Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

TL;DR

This paper establishes a rigidity result for the logarithmic Sobolev inequality on complete metric measure spaces with Bakry-Émery Ricci curvature bounds, showing that equality implies a specific geometric splitting involving a Gaussian shrinking soliton.

Contribution

The paper provides a new proof of the rigidity result for the logarithmic Sobolev inequality on metric measure spaces, previously proved by Ohta and Takatsu in 2019.

Findings

Equality in the logarithmic Sobolev inequality implies the space splits as a product with a Gaussian soliton.

The proof uses a different method from previous work, offering new insights.

The result characterizes the geometric structure of spaces attaining equality in the inequality.

Abstract

In this work, we study the rigidity problem for the logarithmic Sobolev inequality on a complete metric measure space $(M^n,g,f)$ with Bakry-\'Emery Ricci curvature satisfying $Ric_f\geq \frac{a}{2}g$, for some $a>0$. We prove that if equality holds then $M$ is isometric to $\Sigma\times \mathbb{R}$ for some complete $(n-1)$-dimensional Riemannian manifold $\Sigma$ and by passing an isometry, $(M^n,g,f)$ must split off the Gaussian shrinking soliton $(\mathbb{R}, dt^2, \frac{a}{2}|.|^2)$. This was proved in 2019 by Ohta and Takatsu. In this paper, we prove this rigidity result using a different method.