Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

TL;DR

This paper investigates boundary singularities in mean curvature flow, showing that certain tangent flows are non-orientable shrinkers and establishing bounds on total curvature for minimal surfaces bounded by smooth curves.

Contribution

It demonstrates that boundary singularities with embedded shrinkers are non-orientable and improves bounds on total curvature ensuring minimal surfaces are disks.

Findings

Boundary singularities with embedded shrinkers are non-orientable.

Established that total curvature less than 3π guarantees minimal surfaces are disks.

Provided new bounds on total curvature for minimal surfaces in 3-space.

Abstract

For hypersurfaces moving by standard mean curvature flow with 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 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4\pi$. In this paper, we use mean curvature flow to show that $k >3\pi$. We get a slightly larger lower bound for orientable surfaces.