§ 2.36 Flow-generated Platonic and polygonal shrinkers1AF8BDC11238

(Hoffman–Martin–White ray-avoiding shrinkers) Let $Q\subset \mathbb R^3$ be the union of rays from the origin through either the face centers of a regular Platonic solid or the vertices of a regular horizontal $n$-gon, $n\geq 3$. If $q$ is the number of rays, Hoffman–Martin–White construct a smooth embedded shrinker $\Sigma _ Q$ of genus $q-1$ such that

The associated smooth flow is $\sqrt{-t}\, \Sigma _ Q$, $t<0$. This is a construction through singularities of mean curvature flow, not an explicit formula.

• The surface has dimension two and codimension one, is complete, proper, orientable and without boundary, and has all symmetries of $Q$. The genera are $3,5,7,11,19$ for the tetrahedron, cube, octahedron, dodecahedron and icosahedron respectively. For the regular $n$-gon the genus is $n-1$.

• It occurs with multiplicity one as a tangent shrinker of an initially compact, smoothly embedded surface in $\mathbb R^3\setminus Q$. This realization statement is part of the construction, rather than an identification with Ketover's min-max surface.

• The theorem does not determine compactness or the number and types of ends for the Platonic examples or the small-$n$ polygonal examples. The discussion after the theorem states that the polygonal examples are noncompact for sufficiently large $n$ and converge to a multiplicity-two plane as $n\to \infty $; it does not give an exact end count there.

• Each example has finite entropy as a tangent shrinker of a compact initial flow, and $\lambda (\Sigma _ Q)=F_{0,1}(\Sigma _ Q)$. No numerical entropy bound or exact value is recorded here. No uniqueness or equality with the corresponding Ketover or Ilmanen–White constructions is asserted.

See [HMW2026], version 3: Theorem 12 and its discussion (PDF pp. 8–9), with the multiplicity-one tangent-flow convention on p. 4. The compact numerical pictures mentioned in the paper do not establish compactness of the members selected by the theorem. The illustration is an original compact topology model inspired by the construction of initial surfaces in Theorem 24 (PDF pp. 15–16): a thickened spherical edge skeleton has one passage around each ray of $Q$. In the polygonal case the skeleton consists of $n$ meridian semicircles joining the poles, midway between the excluded rays. The inner and outer walls join around the passages, giving genus $q-1$. The cutaway exposes this cavity. These models are not computed shrinkers or simulated flow slices; their compactness is not a claim about the tangent shrinkers. The cubical silhouette also recalls Chopp's numerical picture in Figure 2, without reproducing its mesh.