On the Stability of Cylindrical Singularities of the Mean Curvature Flow

TL;DR

This paper analyzes the stability of cylindrical singularities in mean curvature flow, constructing a stable manifold and proving asymptotic stability for hypersurfaces near these singularities without symmetry assumptions.

Contribution

It introduces an explicit stable manifold for rescaled MCF of hypersurfaces over cylinders and proves stability and uniqueness of singularity profiles in a general setting.

Findings

Constructed a stable manifold for the rescaled MCF.

Proved asymptotic stability of cylindrical singularities.

Provided a simple proof of the uniqueness of tangent flow.

Abstract

We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of finite codimensions in a suitable configuration space. For any initial hypersurface from this stable manifold, we construct a unique global solution to the rescaled MCF, and derive precise asymptotics for these solutions that are valid for all time. Using these asymptotics, we prove asymptotic stability of cylindrical singularities of arbitrary dimensions under generic initial perturbations. As a by-product, for any flow of hypersurfaces evolving according to the MCF that enters this stable manifold at any time and first develops a singularity at a subsequent time, we give a simple proof of the uniqueness of tangent flow, first established by Colding and Minicozzi. Moreover, in this case we show the unique singularity profile is determined by the hypersurface profile when the flow enters the stable manifold. For all results in this paper, there is no symmetry or solitonic assumption.