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Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 4, pp. 613-635.

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 λI(xM), pour un certain λ. 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.

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 λI for all xM, with some λ. 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.

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DOI : https://doi.org/10.5802/afst.1132
@article{AFST_2006_6_15_4_613_0,
     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, Toulouse},
     volume = {Ser. 6, 15},
     number = {4},
     year = {2006},
     doi = {10.5802/afst.1132},
     zbl = {1125.58007},
     mrnumber = {2295207},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1132/}
}
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, Série 6, Tome 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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