Louis Dupaigne

(de l'article arXiv) : 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{é}ron and V{é}ron's 1991 classical work.