A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with type-II curvature blowup

Abstract

We present a numerical study of the local stability of mean curvature flow (MCF) of rotationally symmetric, complete noncompact hypersurfaces with type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs ‘numerically global’ (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviours under MCF. Firstly, there is a ‘near’ class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a type-II rate no slower than (T − t)−1. Secondly, there is a ‘far’ class of initial data which lead to solutions developing a local type-I nondegenerate neckpinch under MCF. These numerical findings further suggest the existence of a ‘critical’ class of initial data which conjecturally lead to MCF of noncompact hypersurfaces forming local type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than (T − t)−1.

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

“…The "Near Class" imposes a small-rotationally-symmetric dimple very close to the tip of each initial surface. The numerical simulations in [GIKW21] show that for Near Class initial data, these dimples disappear as MCF progresses, and Type-II behaviors are found to occur. For solutions originating from Near Class perturbations, as from unperturbed initial data, dilations near the tip approach "bowl solitons" (see Figure 2).…”

“…In previous work [GIKW21], we have used numerical simulations to show that Type-II singularity formations observed in mean curvature flow (MCF) of certain noncompact rotationallysymmetric embedded hypersurfaces [IW19, IWZ20, IWZ] are stable for small perturbations near the tip of each initial embedding, as long as rotational symmetry is retained. The question then arises whether this behavior is stable for small perturbations that are not rotationally symmetric.…”

“…In these solutions, the maximum of the norm of the second fundamental form occurs at the tip (the left-most point on the hypersurface depicted in Figure 1), with the rate of curvature blowup at the tip consistent with the definition of a Type-II singularity. In [GIKW21], we consider numerical simulations of MCF originating from rotationally-symmetric embedded hypersurfaces of this type having "dimple" perturbations of two classes. FIGURE 1.…”

“…For this class of solutions, MCF evolutions develop "neckpinch" singularities that exhibit Type-I behaviors at the neck, with the curvatures at the tip remaining bounded (see Figure 3). As noted above, the main goal of this paper is to determine whether or not the results observed in the numerical simulations of [GIKW21] remain true if we remove rotational symmetry. We have chosen to explore this issue by carrying out numerical simulations of MCF originating from initial data much like that of [GIKW21], but with sinusoidal angular dependence initially imposed.…”

“…As noted above, the main goal of this paper is to determine whether or not the results observed in the numerical simulations of [GIKW21] remain true if we remove rotational symmetry. We have chosen to explore this issue by carrying out numerical simulations of MCF originating from initial data much like that of [GIKW21], but with sinusoidal angular dependence initially imposed. As we show below in Section 4, the asymptotic behavior of these solutions is qualitatively unaffected by this angular dependence.…”