On a non-local area-preserving curve flow
Zezhen Sun, Yuting Wu
TL;DR
This work introduces a new non-local, area-preserving curvature flow for convex planar curves, defined by $\\partial X/\\partial t = (\\kappa - \\lambda(t)/\\kappa) N$ with $\\lambda(t) = \\frac{2\\pi}{\\oint (1/\\kappa) ds}$, which preserves the enclosed area while strictly decreasing the length. The authors reformulate the evolution as a nonlinear differential-integral IVP for the curvature $\\kappa(\\theta,t)$ and the length $L(t)$, establishing local existence via the Leray–Schauder fixed-point theorem, followed by long-time existence through a priori estimates that keep the curvature positive and bounded. They show the isoperimetric deficit $L^{2}-4\\pi A$ decays exponentially and that the flow converges in the Hausdorff sense to a circle of radius $R = \sqrt{A(0)/\\pi}$, with $\\kappa(\\cdot,t) \\to 1/R$ and $\\lambda(t) \\to \\pi/A(0)$. Consequently, the evolving curve converges in $C^{\\infty}$ to the circle, yielding a complete description of the asymptotic shape. The results advance non-local curvature flow theory by providing a concrete area-preserving mechanism driving convex curves to circles and quantifying the convergence rate.
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.