§ 2.35 Ketover self-shrinking doubled planes8423ACEAF2E2

For every integer $g\geq 1$ there exists an embedded self-shrinker $\Sigma _ g\subset \mathbb R^3$ of genus $g$ such that

The normalized smooth flow is $M_ t=\sqrt{-t}\, \Sigma _ g$, $t<0$. There is no explicit formula for $\Sigma _ g$.

• The construction gives a smooth complete properly embedded boundaryless surface of dimension two and codimension one. It is invariant under the prismatic group $D_{g+1}\times \mathbb Z_2$, including reflection across the horizontal plane $P=\{ z=0\} $. The group has order $4(g+1)$.

• For all sufficiently large $g$, the surface is noncompact with exactly two asymptotically conical ends: one in $z>0$ and one in $z<0$, each graphical over the exterior of a disk in $P$. Their cone links converge in $C^0$ to the equator as $g\to \infty $. For these $g$, the topology is a closed orientable genus-$g$ surface with two points removed.

• For the small-genus members, this theorem does not establish noncompactness or an exact number of ends. Numerical evidence for two ends already at $g=1$ is stated separately in Remark 1.2.

• As $g\to \infty $, $\Sigma _ g\to 2P$ as varifolds. From every subsequence one can extract a further subsequence and find a radius $r>0$ such that the surfaces are two normal graphs converging smoothly on each compact subset of $P\setminus C_ r$, where $C_ r=\{ (x,y,0):x^2+y^2=r^2\} $. The radius is part of the subsequential conclusion, not a prescribed geometric parameter.

• For sufficiently large $g$, the level set flow starting from the smooth surface $\Sigma _ g$ fattens (Theorem 1.3, PDF p. 4). The homothetic flow reaches its asymptotic cone at time zero; this statement concerns weak continuation beyond that singular time.

• Entropy minimization among genus-$g$ shrinkers is conjectural (Remark 1.2). Neither uniqueness nor a particular value below two is claimed.

The interactive picture is an original topology schematic informed by Proposition 7.4 (PDF pp. 36–37) and Section 8. In the subsequential large-genus regime of that proposition, $\Sigma _ g\cap P\cap B_ R(0)$ consists of $g+1$ embedded closed curves for each fixed $R>r$ and $g$ sufficiently far along the subsequence. The drawing represents these as the waists of $g+1$ necks joining two sheets near a circle. A cutaway reveals the neck walls and the gap between the sheets; the outer rims are display truncations. The round waists, neck sizes, circle radius and cone slopes are illustrative, not computed geometry. The selectable genera do not specify a threshold for the two-end theorem, and the small-genus two-ended pictures are conjectural. The multiplicity-two plane is a varifold limit rather than a smooth member of the family.