Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

TL;DR

This paper investigates the evolution of shapes under anisotropic curvature flow with volume preservation, proving exponential convergence to Wulff shapes and introducing a novel parametrization method using the Cahn-Hoffman map.

Contribution

It establishes exponential convergence of area-preserving anisotropic mean curvature flow to Wulff shapes and introduces a new boundary parametrization approach using the Cahn-Hoffman map.

Findings

Flow converges exponentially to Wulff shapes of equal area.

Reflection symmetries are preserved during the flow.

Uniform bounds on the distance between the flow profile and initial data are obtained.

Abstract

We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the critical point of the anisotropic perimeter functional. This is an anisotropic analogue of the results in the isotropic case studied in \cite{julin2022}. The novelty of our approach is in using the Cahn-Hoffman map to parametrize boundary components as small perturbations of the Wulff shape. In addition, we show that certain reflection comparison symmetries are preserved by the flat flow, which lets us obtain uniform bounds on the distance between the convergent profile and the initial data.