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Description

Given an optimal transportation problem between two manifolds, Kim and McCann offered a pseudo-Riemannian metric on the product manifold, which captures some of the geometry of the problem. By modifying this metric depending on the mass, the graph of the solution is a minimal surface. It is natural to ask then, how mean curvature flow behaves on these manifolds. Work of Li-Salavessa shows that even in high-codimension, mean curvature flow in pseudo-Riemannian spaces can behave remarkably well. We explore this question in both the background-flat case, and in the curved case, where, not surprisingly, we find the Kim-McCann expression of the Ma-Trudinger-Wang condition.