Translating solutions to Lagrangian mean curvature flow

TL;DR

This paper establishes non-existence results for certain translating solutions in Lagrangian mean curvature flow, showing under specific conditions such solutions must be planes, thereby clarifying the structure of these solutions.

Contribution

It proves that translating solutions with bounded mean curvature or static almost-calibrated solutions are necessarily planes, extending understanding of solution classifications in Lagrangian mean curvature flow.

Findings

Translating solutions with $L^2$ mean curvature bound are planes.

Static almost-calibrated translating solutions are planes.

Conditions are shown to be optimal based on recent work.

Abstract

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.