Ancient mean curvature flows and their spacetime tracks

TL;DR

This paper classifies the convex hulls of ancient solutions to mean curvature flow in Euclidean space, revealing they can only occupy slabs, halfspaces, or entire space, using a bi-halfspace theorem and maximum principle.

Contribution

It introduces a classification of convex hulls for ancient mean curvature flows and proves a bi-halfspace theorem using a parabolic maximum principle.

Findings

Convex hulls are limited to slabs, halfspaces, or all space.

Established a bi-halfspace theorem for ancient solutions.

Applied a parabolic Omori-Yau maximum principle.

Abstract

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic Omori-Yau maximum principle for ancient mean curvature flows.