Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

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–Émery Ricci curvature satisfying ${\textrm{Ric}}_f\ge \frac{a}{2}g$ 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 $({\mathbb {R}}, dt^2, \frac{a}{2}|\cdot |^2)$. This was proved in 2019 by Ohta and Takatsu (Manuscr Math 162:271–282, 2019). In this paper, we prove this rigidity result using a different method.