On Area-preserving and Length-preserving Nonlocal Flow of Convex Closed Plane Curves

TL;DR

This paper investigates specific nonlocal flows of convex plane curves that preserve either length or area, demonstrating they evolve into perfect circles, and discusses related flows for certain parameter ranges.

Contribution

It introduces and analyzes new $k^{ ext{alpha}}$-type nonlocal flows that preserve length or area, showing their convergence to circles in smooth norms.

Findings

Curves evolve into circles under the studied flows.

Flow preserves either length or area during evolution.

Convergence to round circles is proven in $C^{ ext{infinity}}$-norm.

Abstract

For any $\alpha>0,$ we study $k^{\alpha}$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}% $-norm.$\ $Other relevant $k^{\alpha}$-type nonlocal flow is also discussed when $\alpha\geq1.\ $