Finite Time Singularities for Lagrangian Mean Curvature Flow

TL;DR

This paper constructs examples of Lagrangian submanifolds in Calabi-Yau fourfolds that develop finite time singularities under mean curvature flow, challenging existing conjectures about their long-term behavior.

Contribution

It demonstrates the existence of finite time singularities for Lagrangian mean curvature flow within the same Hamiltonian isotopy class, providing a counterexample to a weakened Thomas-Yau conjecture.

Findings

Existence of finite time singularities in Lagrangian mean curvature flow

Counterexample to the long-standing conjecture on flow convergence

Implications for the understanding of Lagrangian geometry in Calabi-Yau manifolds

Abstract

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lagrangian mean curvature flow.