The anisotropic polyharmonic curve flow for closed plane curves

TL;DR

This paper investigates the anisotropic polyharmonic curve flow for closed plane curves within the Minkowski plane, demonstrating long-term existence and exponential convergence to a scaled isoperimetrix under specific initial conditions.

Contribution

It introduces a novel analysis of the curve diffusion flow in Minkowski geometry, establishing existence and convergence results towards the isoperimetrix.

Findings

Curves close to the isoperimetrix exist for all time under the flow.

Such curves converge exponentially fast to a scaled isoperimetrix.

The convergence preserves the enclosed area of the initial curve.

Abstract

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{U}$ (where $\mathcal{U}\subset\mathbb{R}^{2}$ is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix $\tilde{\mathcal{I}}$. This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the flow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged $L^{2}$ sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has enclosed area equal to the enclosed area of the initial immersion.