Centro-affine curvature flows on centrally symmetric convex curves

TL;DR

This paper studies p-centro affine curvature flows on symmetric convex curves, showing finite-time contraction or expansion to ellipses, and proves convergence to the circle and a new affine isoperimetric inequality.

Contribution

It introduces and analyzes p-centro affine flows for convex curves, establishing their asymptotic behavior and providing a new proof of the p-affine isoperimetric inequality.

Findings

Curves shrink to a point under p-contracting flows, converging to ellipses.

Normalized curves converge to the unit circle in Hausdorff metric.

Duality between contracting and expanding flows is established.

Abstract

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $\pi$ converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area $\pi$, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of $p$-affine isoperimetric inequality, $p\geq 1$, for smooth, centrally symmetric convex bodies in $\mathbb{R}^2$.