in \mathbb{N}$ is the geometric dimension of $(X, d, μ)$ and $N\geq n$. We also prove the $K$-concavity of the Shannon entropy power on RCD$(K, N)$ spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam type logarithmic Sobolev inequality on RCD$(0, N)$ spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam type logarithmic Sobolev inequality with sharp constant on noncollapsing RCD$(0, N)$ spaces.

On the $W$-entropy and Shannon entropy power on RCD$(K, N)$ and RCD$(K, n, N)$ spaces

TL;DR

This work generalizes core entropy-analytic concepts from smooth manifolds to non-smooth metric measure spaces with Ricci curvature lower bounds. By establishing entropy dissipation identities, a W-entropy formula and its monotonicity on RCD spaces, as well as the -concavity of Shannon entropy power, the authors extend information-thermodynamic inequalities (entropy isoperimetric and Stam LSI) to the RCD framework, including a rigidity theorem for sharp Stam constants on non-collapsing spaces. They further broaden two classical entropy inequalities—Wang’s and Ye’s formulas—to RCD spaces, enhancing the toolkit for studying heat flow and entropy in singular geometric settings. The results unify geometric analysis and information theory in non-smooth spaces, with implications for isoperimetric-type inequalities, sharp LSI constants, and rigidity phenomena in metric-measure geometry.

Abstract

In this paper, we prove the -entropy formula and the monotonicity and rigidity theorem of the -entropy for the heat flow on RCD and RCD spaces , where , is the geometric dimension of and . We also prove the -concavity of the Shannon entropy power on RCD spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam type logarithmic Sobolev inequality on RCD spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam type logarithmic Sobolev inequality with sharp constant on noncollapsing RCD spaces.

Paper Structure

This paper contains 11 sections, 30 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic facts of RCD spaces
  3. Statement of main results
  4. The entropy dissipation formulas on RCD$(K,N)$ spaces
  5. $W$-entropy formula and its monotonicity on RCD spaces
  6. Shannon entropy power on RCD spaces
  7. Shannon entropy isoperimetric inequality on RCD$(0,N)$ spaces
  8. Rigidity theorem on non-collapsing RCD$(0, N)$ space
  9. Two other entropy inequalities on RCD spaces
  10. Extension of Wang's entropy formula to RCD spaces
  11. Extension of Ye's entropy formula to RCD spaces

Key Result

Theorem 3.1

Let $(X, d, \mu)$ be an RCD space and $u$ be a solution to the heat equation $\partial_t u=\Delta u$. Let $H(u)=-\int_X u \log ud\mu$ be the Shannon entropy. Let $u^\delta$ be the regularization of $u$ which will be precisely defined in Section 4 below. We have (1) Let where $e_\varepsilon :[0,\infty)\rightarrow \mathbb{R}$ with $e'_\varepsilon(r)=\log(\varepsilon+r)+1$ and $e_\varepsilon(0)=0$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2: BS2010
  • Definition 2.3: minimal relaxed gradientAGS2014Invent
  • Definition 2.4: EKS2015KL
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5: KL
  • Definition 3.6
  • ...and 42 more