Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
Cordero-Erausquin, Dario1 ; McCann, Robert J.2 ; Schmuckenschläger, Michael3
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, 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 HessxV+Ricx≥λI(∀x∈M), pour un certain λ∈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.

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 HessxV+Ricx≥λI for all x∈M, with some λ∈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.

DOI : 10.5802/afst.1132
Cordero-Erausquin, Dario 1 ; McCann, Robert J. 2 ; Schmuckenschläger, Michael 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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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {613--635},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
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Cordero-Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael. 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://www.numdam.org/articles/10.5802/afst.1132/

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