Free boundary flow through cylindrical singularities

Abstract

We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types ${\mathbb{R}}^k\times S^{n-k}$, ${\mathbb{R}}_{+}^k\times S^{n-k}$ or ${\mathbb{R}}^k\times S_{+}^{n-k}$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.

20 pages