On a General Linear Nonlocal Curvature Flow of Convex Plane Curves

TL;DR

This paper investigates a broad class of linear nonlocal curvature flows for convex plane curves, establishing short-term existence and convergence results using ODE techniques and heat equation representations.

Contribution

It introduces a general linear nonlocal curvature flow framework and analyzes its short-time existence and asymptotic behavior for convex curves.

Findings

Short-time existence of the flow is proven.

Asymptotic convergence to a circle is established.

The linear structure allows solving via ODE and heat equation methods.

Abstract

Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure of the flow, this partial differential equation problem can be resolved using an ordinary differential equation method, together with the help of representation formula for solutions to a linear heat equation.