Entire solutions of two-convex Lagrangian mean curvature flows

Abstract

Given an entire $C^2$ function $u$ on ${\mathbb{R}}^n$, we consider the graph of $Du$ 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 $D^{2}u$.

21 pages, 3 figures. Typos corrected. Minor modifications. To appear in Calc. Var. Partial Differential Equations