On volume-preserving crystalline mean curvature flow

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...

Motion by volume preserving mean curvature flow near cylinders

We investigate the volume preserving mean curvature flow with Neumann boundary condition for hypersurfaces that are graphs over a cylinder. Through a center manifold analysis we find that initial hypersurfaces sufficiently close to a cylinder of large enough radius, have a flow that exists for all time and converges exponentially fast to a cylinder. In particular, we show that there exist globa...

The Volume Preserving Mean Curvature Flow near Spheres

By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.

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...

Anisotropic and Crystalline Mean Curvature Flow