Entire solutions of two-convex Lagrangian mean curvature flows

TL;DR

This paper studies the evolution of entire convex functions under Lagrangian mean curvature flow, proving long-term existence and convergence under a 2-positivity condition, extending previous results that required stronger assumptions.

Contribution

It establishes long-time existence and convergence for solutions under a weaker 2-positivity condition, broadening the class of functions for which the flow behaves well.

Findings

Proves long-time existence of solutions under 2-positivity.

Shows convergence of the flow to special Lagrangian submanifolds.

Extends previous results from positive Hessian to 2-positivity condition.

Abstract

Given an entire $C^2$ function $u$ on $\mathbb{R}^n$, we consider the graph of $D u$ as a Lagrangian submanifold of $\mathbb{R}^{2n}$, and deform it by the mean curvature flow in $\mathbb{R}^{2n}$. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of $(I+(D^2 u)^2)^{-1}D^2 u$. Such results were previously known only under the stronger assumption of positivity of $D^2 u$.