On short time existence of Lagrangian mean curvature flow

TL;DR

This paper proves short time existence of smooth Lagrangian mean curvature flow starting from certain singular Lagrangian submanifolds in complex Euclidean space, confirming a conjecture of Joyce.

Contribution

It establishes the existence of smooth Lagrangian mean curvature flow for initial data with specific singularities, advancing understanding of flow starting from singular Lagrangians.

Findings

Existence of smooth flow for initial Lagrangians with prescribed singularities.

Flow attains initial Lagrangian as varifolds as time approaches zero.

Flow is smooth away from singularities for positive time.

Abstract

We consider a short time existence problem motivated by a conjecture of Joyce. Specifically we prove that given any compact Lagrangian $L\subset \mathbb{C}^n$ with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains $L$ as $t \searrow 0$ as varifolds, and smoothly locally away from the singularities.