# Rigidity of some functional inequalities on ${\rm RCD}$ spaces

created by han1 on 22 Jan 2020
modified on 10 Feb 2020

[BibTeX]

Preprint

Inserted: 22 jan 2020
Last Updated: 10 feb 2020

Year: 2020
Notes:

A big gap in the previous version is filled.

Abstract:

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Emery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincare inequality in the setting of ${\rm RCD}(K, \infty)$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan etc.. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of $\Phi$-entropy inequalities, the rigidity of the 1-Bakry-Emery inequality are of independent interest even in the smooth setting.