Cohomogeneity-One Lagrangian Mean Curvature Flow

TL;DR

This paper classifies cohomogeneity-one Lagrangian mean curvature flow solutions in complex Euclidean space, including solitons and singularity models, providing explicit examples and a comprehensive understanding of their behavior.

Contribution

It offers a complete classification of cohomogeneity-one self-similar solutions and singularities in Lagrangian mean curvature flow, including explicit examples and new solitons.

Findings

Classified all cohomogeneity-one self-similar solutions.

Described Type I and Type II blowup models explicitly.

Provided new examples of solitons and singularity models.

Abstract

We study mean curvature flow of Lagrangians in $\mathbb{C}^n$ that are cohomogeneity-one with respect to a compact Lie group $G \leq \mathrm{SU}(n)$ acting linearly on $\mathbb{C}^n$. Each such Lagrangian necessarily lies in a level set $\mu^{-1}(\xi)$ of the standard moment map $\mu \colon \mathbb{C}^n \to \mathfrak{g}^*$, and mean curvature flow preserves this containment. We classify all cohomogeneity-one self-similarly shrinking, expanding and translating solutions to the flow, as well as cohomogeneity-one smooth special Lagrangians lying in $\mu^{-1}(0)$. Restricting to the case of almost-calibrated flows in the zero level set $\mu^{-1}(0)$, we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in $\mu^{-1}(0)$, we show it occurs as the Type II blowup model of a Lagrangian MCF singularity. Throughout, we give explicit examples of suitable group actions, including a complete list in the case of $G$ simple. This yields infinitely many new examples of shrinking and expanding solitons for Lagrangian MCF, as well as infinitely many new singularity models.