§ 2.27 Ketover shrinkers with Platonic symmetryB675C6112328

There are smooth connected embedded self-shrinking surfaces $\Sigma \subset \mathbb R^3$ with the following pairs of genus and rotational symmetry group:

Here the subscripts give the orders of the tetrahedral, octahedral, and icosahedral rotation groups. The normalization is $\vec H=-X^\perp /2$, and $M_ t=\sqrt{-t}\, \Sigma $ for $t<0$. No explicit parametrization is supplied.

• The sweepout joins two spherical sheets by one neck for each face of a tetrahedron, cube, octahedron, dodecahedron, or icosahedron, respectively. The numbers of necks are $4,6,8,12,20$; the first neck connects the components, so the genus is one less than this number. This is a construction description, not an exact metric model.

• The min-max limit is multiplicity one. It is complete and proper, has no boundary, and retains the stated genus and symmetry. The construction controls the genus but does not determine compactness, the number of ends, or end asymptotics. In particular, a compact picture suggested by numerical work must not be used to assign zero ends to these proved examples.

• The proof gives a useful strict Gaussian-area bound:

  \[ \frac4e < F_{0,1}(\Sigma )=\lambda (\Sigma )<\frac8e. \]

  For the doubled cube, the first equivariant width is $4/e$, the second is strictly below twice the first by Lemma 2.4, and the equality of the two widths is excluded in Remark 2.7. Remark 2.5 gives the same construction for the other four symmetry types. These are bounds, not computed entropy values.

• No uniqueness or stability classification is asserted. The two-parameter equivariant construction alone does not give an index bound for unrestricted variations. Later constructions with similar symmetries need not be different surfaces, and no identification is assumed here.

Sources: [Ketover2016], Theorem 0.1 and the compactness discussion, PDF pp. 2–3; Theorems 1.1 and 1.3, p. 5; Lemma 2.4 and multiplicity-one argument, p. 12; Remark 2.5 and Remark 2.7, pp. 14–15. The illustration shows the neck counts and symmetry of the sweepout, not a numerical solution or a proof of compactness of the limit.