Regularity of minimal submanifolds and mean curvature flows with a common free boundary

TL;DR

This paper proves smoothness of common boundaries and regularity of minimal submanifolds and mean curvature flows with multiple boundary components, extending previous results to higher codimensions and dynamic settings.

Contribution

It establishes boundary regularity for unions of multiple submanifolds with common boundary and extends these results to mean curvature flows with multiple boundary components.

Findings

Common boundary $\Gamma$ is smooth for unions of three or more submanifolds.

Submanifolds are smooth up to the boundary, real-analytic if $N$ is real-analytic.

Mean curvature flows with multiple boundary components are smooth in space and time.

Abstract

Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,\mu}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $\Gamma$, then $\Gamma$ is smooth and each $M_k$ is smooth up to $\Gamma$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l = 1$. We additionally show that if $\{V_t\}_{t \in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t} \subset N$ with a common boundary $\Gamma_t$ and with parabolic $C^{2+\mu}$ regularity in time-space, then $\{\Gamma_t\}_{t \in (-1,1)}$ and $\{M_{k,t}\}_{t \in (-1,1)}$ are smooth (second Gevrey with real-analytic time-slices in the case $N$ is real-analytic).