Generic neck pinch singularities along 2D Lagrangian mean curvature flow

TL;DR

This paper introduces and analyzes nondegenerate neck pinch singularities in Lagrangian mean curvature flow, demonstrating their stability, occurrence, and relation to existing questions in the field.

Contribution

It defines nondegenerate neck pinch singularities, proves their stability and generic occurrence, and rules out certain singularities for embedded flows.

Findings

Nondegenerate neck pinch singularities can occur in Lagrangian mean curvature flow.

Such singularities are stable under small perturbations.

Nondegenerate teardrop singularities cannot occur for embedded flows.

Abstract

We introduce a notion of nondegenerate neck pinch singularity along the Lagrangian mean curvature flow of surfaces in a Calabi-Yau surface. We show that such singularities can occur, are stable under small perturbations, and any neck pinch singularity can be perturbed to such a nondegenerate singularity near the singular time. Using this we answer some questions raised by Neves and Joyce. We also introduce nondegenerate teardrop singularities and show that these cannot occur for embedded flows.