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Pré-Publication, Document De Travail Année : 2020

Sobolev's inequality under a curvature-dimension condition

Louis Dupaigne
Ivan Gentil
Simon Zugmeyer
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Résumé

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-Véron and Véron's 1991 classical work.
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Dates et versions

hal-03006155 , version 1 (15-11-2020)
hal-03006155 , version 2 (17-11-2020)
hal-03006155 , version 3 (19-01-2021)
hal-03006155 , version 4 (19-12-2023)
hal-03006155 , version 5 (08-01-2024)

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Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. Sobolev's inequality under a curvature-dimension condition. 2020. ⟨hal-03006155v2⟩
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