Boundary singularities in mean curvature flow and total curvature of minimal surface boundaries

Abstract

For hypersurfaces moving by standard mean curvature flow with fixed boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in $\bf{R}^3$ that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a non-empty open set of such initial surfaces. Let $\kappa$ be the largest number with the following property: if $M$ is a minimal surface in $\bf{R}^3$ bounded by a smooth simple closed curve of total curvature $< \kappa$, then $M$ is a disk. Examples show that $\kappa<4\pi$. In this paper, we use mean curvature flow to show that $\kappa \ge 3\pi$. We get a slightly larger lower bound for orientable surfaces.