Crystalline Mean Curvature Flow of Convex Sets

Crystalline mean curvature flow of convex sets

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, sa...

Singularity Structure in Mean Curvature Flow of Mean Convex Sets

In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres. In this note we announce results on the mean curvature flow of mean convex sets; all the statements below have ...

The volume preserving crystalline mean curvature flow of convex sets in R

We prove the existence of a volume preserving crystalline mean curvature flat flow starting from a compact convex set C ⊂ R and its convergence, modulo a time-dependent translation, to a Wulff shape with the corresponding volume. We also prove that if C satisfies an interior ball condition (the ball being the Wulff shape), then the evolving convex set satisfies a similar condition for some time...

Anisotropic and Crystalline Mean Curvature Flow

The Size of the Singular Set in Mean Curvature Flow of Mean-convex Sets

In this paper, we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point. It is most convenient to describe the results in terms of the level set flow (also called “biggest flow” [I2]) of Chen-Giga-Goto [CGG] and Evans-Spruck [ES]. Under the level set flow, any closed s...