On a non-local area-preserving curve flow

TL;DR

This paper introduces a novel non-local curvature flow for convex planar curves that preserves area, reduces length, and causes the curves to become increasingly circular, ultimately converging to a perfect circle.

Contribution

The paper proposes a new non-local area-preserving curvature flow and proves its convergence to a circle for convex curves.

Findings

Curve length decreases over time.

Curves become more circular during evolution.

Curves converge smoothly to a circle as time approaches infinity.

Abstract

In this paper, we study a new area-preserving curvature flow for closed convex planar curves. This flow will decrease the length of the evolving curve and make the curve more and more circular during the evolution process. And finally, the curve converges to a finite circle in $C^{\infty}$ sense as time goes to infinity.