Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Arunima Bhattacharya, Micah Warren, Daniel Weser

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 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.

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

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 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.
Paper Structure (23 sections, 20 theorems, 322 equations)

This paper contains 23 sections, 20 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. Relation to the Sinkhorn Algorithm
  3. Organization
  4. Geometric Background
  5. Kim-McCann Manifold
  6. Para-complex Geometry
  7. Almost Calabi-Yau Manifold
  8. Lagrangian Submanifolds of Almost Calabi-Yau Manifolds
  9. The Lagrangian Angle
  10. Expressing $\theta$ in terms of the graphing Jacobian
  11. Short-time Existence and Preservation of Lagrangian Condition
  12. Special Coordinates
  13. Totally Real Submanifolds
  14. Differential of the Mean Curvature Form in the Totally Real Setting
  15. Long-time Existence and Regularity
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose that $M \times \bar{M}$ is a Kim-McCann manifold endowed with a para-holomorphic $n$-form $\Omega$ and $L(t)$ is a family of immersed Lagrangian submanifolds flowing by the generalized mean curvature flow. If $L(0)$ is Lagrangian, then so is $L(t)$ for $t>0$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 48 more