Equality in the logarithmic Sobolev inequality

TL;DR

This paper proves a rigidity result for the logarithmic Sobolev inequality on weighted Riemannian manifolds with positive Ricci curvature, showing that equality implies a Gaussian splitting, using needle decomposition techniques.

Contribution

It establishes a new rigidity theorem for the logarithmic Sobolev inequality under curvature conditions, extending known results to this setting.

Findings

Equality in the inequality implies a Gaussian splitting.

The needle decomposition method is effective for proving rigidity.

Open problems related to the inequality are discussed.

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.