Area-preserving crystalline curvature flow in two dimensions

Abstract

We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on $ϕ$ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of \cite{Almgren1995} to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, \emph{i.e.} without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz $ϕ$-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in \cite{Ambrosio2002regularity}. The novelty of our approach lies in the application of $ϕ$-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.