Second-mode perturbation of the Angenent torus and a two-ended genus-one candidate

The flow starts at t = 0 from the rotationally symmetric shrinking torus perturbed by a normal graph ε[−cos(πs) φ₀ + sin(πs) φ₂]ν with ε = 0.4. Here φ₀ is the positive principal unstable Jacobi eigenfunction and φ₂ = f₂(ℓ) cos 2θ is the principal second-angular-mode eigenfunction (eigenvalue ≈ 0.4876). The one tuned parameter is s ≈ 0.5009123127, chosen target-free, without using the stationary surface, to sit between “paired solid-tube pinching” and “central-hole closing”. The perturbation keeps all three coordinate reflections; the surface is therefore drawn by reflecting one eighth.

The picture shows the rescaled surface Y = X/√(T − t) with T = 0.971915054035097, the single physical time the note fitted on three preselected frames. The centre stays at the origin by symmetry. The slider runs in τ = −ln(T − t), which stretches the final instants. The torus axis is vertical (z); the two ends of the candidate point along ±x.

• Golden surface: the flow. Colour (default) is each point’s distance to the stationary candidate: blue is close, red is far.
• Teal ghost: the stationary candidate, a genus-one surface with two non-circular conical ends and Gaussian area ≈ 1.7901 (degree-48 polynomial model with its separately solved exterior, shown truncated). It is never used by the flow.
• Orange wireframe: where the unperturbed torus would be at the same time, in the same rescaled frame. It separates steadily from the flow (RMS distance 0.11 at t = 0, above 0.5 later, in torus units).

I replayed the note’s selected deeper-cutoff 64 × 128 parametric trajectory (fixed topology, fourth-order finite differences, RK4, ordinary mean curvature flow with a tangential gauge) with the archive’s own solver and saved 98 states on a grid equally spaced in τ. The replay is bit-for-bit identical to the archived frames: all 49 archived frames match with zero difference, at the same step counts, and the final state equals the archived last state (t = 0.97184934). The snapshots never change the time step.

Residual, Gaussian area and σ use the note’s own estimator (doubled-grid Fourier geometry at the fixed T). For the note’s six audited frames (42–47) my values agree with its audit table to all printed digits, for example the literal residual 0.04804739, 0.04065169, 0.03429151 | 0.02917890, 0.02849774, 0.05975823. The distance to the candidate is my own addition: the Gaussian-weighted RMS, over |Y| < 3, of the distance from the flow’s native vertices (with lumped areas, no interpolation) to the stationary surface, evaluated by projecting onto the analytic degree-48 model and cross-checked against the exported mesh. A version computed on the Fourier-resampled surface agrees to within 5% at every frame and gives the same closest approach.

Residual and Gaussian density are shown through frame 92 (τ ≈ 9.23). After that the parametrization degenerates (minimum area density below 2×10⁻⁵, falling to about 10⁻⁷), the Fourier geometry breaks down (the residual jumps from 0.17 to 2.5 and 14), and those two quantities are marked n/a. The surfaces and the native-vertex distance are still shown.

• No convergence claim. The note says explicitly that at this checkpoint no flow-based convergence to the candidate is claimed, and that stationary accuracy alone does not settle the dynamical identification. The flow here approaches and then departs; the closest approach is marked on the plots.
• The closest approach is not very close in absolute terms: the distance only reaches about 0.05 (against 0.32 at t = 0), and the lowest residual is about 0.028. The final frames are resolution-limited in the note’s own words: its last audited row’s curvature estimate changes from 0.0376 to 0.0598 to 0.0634 as the Fourier quadrature is refined, while the native finite-difference value is 0.0287. The shaded region marks σ below the last audited scale; the rise of the residual there is therefore not a clean measurement of departure.
• The critical parameter depends on the grid, and the note itself says the fixed-parameter trajectory eventually leaves because of finite tuning. The parametric flow cannot continue through the topological singularity.
• Surfaces are cut at the chosen view radius. The distance is a statement about the core (|Y| < 3); the candidate is itself a numerical approximation (sampled residual about 2×10⁻⁸, certified geometry for the stored approximation only), not a proved exact shrinker.
• This is one numerical experiment (second angular mode, ε = 0.4). Uniqueness, the identification of the flow’s tangent with the candidate, and the Morse index are not established.