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Singularity formations in Lagrangian mean curvature flow

Yang Li, Gábor Székelyhidi

Abstract

We study singularities along the Lagrangian mean curvature flow with tangent flows given by multiplicity one special Lagrangian cones that are smooth away from the origin. Some results are: uniqueness of all such tangent flows in dimension two; uniqueness in any dimension when the link of the cone is connected; the existence of nontrivial special Lagrangian blowup limits. We also prove a singular version of Imagi-Joyce-dos Santos's uniqueness result of the Lawlor neck. As an application we prove that in any dimension, singularities that admit a tangent flow given by the union of two transverse planes is modeled on shrinking Lawlor necks at suitable scales.

Singularity formations in Lagrangian mean curvature flow

Abstract

We study singularities along the Lagrangian mean curvature flow with tangent flows given by multiplicity one special Lagrangian cones that are smooth away from the origin. Some results are: uniqueness of all such tangent flows in dimension two; uniqueness in any dimension when the link of the cone is connected; the existence of nontrivial special Lagrangian blowup limits. We also prove a singular version of Imagi-Joyce-dos Santos's uniqueness result of the Lawlor neck. As an application we prove that in any dimension, singularities that admit a tangent flow given by the union of two transverse planes is modeled on shrinking Lawlor necks at suitable scales.

Paper Structure

This paper contains 24 sections, 55 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. Previous works
  3. Main results
  4. Outline of the proofs: LMCF ingredients
  5. Outline of the proof: Lawlor neck
  6. Organisation
  7. Preliminaries
  8. Special Lagrangian cones
  9. The drift heat equation on cones
  10. Special Lagrangian currents asymptotic to cones
  11. Preliminaries for the Lagrangian mean curvature flow
  12. Graphicality and linearization
  13. Estimates for a given time slice in a ball
  14. Decay estimate along the flow
  15. Characterisation of the Lawlor necks
  16. ...and 9 more sections

Key Result

Theorem 2

There exists some type II blowup of the flow given by an exact special Lagrangian current $L_\infty$ which has tangent cone $W_0$ at infinity, but which is not equal to a translate of $W_0$.

Theorems & Definitions (111)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Definition 9
  • Lemma 10
  • ...and 101 more