Singularity formations in Lagrangian mean curvature flow

TL;DR

This paper investigates singularities in Lagrangian mean curvature flow, establishing uniqueness of tangent flows under certain conditions, existence of special Lagrangian blowups, and characterizing singularities modeled on Lawlor necks.

Contribution

It provides new uniqueness results for tangent flows, constructs nontrivial blowup limits, and characterizes singularities modeled on Lawlor necks in Lagrangian mean curvature flow.

Findings

Uniqueness of tangent flows in dimension two.

Uniqueness when the cone's link is connected.

Existence of nontrivial special Lagrangian blowup limits.

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.