Cohomogeneity-One Lagrangian Mean Curvature Flow

Abstract

We study mean curvature flow of Lagrangians in ${\mathbb{C}}^n$ that are cohomogeneity-one with respect to a compact Lie group $G\le \mathrm{SU}(n)$ acting linearly on ${\mathbb{C}}^n$. Each such Lagrangian necessarily lies in a level set ${\mu }^{-1}(\xi )$ of the standard moment map $\mu :{\mathbb{C}}^n\to {\mathfrak{g}}^{\ast }$, 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.

arXiv comment53 pages, 3 figures; v2: minor changes, corrected erroneous proofs of Proposition 3.13 and Corollary 3.15