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

We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in 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.