Convex ancient solutions of the mean curvature flow

TL;DR

This paper investigates convex ancient solutions to the mean curvature flow, establishing conditions under which such solutions are spherical, and explores their behavior and generalizations in different geometric contexts.

Contribution

It provides new criteria for classifying convex ancient solutions as shrinking spheres and analyzes their properties in various geometric settings.

Findings

Convex ancient solutions are spheres under certain curvature pinching conditions.

Growth bounds on diameter imply solutions are shrinking spheres.

Analysis of k-convex solutions and solutions in spherical ambient spaces.

Abstract

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.