Neck pinches along the Lagrangian mean curvature flow of surfaces

TL;DR

This paper investigates the behavior of Lagrangian mean curvature flows in Calabi-Yau surfaces, proving tangent flow uniqueness at singularities and conditions preventing certain singularities for stable initial spheres.

Contribution

It establishes the uniqueness of tangent flows at singularities and demonstrates the continuation of the flow past singularities in specific geometric settings.

Findings

Tangent flow at the first singularity is unique.

Flow can be continued past singularities as a smooth immersed Lagrangian.

Stable spheres in the Thomas-Yau sense do not develop certain singularities.

Abstract

Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.