Calculus of Variations and Geometric Measure Theory

T. Rajala - K. T. Sturm

Non-branching geodesics and optimal maps in strong $CD(K,\infty)$-spaces

created by rajala1 on 29 Jul 2012
modified on 15 Jul 2013


Accepted Paper

Inserted: 29 jul 2012
Last Updated: 15 jul 2013

Journal: Calc. Var. Partial Differential Equations
Year: 2012


We prove that in metric measure spaces where the entropy functional is $K$-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map.

The results are applicable in metric measure spaces having Riemannian Ricci-curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci-curvature bounded from below by some constant.

Tags: GeMeThNES
Keywords: Ricci curvature, metric measure spaces, branching geodesics