Abstract

We present a probabilistic formulation of rigidity for the sharp log–Sobolev inequality under a Bakry–Émery curvature lower bound. Interpreting entropy externally—as a quantity optimized relative to a class of measures with fixed curvature constraints—we show that saturation of the log–Sobolev constant forces isotropy of the associated diffusion operator. As a consequence, equality implies a Gaussian factor or splitting structure. The argument emphasizes equality cases of convex functional inequalities rather than measure classification or algebraic rigidity, providing a minimal and portable mechanism underlying entropy rigidity phenomena.

1. Introduction

Rigidity results in geometry and dynamics frequently arise from extremal entropy phenomena: a system maximizes entropy among a constrained class, and this extremality forces strong structural conclusions. Classical approaches typically rely on global classification tools—homogeneous dynamics, representation theory, or invariant-measure classification—to identify the extremal models.

In this note we isolate a simpler and more local mechanism. We view entropy as an external quantity: not merely an invariant of a single system, but a functional optimized relative to a family of systems satisfying fixed coarse constraints. Under this viewpoint, rigidity becomes a statement about equality cases of sharp functional inequalities.

We demonstrate this mechanism concretely in the setting of the log–Sobolev inequality under a Bakry–Émery curvature bound. This setting provides a clean prototype: the inequality is sharp, its extremizers are well understood, and the curvature–concentration correspondence is explicit.