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Equality in the logarithmic Sobolev inequality

Shin-ichi Ohta, Asuka Takatsu

Abstract

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems.

Equality in the logarithmic Sobolev inequality

Abstract

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying . Assuming equality holds, we show that the -dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems.

Paper Structure

This paper contains 10 sections, 6 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries for optimal transport theory
  3. Wasserstein distance
  4. Curvature-dimension condition
  5. Needle decompositions
  6. Proof of Theorem \ref{['main']}
  7. Decomposition into needles
  8. Analysis on needles
  9. Integration in needles
  10. Further problems

Key Result

Theorem 1.1

Let $(M,g,\omega)$ be a connected, complete, weighted $\mathcal{C}^{\infty}$-Riemannian manifold of dimension $n \ge 2$ such that $\mathrm{Ric}_{\infty} \ge K>0$. Assume that there is a nonconstant, nonnegative, locally Lipschitz function $\rho: M \longrightarrow [0,\infty)$ such that $\|\rho\|_{L^1 Then we have the following.

Theorems & Definitions (6)

  • Theorem 1.1: Main theorem
  • Corollary 1.2
  • Theorem 2.1: Optimal transports
  • Theorem 2.2: Curvature-dimension condition
  • Proposition 2.3: Talagrand, HWI inequalities
  • Theorem 2.4: Needle decompositions