Sobolev's inequality under a curvature-dimension condition

TL;DR

This paper offers a concise, robust proof of Sobolev's inequality under Ricci curvature bounds, utilizing a gradient-flow approach, and reviews its historical development and computational techniques.

Contribution

It introduces a simplified, gradient-flow-based proof of Sobolev's inequality under curvature-dimension conditions, clarifying previous complex computations.

Findings

Provides a short, robust proof of Sobolev's inequality

Clarifies the gradient-flow interpretation of the inequality

Reviews historical and computational aspects of the inequality

Abstract

In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{\'e}ron and V{\'e}ron's 1991 classical work.