On Neck Singularities for 2-Convex Mean Curvature Flow

TL;DR

This paper advances the understanding of neck singularities in 2-convex mean curvature flow by refining neck detection, parametrisation, and uniqueness properties, especially in the context of surgeries and infinite ends.

Contribution

It introduces a detailed analysis of neck detection and parametrisation techniques, including harmonic spherical parametrisation and the concept of normal and maximal necks, for better control in mean curvature flow surgeries.

Findings

Neck detection lemma effectively identifies neck regions with $S^{n-1}$ cross sections.

Harmonic spherical parametrisation provides full control over neck parametrisation.

Infinite ends in necks imply a solid tube structure $S^{n-1} imes S^1$.

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 $S^{n-1}\times S^1$.