Mean curvature flow singularities for mean convex surfaces

Abstract

We study the evolution by mean curvature of a smooth n–dimensional surfaceM Rn+1, compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case n = 2

Mean curvature flow with surgeries of two–convex hypersurfaces

“…3 we introduce the standard surgery procedure for necks, which replaces a piece of a neck close to some cylinder by two convex spherical caps, while keeping track of all curvature quantities. Section 4 first revisits the convexity estimates established in [18,19] for smooth mean curvature flow and establishes that they are retained by the surgery procedure if the surgery parameters are chosen appropriately. We state the estimates in a non-technical form which is made precise in This result implies that any surface obtained by rescaling our flow around a singularity is convex.…”

“…Let us start with the case m = 2 which was first done in [18] and is technically simpler. We introduced the following function on surfaces with positive mean curvature.…”

Stability of translating solutions to mean curvature flow

“…These solutions arise as parabolic rescalings of type II singularities [11]. They have constant time derivativesU > 0.…”

“…They can be obtained by rescaling a type II singularity parabolically [11]. Xu-Jia Wang [14] found other convex translating solutions without rotational symmetry.…”

Stability of the surface area preserving mean curvature flow in Euclidean space

“…Now using the fact that |tr A 3 | ≤ |A| 3 (see Lemma 2.2 of [HS99]), and Kato's inequality |∇|A|| ≤ |∇A|, we derive from (ii) of Corollary 2.6 to find…”