Alexander Majchrowski

Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in ${\mathbb{R}}^{n+1}$ undergoing this $G$-flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex surfaces undergoing $G$-flow.

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