Numerical references

Höskuldur P. Halldórsson

The planar classification, Sections 2–5 and Figures 4–19, supplies the ODEs and visual references for the annulus-dense shrinkers, rotators, expanding spirals and rotating shrinking spirals. The helicoidal paper, equations (3.3), (4.1) and (5.3), and Figures 1–4, supplies the helicoidal translator ODE and the classical minimal-surface formula attributed there to Wunderlich.

All displayed geometry is newly generated by scripts/generate_halldorsson_geometry.py; no paper figures or source meshes are redistributed. SciPy DOP853 integrates the arclength systems. A scalar Brent solve targets the irrational turning condition; finite precision does not certify irrationality. A comet preset uses finite-radius asymptotic shooting and is explicitly an approximation to the special orbit. Refinement and equation diagnostics are recorded in the geometry audit, without rigorous error bounds. Minimal helicoids are sampled from the exact formula. The viewers apply exact similarity motions to finite patches, and use equal spatial scales. Dashed circles mark accumulation sets, not extra flow components.

David L. Chopp

Chopp's Computation of Self-Similar Solutions for Mean Curvature Flow, Experimental Mathematics 3 (1994), 1–15 supplies the numerical candidates and visual references for the five Chopp entries: Figures 8–12 (cubical), 13–15 (octahedral), 16–19 (saddle), 20–23 (two-handled cone), and 26–28 (four-handled cone). The latter also appears in Angenent–Ilmanen–Chopp, A Computed Example of Nonuniqueness of Mean Curvature Flow in R3 (1995), p. 1943.

Chopp is also credited for Figure 1 in Ilmanen–White, Fattening in Mean Curvature Flow, which informs our two-ended topology drawing, and Figure 2 in Hoffman–Martín–White, a visual reference for the cubical silhouette. Our corresponding models are schematics, not recomputations of Chopp's numerical surfaces. Source review and drawing provenance.

Tom Ilmanen

The conjectural genus-eight picture credited to Ilmanen in Kapouleas–Kleene–Møller, Mean curvature self-shrinkers of high genus: Non-compact examples, Figure 1, is a visual reference for the sphere–plane configuration. Our KKM drawing uses the paper's Scherk equation and a project-specific drawing map; it is not a solved shrinker or the paper's matched initial surface. Drawing provenance.

O. Michael Melko

Melko's Visualizing Minimal Surfaces (2010), equations (16)–(19), provides the stationary Costa parametrization used for the ancient-Costa illustration. The project evaluates this parametrization and adds a truncated linearized normal mode. Those perturbed frames are not a numerical construction of the nonlinear ancient flow. Implementation and limitations.

Su–Tsai–Wood: infinite-time Lagrangian neck collapse

The signed-profile time plate and interactive comparison are newly drawn from Su, Tsai and Wood, Infinite-Time Singularities of the Lagrangian Mean Curvature Flow: Section 2.3 gives the Lawlor neck, equation (1.1) gives its leading-order scale dynamics, and Proposition 9.5 identifies the rescaled limit. Figures 1–2 explain the desingularization and compact topology. No paper figure or mesh is copied. The local equal-angle profiles are exact; the time-dependent scaling and global topology diagram illustrate the construction and do not compute the compact flow. Formulas, scope, and reproducibility.

Lee–Wang quadric flow

The shrinker time plate and circle-action viewer are original drawings from Lee and Wang, Hamiltonian stationary cones and self-similar solutions in higher dimension, Theorem 1.1 and Sections 2–3. They sample the exact unit-weight coordinates in C³; no PDE is solved. The displayed phase leaves and projected orbit curves are labeled as such. The SVG and browser canvas use the same coordinate generator, with a fixed ambient crop and scale across times. No figure, mesh, or software from the paper is reproduced.

The companion expander illustration samples the exact real slices x₃ = ±√(x₁² + x₂² + 2t) at positive times and the singular cone at t = 0. The comparison keeps one spatial scale and a fixed radial display cutoff. Its circle-orbit diagram is drawn in the exact two-plane spanned by x and i(x₁,x₂,−x₃) in R⁶; changing the circle phase keeps flow time fixed. These real slices are not themselves mean curvature flows in R³. See the expander geometry and verification notes.

Mario B. Schulz

The illustration for the work of Reto Buzano, Huy The Nguyen, and Mario B. Schulz follows the qualitative geometry of Section 4, Figures 3–4 of Noncompact self-shrinkers for mean curvature flow with arbitrary genus, version 2. These are visual references as well as mathematical source material. The authors produced their simulations with Kenneth Brakke's Surface Evolver. Our mesh instead samples an illustrative implicit field. BNS drawing provenance.

Schulz is individually credited for Figure 5, the genus-five numerical picture in Hoffman–Martín–White, Generating Shrinkers by Mean Curvature Flow, version 3. It informs our sphere–plane drawing and its interior view, together with Torralbo's figures below.

Schulz's Geometric Analysis Gallery, especially its self-shrinker collection, is an additional resource for these visualizations. The documented sources used for our drawings are the paper figures identified above; the gallery link does not assert that the website itself was consulted during the original implementation, or that its images or meshes were reused.

Ao Sun: supplied Chu–Sun numerical replay pages

The two supplementary Chu–Sun experiment viewers were supplied by Ao Sun with their numerical-note and archive provenance retained. Both use three.js 0.128.0; the first also uses pako 2.1.0 to decompress stored frames. These are replays of supplied computations, not numerical experiments independently reproduced by the website integration. See the integration and provenance record and the retained three.js and pako license notices. Private notes and solver archives are not exported.

Francisco Torralbo

Torralbo is credited for the numerical genus-one through genus-six illustrations in Hoffman–Martín–White, Figure 1. They inform the qualitative appearance of our sphere–plane model, alongside Schulz's Figure 5. The displayed MCF-stack surface is a topology schematic. Sphere–plane drawing provenance.

Matthias Weber

Weber is credited for Hoffman–Martín–White, Figure 7, a visual reference for our three-ended model. The source describes a conjectural qualitative genus-two picture, not a numerical shrinker approximation. Our separately generated mesh retains that distinction. Three-ended drawing provenance.