Lagrangian Mean Curvature flow for entire Lipschitz graphs

TL;DR

This paper proves that entire Lipschitz continuous Lagrangian graphs under mean curvature flow become instantly smooth, exist globally, and decay in curvature, with applications to Bernstein-type theorems for translating solitons.

Contribution

It establishes long-time existence and smoothness of solutions for Lipschitz initial data in Lagrangian mean curvature flow, and proves a Bernstein-type theorem for translating solitons.

Findings

Solutions become smooth immediately under the flow.

The second fundamental form decays to zero as time approaches infinity.

Any entire Lagrangian translating soliton must be a flat plane.

Abstract

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\to \infty$. Our assumption on the Lipschitz norm is equivalent to the assumption that the underlying Lagrangian potential $u$ is uniformly convex with its Hessian bounded in $L^\infty$. We apply this result to prove a Bernstein type theorem for translating solitons, namely that if such an entire Lagrangian graph is a smooth translating soliton, then it must be a flat plane. We also prove convergence of the evolving graphs under additional conditions.