Local non-collapsing of volume for the Lagrangian mean curvature flow

TL;DR

This paper establishes optimal volume control for a reparametrized Lagrangian mean curvature flow in Calabi-Yau manifolds and classifies certain translating solitons in complex Euclidean space.

Contribution

It introduces a new optimal control result for volume measures under a reparametrized flow and classifies Lagrangian translating solitons in $ ext{C}^m$.

Findings

Optimal volume control under the flow

Classification of Lagrangian translating solitons

Insights into the behavior of almost calibrated submanifolds

Abstract

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in $\mathbb{C}^m$ that evolve by this reparametrized flow