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

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 Kähler geometry, our work shows that techniques from para-Kähler 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.