On Neck Singularities for 2-Convex Mean Curvature Flow

Abstract

In this paper we are dealing with mean curvature flow with surgeries of two-convex hypersurfaces. The main focus is to expand on the discussion in Section $3$ of Mean Curvature Flow with Surgeries of Two-Convex Hypersurfaces by Huisken and Sinestrari. Firstly we wish to establish how the neck detection lemma allows us to detect necks where the cross sections will be diffeomorphic to $S^{n-1}$. We then want to see how we are able to glue these cross sections together with full control on their parametrisation - for this we will show we can use a harmonic spherical parametrisation using the techniques from Hamiltons paper, Four-manifolds with Positive Isotropic Curvature. We then introduce the notion of a normal and maximal necks, this allows us to obtain uniqueness, existence and overlapping properties for normal parametrisations on $(\epsilon,k)$-cylindrical hypersurface necks. Lastly given a neck $N:S^{n-1}\times[a,b]\to\mathcal{M}$ we want to see that in the case that either $a=\infty$ or $b=\infty$ that this forces them to both to be $\infty$ and that we are left with a solid tube $S^{n-1}\times S^1$.