Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger

doi:10.5802/afst.1132

Dario Cordero-Erausquin ¹; Robert J. McCann ²; Michael Schmuckenschläger ³

¹ Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 77454 Marne la Vallée Cedex 2, France.
² Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 3G3.
³ Institut für Analysis und Numerik, Universität Linz, A-4040 Linz, Österreich.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 613-635.

Abstract
Abstract

We investigate Prékopa-Leindler type inequalities on a Riemannian manifold $M$ equipped with a measure with density $e^{- V}$ where the potential $V$ and the Ricci curvature satisfy ${Hess}_{x} V + {Ric}_{x} \geq λ I$ for all $x \in M$ , with some $λ \in ℝ$ . As in our earlier work [14], the argument uses optimal mass transport on $M$ , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.

Nous étudions l’extension d’inégalités de type Prékopa- Leindler au cas d’une variété riemannienne $M$ équipée d’une mesure ayant une densité $e^{- V}$ où le potentiel $V$ et la courbure de Ricci vérifient ${Hess}_{x} V + {Ric}_{x} \geq λ I (\forall x \in M)$ , pour un certain $λ \in ℝ$ . Nous ferons appel, comme dans notre travail précédent [14], au transport optimal de mesure. Mais nous exploiterons plus encore son lien avec les champs de Jacobi, ce qui permettra de ramener la discussion à l’étude du déterminant d’une matrice de champs de Jacobi. Nous présentons également d’autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de Bakry-Emery) et à l’étude de la convexité de déplacement de la fonctionnelle entropie.

MR Zbl

DOI: 10.5802/afst.1132

Author's affiliations:

Dario Cordero-Erausquin ¹; Robert J. McCann ²; Michael Schmuckenschläger ³

¹ Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 77454 Marne la Vallée Cedex 2, France.
² Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 3G3.
³ Institut für Analysis und Numerik, Universität Linz, A-4040 Linz, Österreich.

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     author = {Dario Cordero-Erausquin and Robert J. McCann and Michael Schmuckenschl\"ager},
     title = {Pr\'ekopa{\textendash}Leindler type inequalities on {Riemannian} manifolds, {Jacobi} fields, and optimal transport},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {613--635},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {4},
     year = {2006},
     doi = {10.5802/afst.1132},
     mrnumber = {2295207},
     zbl = {1125.58007},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1132/}
}

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Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger. Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 613-635. doi : 10.5802/afst.1132. https://afst.centre-mersenne.org/articles/10.5802/afst.1132/

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