Cohomogeneity-One Lagrangian Mean Curvature Flow

Jesse Madnick, University of Oregon

In C^n, mean curvature flow preserves the class of Lagrangian submanifolds, a fact known as "Lagrangian mean curvature flow" (LMCF). As LMCF typically forms finite-time singularities, it is of interest to understand the blowup models of such singularities, as well as the soliton solutions.

In this talk, we'll consider the mean curvature flow of Lagrangians that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set \mu^{-1}(c) of the moment map \mu, and mean curvature flow preserves this containment. Using this, we'll classify all cohomogeneity-one shrinking, expanding, and translating solitons. Further, in the zero level set \mu^{-1}(0), we'll classify the Type I and Type II blowup models of cohomogeneity-one LMCF singularities.

Finally, given any cohomogeneity-one special Lagrangian in \mu^{-1}(0), we'll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models. This is joint work with Albert Wood.

MC 5417