JournalsifbVol. 7 , No. 3DOI 10.4171/ifb/123

2-Dimensional flat curvature flow of crystals

  • David G. Caraballo

    Georgetown University, Washington, USA
2-Dimensional flat curvature flow of crystals cover

Abstract

In the impressive and seminal paper \cite{ATW}, Fred Almgren, Jean Taylor, and Lihe Wang introduced flat curvature flow in Rn\mathbb{R}^{n}, a variational time-discretization scheme for (isotropic or anisotropic) mean curvature flow. Their main result asserts the H\"{o}lder 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 n3.n\geq 3. Similar estimates for Brownian flows, from important work by N.K Yip on stochastic mean curvature flow \cite{Yip}, also rely on this pivotal regularity result. In this paper, we complete these analyses for the case n=2n=2 by establishing the necessary uniform lower density ratio bounds. MSC 2000 subject classification: 53C44, 49Q20, 49N60.