Singularities of Lagrangian mean curvature flow: monotone case

TL;DR

This paper investigates singularity formation in the mean curvature flow of monotone Lagrangians in complex space, showing tangent flows decompose into area-minimizing cones and providing specific examples of such singularities.

Contribution

It demonstrates that singularities lead to tangent flows decomposing into area-minimizing Lagrangian cones, with improved results for the case when n=2, and provides explicit examples.

Findings

Singularities result in tangent flows as unions of area-minimizing cones.

For n=2, connected components converge to a single area-minimizing cone.

Explicit examples of singularity formation are constructed.

Abstract

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this result by showing that connected components of the rescaled flow converge to an area-minimizing cone, as opposed to possible non-area minimizing union of Slag cones. In the last section, we give specific examples for which such singularity formation occurs.