The polyharmonic heat flow of closed plane curves

TL;DR

This paper studies the evolution of closed plane curves under polyharmonic heat flow, showing that small initial deviations lead to exponential convergence to a circle, with insights into convexity loss periods.

Contribution

It provides a new analysis of polyharmonic heat flow for closed curves, including convergence rates and convexity behavior, which was not previously established.

Findings

Flow converges exponentially fast to a circle for small initial oscillations.

Quantifies the duration when the flow is not strictly convex.

Characterizes the flow's behavior in terms of curvature and isoperimetric defect.

Abstract

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of the total amount of time during which the flow is not strictly convex, quantifying in a sense the failure of the maximum principle.