2022/08/02 by Madnick, Jesse, Wood, Albert · 1 citation
#53E10 53D12 (Primary) 53D20 (Secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.01574
We study mean curvature flow of Lagrangians in ℂn that are cohomogeneity-one with respect to a compact Lie group G ≤ SU(n) acting linearly on ℂn. Each such Lagrangian necessarily lies in a level set μ-1(ξ) of the standard moment map μ\colon ℂn → \mathfrakg^*, 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 μ-1(0). Restricting to the case of almost-calibrated flows in the zero level set μ-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 μ-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.