https://neckpinch.com/authors/alexander-majchrowski/New arXiv papers by Alexander Majchrowski, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1803.09878/arxiv:1803.09878Tue, 27 Mar 2018 00:00:00 GMTAlexander 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 M₀ in ℝⁿ⁺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.Mean curvature flowSurgery https://neckpinch.com/papers/1706.02818/arxiv:1706.02818Fri, 09 Jun 2017 00:00:00 GMTAlexander Majchrowski — 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]→𝓜 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.Mean curvature flowSingularitiesSurgery4-manifoldsSphere theoremsStability & uniqueness