A concavity estimate is derived for interpolations between L 1 (M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.

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 ≥ λ I 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. RésuméNousétudions l'extension d'inégalités de type Prékopa-Leindler au cas d'une variété riemannienne Méquipée d'une mesure ayant une densité e −V où le potentiel V et la courbure de Ricci vérifient Hess x V + Ric x ≥ λ I (∀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 discussionà l'étude du déterminant d'une matrice de champs de Jacobi. Nous présentonségalement d'autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de BakryEmery) età l'étude de la convexité de déplacement de la fonctionnelle entropie.

No abstract

No abstract