Ancient solutions and translators of Lagrangian mean curvature flow

TL;DR

This paper classifies certain ancient solutions to Lagrangian mean curvature flow in complex Euclidean space, showing that under specific conditions they are translators or special Lagrangian solutions, especially in dimension two.

Contribution

It proves that ancient solutions with particular blow-downs are translators and classifies low-entropy solutions in c2b2 as special Lagrangian, unions of planes, or translators.

Findings

Ancient solutions with specific blow-downs are translators.

In c2b2, low-entropy solutions are classified as special Lagrangian, planes, or translators.

Abstract

Suppose that $\mathcal{M}$ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbb{C}^n$. We show that if $\mathcal{M}$ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then $\mathcal{M}$ is a translator. In particular in $\mathbb{C}^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.