A density function and the structure of singularities of the mean curvature flow

Abstract

We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ℝ n+1, primarily in the boundaryless setting. We concentrate on the so-called “Type I” case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of “slow-forming singularities” of the flow. The main new ingredient introduced, to address these issues, is a certain “density function”, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.