Universality in mean curvature flow neckpinches

Abstract

Abstract. We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is C 3 -close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.

Sphere bundles with 1/4-pinched fiberwise metrics

“…Note that we cannot use standard results here, because we need uniform control on how far each metric moves in the orbit of the diffeomorphism group. Instead, we adapt methods developed in [6,7] for controlling coordinate parameterizations of mean curvature flow solutions. Then we use a fiberwise version of the fact that each round metric is isometric to the canonical round metric on the sphere.…”

“…Our modified linearization argument follows ideas introduced in [6] and [7]. At a sequence of times T γ ր ∞, with T 0 = T * , we construct DeTurck background metricsĝ γ such that all b i (T γ ) = 0, and all b i (t) remain small for T γ ≤ t ≤ T γ+1 .…”

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

“…For very loose cinching, the flow converges to the shrinking round sphere with its usual global Type I singularity. For very tight cinching, it has been shown (see [19] and [3] for the case of rotationally symmetric embeddings, and see [14,15] for embeddings which are nearly rotationally symmetric) that the equator shrinks more rapidly than the two "dumbbell" hemispheres, and forms a Type-I "neckpinch". To obtain a mean curvature flow which develops a Type-II singularity, one starts the flow at the embedding with the parameter value at the threshold between those embeddings flowing to Type-I neckpinches and those flowing to Type I sphere collapses.…”

Ricci flow neckpinches without rotational symmetry

“…For example, rotationally-symmetric solutions of Ricci flow that develop neckpinch singularities asymptotically acquire the additional translational symmetry of the cylinder soliton [2,3]. More recently, it has been shown that any complete noncompact 2-dimensional solution of mean curvature flow that is sufficiently C 3 -close to a standard round neck at some time will develop a finite-time singularity and become asymptotically rotationally symmetric in a space-time neighborhood of that singularity [8,9]. As well, numerical experiments support the expectation that broader classes of mean curvature flow solutions asymptotically develop additional local symmetries as they become singular [10].…”