Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

TL;DR

This paper develops a geometric framework connecting mean curvature flow of Lagrangian submanifolds in Kim-McCann metrics with optimal transport theory, demonstrating convergence under certain curvature conditions and linking to special Lagrangian geometry.

Contribution

It introduces a novel approach to analyze Lagrangian mean curvature flow within Kim-McCann manifolds using para-K"ahler geometry, establishing convergence results and connections to optimal transport.

Findings

Lagrangian condition is preserved along the flow.

Flows converge smoothly under cross-curvature positivity.

Lagrangian submanifolds asymptotically approach graphs of optimal transport maps.

Abstract

We express the mean curvature flow of Lagrangian submanifolds in pseudo-Riemannian manifolds endowed with the Kim-McCann-Warren metric within the framework of generalized mean curvature flow on Kim-McCann manifolds. While generalized mean curvature flow has been studied in K\"ahler geometry, our work shows that techniques from para-K\"ahler geometry arise naturally in the Kim-McCann setting. Using this perspective, we prove that the Lagrangian condition is preserved along the flow. By identifying generalized mean curvature flow with Lagrangian mean curvature flow, we show that the Ma-Trudinger-Wang regularity theory applies to this setting. In particular, the cross-curvature positivity condition of Kim-McCann yields smoothly converging flows of Lagrangian submanifolds. Under the cross-curvature condition, any Lagrangian submanifold avoiding the cut locus converges exponentially to a stationary submanifold, which locally arises as the graph of an optimal transport map. Our framework substantiates the analogy between special Lagrangian geometry in almost Calabi-Yau manifolds and optimal transport theory in the Kim-McCann setting. In particular, we show that Kim-McCann manifolds equipped with a para-holomorphic volume form serve as the natural counterpart to almost Calabi-Yau manifolds.