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

Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger — 2006

Annales de la faculté des sciences de Toulouse Mathématiques

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 [], 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...

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