2-Dimensional flat curvature flow of crystals

Abstract

In the impressive and seminal paper [5], Fred Almgren, Jean Taylor, and Lihe Wang introduced flat curvature flow in ${\mathbb{R}}^n$, a variational time-discretization scheme for (isotropic or anisotropic) mean curvature flow. Their main result asserts the Hölder continuity, in time, of these flows. This essential estimate requires a boundary regularity result, a uniform lower density ratio bound condition, which they proved for each $n\ge 3.$ Similar estimates for Brownian flows, from important work by N. K Yip on stochastic mean curvature flow [30], also rely on this pivotal regularity result.

In this paper, we complete these analyses for the case $n=2$ by establishing the necessary uniform lower density ratio bounds.