Short-time existence of Lagrangian mean curvature flow

TL;DR

This paper proves short-time existence of Lagrangian mean curvature flow for certain desingularizations of immersed Lagrangians, using PDE methods and stronger convergence notions.

Contribution

It establishes a short-time existence result for Lagrangian mean curvature flow with stronger convergence, extending previous varifold-based results.

Findings

Existence of solutions with convergence in the sense of manifolds with corners.

Construction of desingularisations using JLT expanders.

Application of PDE techniques similar to network flow proofs.

Abstract

In his paper `Conjectures on Bridgeland Stability', Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez.