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Generic neck pinch singularities along 2D Lagrangian mean curvature flow

Gábor Székelyhidi

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.

Generic neck pinch singularities along 2D Lagrangian mean curvature flow

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.
Paper Structure (8 sections, 18 theorems, 78 equations)

This paper contains 8 sections, 18 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lagrangians and the mean curvature flow
  4. Lawlor necks and connected sums
  5. The local model
  6. Existence of singularities
  7. Nondegenerate neck pinches
  8. Teardrop singularities

Key Result

Theorem 1

Suppose that for $t\in [0,T)$, $L_t$ is a rational, graded Lagrangian mean curvature flow in a compact Calabi-Yau surface (see Section sec:LMCFprelim for definitions). Suppose that the flow develops a first time singularity at $(x_0, T)$ with tangent flow given by the transverse union of two planes.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • Theorem 10: Seidel Sei99
  • ...and 26 more