Cohomogeneity-one Lagrangian mean curvature flow

We study mean curvature flow of Lagrangians in \(\mathbb {C}^n\) ${\mathbb{C}}^n$ that are cohomogeneity-one with respect to a compact Lie group \(G \le \textrm{SU}(n)\) $G\le \text{SU}(n)$ acting linearly on \(\mathbb {C}^n\) ${\mathbb{C}}^n$ . Each such Lagrangian necessarily lies in a level set \(\mu ^{-1}(\xi )\) ${\mu }^{-1}(\xi )$ of the standard moment map \(\mu :\mathbb {C}^n \rightarrow \mathfrak {g}^*\) $\mu :{\mathbb{C}}^n\to {\mathfrak{g}}^{\ast }$ , 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 \(\mu ^{-1}(0)\) ${\mu }^{-1}(0)$ . Restricting to the case of almost-calibrated flows in the zero level set \(\mu ^{-1}(0)\) ${\mu }^{-1}(0)$ , we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in \(\mu ^{-1}(0)\) ${\mu }^{-1}(0)$ , 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.