Non-homothetic convex ancient solutions for flows by high powers of curvature

TL;DR

This paper establishes the existence of convex ancient solutions to curvature flows driven by high powers of curvature, which become increasingly oval over time, generalizing previous results on mean curvature flow.

Contribution

It introduces new convex ancient solutions for curvature flows with high power speeds, extending prior work on one-homogeneous flows and providing a convergence theorem for symmetric hypersurfaces.

Findings

Existence of convex ancient solutions with high power curvature speeds.

Solutions become more oval as time approaches negative infinity.

A new convergence theorem for symmetric hypersurfaces to round points.

Abstract

We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree greater than one. This generalises previous work on the mean curvature flow and other one-homogeneous curvature flows. As an auxiliary result, we prove a new theorem on the convergence to a round point of convex rotationally symmetric hypersurfaces satisfying a suitable constraint on the curvatures.