Existence and uniqueness of rectilinear slit maps

Existence and Uniqueness of Optimal Transport Maps

Let (X, d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X, d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure con...

Existence and Uniqueness of Monotone Measure - Preserving Maps

Existence, Uniqueness, and Regularity of Optimal Transport Maps

Adapting some techniques and ideas of McCann [8], we extend a recent result with Fathi [6] to yield existence and uniqueness of a unique transport map in very general situations, without any integrability assumption on the cost function. In particular this result applies for the optimal transportation problem on a n-dimensional non-compact manifold M with a cost function induced by a C2-Lagrang...

Flowing Maps to Minimal Surfaces: Existence and Uniqueness of Solutions

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region o...

Existence and uniqueness of optimal maps on Alexandrov spaces

The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov’s curvature bounded below, Monge’s transport problem for the quadratic cost admits a unique solution.