2017/06/09 by Alexander Majchrowski, Majchrowski, Alexander
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1706.02818
openalex publication_date 2017/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sn-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 (ε,k)-cylindrical hypersurface necks. Lastly given a neck N:Sn-1×[a,b]\toM we want to see that in the case that either a=∞ or b=∞ that this forces them to both to be ∞ and that we are left with a solid tube Sn-1× S1.