Area-preserving anisotropic mean curvature flow in two dimensions

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 Julin et al. (Math Ann 387(3):1969–1999, 2022). 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.