Mean curvature flow near a peanut solution

TL;DR

This paper investigates the instability of peanut-shaped mean curvature flow solutions, showing that small perturbations can lead to different singularity types and that rescaled limits of certain solutions converge to ancient oval solutions.

Contribution

It demonstrates the high instability of peanut solutions and characterizes the limits of solutions with spherical singularities as ancient oval solutions.

Findings

Peanut solutions are highly unstable under perturbations.

Small perturbations can lead to spherical or neckpinch singularities.

Rescaled limits of solutions with spherical singularities converge to ancient oval solutions.

Abstract

It was shown by Angenent, Altschuler and Giga, and by Angenent and Velazquez that there exist closed mean curvature flow solutions that extinct to a point in finite time, without ever becoming convex prior to their extinction. These solutions develop a degenerate neckpinch singularity, meaning that the tangent flow at a singularity is a round cylinder, but at the same time for each of these solutions there exists a sequence of points in space and time, so that the pointed blow up limit around this sequence is the Bowl soliton. These solutions are called peanut solutions and they were first conjectured to exist by Richard Hamilton, while the existence of those solutions was shown by Angenent, Altschuler and Giga. In this paper we show that this type of solutions are highly unstable, in the sense that in every small neighborhood of any such peanut solution we can find a perturbation so that the mean curvature flow starting at that perturbation develops spherical singularity, and at the same time we can find a perturbation so that the mean curvature flow starting at that perturbation develops a nondegenerate neckpinch singularity. We also show that appropriately rescaled subsequence of any sequence of solutions whose initial data converge to the peanut solution, and all of which develop spherical singularities, converges to the Ancient oval solution.