Singularities of the Lagrangian Mean Curvature Flow

TL;DR

This paper studies the formation of singularities in Lagrangian mean curvature flow in complex Euclidean spaces, revealing conditions for Type I singularities and characterizing Type II singularities via special Lagrangian cones.

Contribution

It provides a detailed analysis of singularity types in Lagrangian mean curvature flow and links Type II singularities to special Lagrangian cones, advancing understanding of flow behavior.

Findings

Type I singularities occur at specific times linked to cohomology invariants.

Type II singularities are modeled by unions of special Lagrangian cones.

Smooth singularity models describe the asymptotic behavior near singularities.

Abstract

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asymptotic to special Lagrangian cones; hence all type II singularities are modeled by unions of special Lagrangian cones.