Mean Curvature Flows of Two-Convex Lagrangians

TL;DR

This paper proves regularity, global existence, and convergence of Lagrangian mean curvature flows for two-convex cases, extending known results from convex cases using a new monotone quantity.

Contribution

It introduces a novel monotone quantity that controls two-convexity, enabling the extension of results to a broader class of Lagrangian submanifolds.

Findings

Proves regularity and convergence of two-convex Lagrangian mean curvature flows.

Establishes a new monotone quantity controlling two-convexity.

Extends results to area-decreasing Lagrangian submanifolds via a unitary transformation.

Abstract

We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case. Such results were previously only known in the convex case, of which the current work represents a significant improvement. The proof relies on a newly discovered monotone quantity that controls two-convexity. Through a unitary transformation, same result for the mean curvature flow of area-decreasing Lagrangian submanifolds were established.