§ 2.14 Chopp cubical shrinker candidate90FE05D9053E

(Numerical genus-five cube) The cubical candidate is a two-dimensional hypersurface in $\mathbb R^3$, with the topology of a closed orientable surface of genus five and cubical symmetry. Chopp starts with the boundary of a ball bored along the three coordinate axes. The connected central cavity has six openings; this is not a genus-three solid with three disjoint tunnels.

The algorithm evolves a family of level sets, selects a level using a geometric ratio, and dilates to maintain a chosen volume. It gives an approximation, with no closed parametrization or exact implicit equation for the shrinker. The target equation and associated flow, in our convention, are

Chopp's equation (3.1) instead uses $H=\langle X,\nu \rangle /(2T)$ for an unrescaled surface $S$; the corresponding normalized surface would be $\Sigma =T^{-1/2}S$. His constant-volume display is not automatically the slice at $t=-1$.

The intended surface is smooth, compact, embedded and without boundary; if such a surface exists it is complete and proper and has no ends. The computation does not establish this existence statement. The reported self-similarity diagnostic is $E=0.01869$, and subsequent ordinary MCF approximately preserves the computed shape. This is a finite-grid check, not a certified error bound. No uniqueness, stability spectrum, Gaussian area or entropy value is established for this particular candidate. The diagnostic $E$ is not Gaussian area or entropy.

Source: [Chopp1994], Sections 2–3 and 4.1, pp. 2–8, Figures 8–12. The later constructions in Examples 2.36 and 2.27 have related symmetries; no identification with this numerical surface is asserted. The rotatable illustration is a compact topology schematic with six passages into a cavity, not Chopp's computed mesh or a newly solved shrinker.