Finsler interpolation inequalities

Abstract

We extend Cordero-Erausquin et al.'s Riemannian Borell-Brascamp-Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani's curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest. Mathematics Subject Classification IntroductionOptimal transport theory is making rapid and breathtaking progress in recent years as it finds a large number of connections with various fields. Villani's massive lecture notes [42] provide the global picture, and a great deal of new insight as well. One of the milestones of the theory is a Riemannian Borell-Brascamp-Lieb inequality due to Cordero-Erausquin, McCann and Schmuckenschläger [11] that we will extend to Finsler manifolds. Their work offers deep inspiration as well as a technical breakthrough in the investigation of optimal transport in curved spaces. Furthermore, the Riemannian Borell-Brascamp-Lieb inequality has many meaningful applications from its ancestors (the Prékopa-Leindler inequality and the Brunn-Minkowski inequality) to the attractive curvature-dimension condition. We will also extend them to Finsler manifolds.Our generalization is twofold. We consider a Finsler manifold where one naturally encounters a nonsymmetric distance function, and we equip it with an arbitrary measure. Then the main difficulty arises from the lack of a good notion of the Hessian, and we always have to

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

“…We review the basics of Finsler geometry (we refer to [BCS] and [Sh1] for further reading), and introduce the weighted Ricci curvature and the nonlinear Laplacian studied in [Oh3] and [OS1] (see also [GS]). Throughout the article, let M be a connected, n-dimensional C ∞ -manifold without boundary such that n ≥ 2.…”

Comparison Theorems in Finsler Geometry With Weighted Curvature Bounds and Related Results

“…The following theorem was first obtained in [9], and here we provide another proof using Laplacian comparison theorem. …”

“…(1) The notion of weighted Ricci curvature Ric N (N ∈ (n, ∞)) was first introduced by Ohta from view point of curvature-dimension condition [9]. Here we introduce the weighted Ricci curvature based on the weighted flag curvature from view point of comparison geometry.…”

“…Here we introduce the weighted Ricci curvature based on the weighted flag curvature from view point of comparison geometry. When N ∈ (n, ∞), the notion of Ric N above coincides with that of [9].…”

“…It should be noted here that by utilizing the weighted Ricci curvature condition, Ohta and Sturm [9,10] gave another version of Laplacian comparison theorem and volume comparison theorem, which absorb the S-curvature into the weighted Ricci curvature assumption and remove the S-curvature in the conclusion. In this paper we shall continue investigations in this direction.…”