Centro-affine normal flows on curves: Harnack estimates and Ancient solutions

TL;DR

This paper proves that the only compact, origin-symmetric, strictly convex ancient solutions to planar p-centro-affine normal flows are contracting ellipses centered at the origin, providing a classification result.

Contribution

It establishes a uniqueness classification for ancient solutions of a specific geometric flow, showing they must be ellipses.

Findings

Ancient solutions are only contracting origin-centered ellipses.

The result applies to planar p-centro-affine normal flows.

Provides Harnack estimates for these flows.

Abstract

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses.