Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

TL;DR

This paper investigates how locally convex closed curves evolve under nonlocal curvature flows, identifying conditions that lead to singularities or convergence to specific shapes, with implications for geometric analysis.

Contribution

It establishes sufficient conditions on initial curves for singularity formation or convergence in area and length preserving curvature flows.

Findings

Positivity of enclosed algebraic area influences finite-time singularity in area-preserving flow.

Positivity of initial energy determines long-term behavior in length-preserving flow.

Curves can develop singularities or converge to m-fold circles based on initial geometric properties.

Abstract

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the length-preserving flow, it is the positivity of an energy associated with initial curve that plays such a role.