Degenerate Neckpinch Singularity

• Degenerate neckpinch singularity is a phenomenon in mean curvature flow where a hypersurface develops dual blow-up limits: a cylindrical tangent flow and off-axis Bowl soliton rescalings.
• The peanut solution is constructed using matched asymptotic expansions, spectral decomposition with Hermite modes, and finite-dimensional reduction techniques.
• Instability in these solutions causes divergent behaviors, with perturbations leading either to spherical extinction or nondegenerate neckpinches, emphasizing sensitivity to initial conditions.

A degenerate neckpinch singularity is a phenomenon in the evolution of hypersurfaces under mean curvature flow (MCF) in ℝⁿ⁺¹. Unlike conventional neckpinch singularities, where the tangent flow at the singularity is cylindrical and all blow-up limits are cylinders, a degenerate neckpinch—exhibited by the so-called "peanut solution"—has the property that the tangent flow at the singularity is the round cylinder Sⁿ⁻¹×ℝ, but certain rescaled blow-up limits along off-axis sequences yield the translating Bowl soliton. The peanut solution is a closed, rotationally symmetric, non-convex hypersurface that evolves under MCF, shrinks to a single point, and develops a singularity characterized by this dual limit structure. These solutions, first conjectured by Hamilton and constructed by Angenent, Altschuler, and Giga, provide a precise geometric manifestation of degenerate neckpinch singularities and have been shown to be highly unstable, admitting perturbations leading to either spherical singularities or nondegenerate neckpinches (Angenent et al., 4 Dec 2025).

1. Definition and Geometric Characterization

A peanut solution under mean curvature flow is a closed, rotationally symmetric hypersurface M(t) in ℝⁿ⁺¹ evolving via

$\partial_t X = -H n,$

where H is the mean curvature and n is the unit normal. The defining properties are:

• M(t) becomes extinct at a single point in finite time T.
• The hypersurface remains non-convex throughout [0, T).
• At the unique singularity, the tangent flow is Sⁿ⁻¹×ℝ (a round shrinking cylinder of multiplicity one).
• There exists at least one sequence of space–time points (x_k, t_k)→(0, T) with associated rescalings

$$X^{(k)}(y,s) = \lambda_k \big(X(t_k + \lambda_k^{-2} s) - x_k\big), \quad \lambda_k\to\infty,$$

whose pointed blowup limits are the Bowl soliton. This combination of tangent flows is called a degenerate neckpinch.

2. Existence Theory and Construction

Angenent, Altschuler, and Giga (and later Angenent and Velázquez) constructed these degenerate neckpinch/peanut solutions using matched asymptotic expansions in rescaled MCF. The peanut is built by:

• Starting with a super-ellipsoid type outer profile:

$$U_{\text{out}}(y, \tau_0) = \sqrt{2(n-1) - K_0 y^m e^{-(m/2)\tau_0}},$$

where m is even and τ_0 ≫ 1.

• Gluing in combinations of Hermite modes, H₀,…,H_{m–2}, in the parabolic region |y|=O(e^{τ/4}).
• Near the "tips," scaling in a Bowl soliton profile to ensure smooth embedding.
• The construction is justified by the dynamical system induced by the rescaled MCF equation

$$u_\tau = \frac{u_{yy}}{1 + u_y^2} - \frac{y}{2}u_y - \frac{n-1}{u} + \frac{u}{2}.$$

A shooting method on the finite-dimensional unstable manifold (supported by the Hermite modes) identifies initial data that generate a solution approaching the desired degenerate neckpinch (Angenent et al., 4 Dec 2025).

3. Tangent Flows and Blow-up Analysis

At the singularity (0, T), parabolic dilations yield tangent flows that are round cylinders Sⁿ⁻¹×ℝ, as dictated by Huisken’s monotonicity. However, for sequences focusing away from the central axis (off-axis rescalings), the pointed limits are Bowl solitons W that solve

$\frac{W''}{1 + (W')^2} + (n-1)\frac{W'}{\xi} = 1.$

This dichotomy, with both cylindrical and Bowl soliton blow-up behaviors, is the signature of the degenerate neckpinch.

4. Instability and Dynamics Near the Peanut

The degenerate neckpinch (peanut) solution is highly unstable. Perturbations in the unstable Hermite polynomial modes (H₀, H₂) at a rescaled time τ₀, with

$$u(y, \tau_0) = \hat{u}(y, \tau_0) + \epsilon \eta_0(y/\ell_0)[\Omega_0 H_0(y) + \Omega_2 H_2(y)],$$

generate divergent behaviors depending on the direction in the two-dimensional unstable subspace:

• Spherical (S-case): perturbation projects negatively on H₂; convexity results and the flow shrinks to a round point.
• Cylindrical (C-case): positive direction grows, stabilizing a nondegenerate neckpinch, and the flow pinches off with two persistent lobes. Barrier arguments, spectral decomposition, and maximum principle techniques are used to rigorously track these instabilities via inner-outer estimates and finite-dimensional reduction (Angenent et al., 4 Dec 2025).

5. Convergence to Ancient Ovals

If the peanut solution is perturbed so that the subsequent flow yields a spherical singularity, then under proper rescaling—specifically, at times when convexity is attained—the solution converges in the smooth topology to the unique closed, convex ancient oval constructed previously by Hamilton–White, Haslhofer–Hershkovits. Uniform estimates for the deviation

$$|q| + |q_y| + |q_{yy}| \leq C \epsilon_k (1 + |y|^4)$$

in exponentially large domains confirm this convergence.

6. Analytical and Spectral Structure

The analysis leverages the spectral decomposition of the drift Laplacian

$L = v_{yy} - \frac{y}{2}v_y + v$

on L²(e^{-y²/4} dy), whose eigenfunctions are Hermite polynomials H_{2j} with spectrum λ_{2j} = 1 - j. Key techniques include:

• Matched asymptotic expansions in parabolic and tip regions.
• Barrier methods to control unstable directions.
• Energy estimates in truncated and full regions.
• Degree-theoretic arguments for existence of critical perturbation directions.

7. Significance and Implications

Degenerate neckpinch singularities in MCF, as exemplified by peanut solutions, illustrate the possibility of non-unique blow-up behavior at geometric singularities—contrary to the simplified cylinder- or sphere-only scenarios. This demonstrates that the landscape of mean curvature singularities is more intricate, and resolution of such singularities is highly sensitive to initial data. These peanut solutions act as organizing centers in the space of MCF evolutions, separating flows that end in spherical extinction from those that develop nondegenerate neckpinches, and their instability reveals the prevalence and accessibility of both outcomes in high-codimension perturbations (Angenent et al., 4 Dec 2025).