On an area-preserving inverse curvature flow for plane curves

TL;DR

This paper investigates a non-local, area-preserving inverse curvature flow for convex plane curves, proving global existence, non-increasing length, and convergence to a circle as time progresses.

Contribution

It introduces a new class of inverse curvature flows that preserve area and demonstrates their long-term behavior and convergence properties.

Findings

Flow exists globally for all time.

Curve length is non-increasing under the flow.

Evolving curves converge to a circle in the smooth topology.

Abstract

In this paper, we study a $1/\kappa^{n}$-type area-preserving non-local flow of convex closed plane curves for any $n>0$. We show that the flow exists globally, the length of evolving curve is non-increasing, and the limiting curve will be a circle in the $C^{\infty}$ metric as time $t\to\infty$.