Mean curvature flow near a peanut solution

TL;DR

This work analyzes mean curvature flow near Hamilton-constructed peanut solutions, showing a striking instability: arbitrarily small perturbations near a peanut can produce either spherical singularities or nondegenerate neckpinch singularities, rather than preserving the original degenerate neckpinch. Central to the analysis is a multi-scale, barrier-based framework combining inner, intermediate, and outer regions, with a drift-Laplacian spectral decomposition into unstable and stable Hermite modes. The authors develop inner-outer L^2 estimates, construct sophisticated sub- and super-solutions in the cylindrical regime, and employ funnel/exiting arguments to prove the dual instability scenarios; they also connect sequences of such perturbations to the Ancient oval as a limiting rescaled flow. Together these results illuminate the fine structure and rigidity (or lack thereof) of degenerately neckpinching MCFs and establish precise blow-up behavior for nearby flows. The findings have implications for understanding singularity formation, stability, and the asymptotic spectrum of MCF near complex neckpinches.

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.

Figures (2)

• Figure 1: The peanut and neighbors. In the 1980ies Hamilton suggested an atypical singularity in a solution to MCF. He considered a one-parameter family of initial surfaces $\{M_\epsilon \mid \epsilon\in{\mathbb R}\}$ each of which is a sphere. For $\epsilon\ll0$ the surface $M_\epsilon$ has a neck that is so narrow that it will pinch before the whole solution can vanish in a point. For $\epsilon\gg0$ the surface $M_\epsilon$ is convex, or so close to being convex that it quickly becomes convex and shrinks to a point, obeying Huisken's theorem Hu. Observing that continuous dependence on initial data should imply the existence of at least one parameter value $\epsilon_*\in{\mathbb R}$ whose corresponding solution $M_{\epsilon_*}(t)$ forms a neck pinch, but does not ever become convex. A rigorous version of Hamilton's arguments appeared in AAG
• Figure 2: Construction of the initial peanut

Theorems & Definitions (94)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• proof
• Definition 3.2: The first exit time
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5