Uniqueness of convex ancient solutions to hypersurface flows

TL;DR

This paper classifies convex ancient solutions to mean curvature flow and similar curvature flows, showing they are either spherical, cylindrical, or planar, without assuming interior noncollapsing.

Contribution

It provides a complete classification of convex ancient solutions for a broad class of curvature flows, extending previous results.

Findings

Convex ancient solutions are spherical, cylindrical, or planar.

Classification holds for flows with convex or concave curvature functions.

Results do not require interior noncollapsing assumption.

Abstract

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.