Convexity estimates for surfaces moving by curvature functions

Convexity estimates for hypersurfaces moving by convex curvature functions

We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Hui...

Geometric Smoothing of 3d Surfaces and Non-linear Diiusion of 3d Images

In this paper we present a geometric smoothing technique for three-dimensional surfaces and images. The technique relies on curvature-dependent deformations of surfaces and the intuition that highly curved regions should move into their convexity by an amount proportional to their curvature. While this intuition in 2D has lead to a well-de ned process and formal theorems about its smoothing pro...

Surveys in Differential Geometry XII Singularities of Mean Curvature Flow and Flow with Surgeries

We collect in this paper several results on the formation of singularities in the mean curvature flow of hypersurfaces in euclidean space, under various kinds of convexity assumptions. We include some recent estimates for the flow of 2-convex surfaces, i.e. the surfaces where the sum of the two smallest principal curvatures is positive everywhere. Such results enable the construction of a flow ...

Cylindrical Estimates for Hypersurfaces Moving by Convex Curvature Functions

Regularity Properties of a Certain Class of Surfaces

Introduction. Convex curves in the plane and convex surfaces in space have many at first sight unexpected regularity properties. The stringency of the convexity condition makes it desirable to find a geometric property of convex curves or surfaces which has similar analytical implications but applies to a wider class of geometrical objects. The first and most studied generalization of convexity...