Uniqueness of two-convex closed ancient solutions to the mean curvature flow

TL;DR

This paper proves the uniqueness of certain ancient solutions to the mean curvature flow, showing they are all equivalent to a known rotationally symmetric solution up to translations and scaling.

Contribution

It establishes the uniqueness of two-convex closed ancient solutions to the mean curvature flow, linking them to a previously constructed symmetric solution.

Findings

Any two such solutions are identical up to translations and scaling

They coincide with the known rotationally symmetric solution

The result applies to non-collapsed, uniformly two-convex ancient solutions

Abstract

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).