A Liouville Theorem for Mean Curvature Flow

TL;DR

This paper proves a Liouville-type theorem that classifies ancient, type-I, non-collapsing two-dimensional mean curvature flows as either spheres or cylinders, aiding the understanding of blow-up limits in geometric analysis.

Contribution

It introduces a Liouville theorem that restricts ancient solutions of mean curvature flow to specific geometric shapes, enhancing classification efforts.

Findings

Ancient solutions are restricted to spheres or cylinders.

The theorem applies to type-I, non-collapsing two-dimensional flows.

Results help in understanding blow-up limits in geometric flows.

Abstract

Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.