From a perturbed Angenent torus to another genus-one shrinker

The flow starts at t = 0 from the rotationally symmetric shrinking torus perturbed by a normal graph ε[−cos(πs) φ₀ + sin(πs) φ₁]ν with ε = 0.2 and s ≈ 0.50041. Here φ₀ is the positive principal unstable Jacobi eigenfunction and φ₁ ∝ e^(r²+z²)/4 cos θ / r is the explicit first-angular-mode eigenfunction chosen in the note. The one tuned parameter s sits between “inward tube collapse” (smaller s) and “outward hole closure” (larger s), the interpolation mechanism of Chu–Sun’s Theorem 1.1. The flow develops a genus-one singularity on the x-axis at time T ≈ 0.90768.

The picture shows the rescaled surface Y = (X − c)/√(T − t) with c ≈ (−0.3345, 0, 0). The slider runs in τ = −ln(T − t), which stretches the final instants so that all eight spatial zoom stages of the computation are visible. The torus axis is vertical (z); the surface is mirror-symmetric in y and z, which is why it is drawn by reflecting one quarter.

• Golden surface: the flow. Colour (default) is each point’s distance to the stationary candidate: blue is close, red is far.
• Teal ghost: the independent stationary candidate (genus one, one apparent conical end; Gaussian area ≈ 1.7143), rotated into the flow’s frame by the note’s fixed rotation. It is not used by the flow.
• Orange wireframe: where the unperturbed torus would be at the same time, shown in the same rescaled frame. It shrinks to a point at t = 1 and so drifts away; the perturbed flow does not follow it.

I replayed the note’s exact ε = 0.2 critical trajectory (eight adaptive spatial zooms) with the archive’s own level-set solver and saved the field at 129 times equally spaced in τ. The replayed stage fields match the archived ones to within 2×10⁻⁴ of a grid cell. Surfaces are extracted from the unsmoothed scalar field (marching cubes).

The residual and distance use the note’s own estimators (quartic Savitzky–Golay derivatives with window 7; Gaussian-weighted over the ball |Y| < 3; face-centroid distances to the stationary mesh). At the last stage my common-frame residual is 0.00844, matching the 0.008439 reported in the note for the same fitted centre and time. All frames use that one common (c, T); per-frame fits shift the numbers slightly.

• This is one numerical experiment (first angular mode, ε = 0.2). The note itself does not prove convergence of the discretised flow, uniqueness of the selected surface, the structure of its end, or its Morse index.
• The surface is cut at the chosen view radius, and each stage’s computational box is cropped from the previous one, so later frames show only the central region. Points near that cut are not meaningful.
• The stationary candidate is itself a numerical (reconstructed) mesh, not an exact solution. The residual and distance shown never reach zero.
• The residual first rises as the flow leaves the Angenent torus and only then falls. At the very last stage it rises slightly again; the note reads this as consistent with departure from an unstable shrinker after finite parameter tuning, and says explicitly that this is not a stability theorem.
• The agreement is a statement about the core. The note’s comparison is restricted to the ball |Y| < 3; farther out, even the last frames are visibly farther from the candidate (greener in the colouring), and the computed domain does not extend far enough to say what the surface does beyond that.
• The Gaussian-density curve is computed on box-truncated surfaces; its small non-monotone wiggles are truncation effects, not violations of Huisken’s monotonicity.