§ 2.32 Noncompact shrinkers of every positive genus (Buzano–H. T. Nguyen–Schulz, 2021; Chu–Sun, 2023)0E02104583BA

For every integer $g\geq 1$ there exists a complete embedded noncompact self-shrinker $\Theta _ g\subset \mathbb R^3$ of genus $g$. The smooth flow is $\sqrt{-t}\, \Theta _ g$ for $t<0$, using $\vec H=-X^\perp /2$. There is no explicit parametrization.

• These surfaces have dimension two and codimension one, are proper and boundaryless, and have a finite number of annular ends. Properness and the end topology are recorded in Corollary 2.8, PDF p. 8. They have at least one asymptotically conical end. Any additional ends are asymptotically conical or cylindrical.

• The proved symmetry is invariance under the dihedral group $D_{g+1}$ of order $2(g+1)$, generated by half-turns about $g+1$ horizontal axes. The surface contains these axes and meets the vertical axis orthogonally. Additional reflection symmetry is proposed, rather than proved, in Section 2, PDF pp. 3–4.

• For all sufficiently large $g$, there is exactly one asymptotically conical end. As $g\to \infty $, $\Theta _ g$ converges as a varifold to the union of the horizontal plane and $S^2(2)$, smoothly locally away from their intersection circle. Theorem 1.3 does not establish exactly one end for every small genus.

• The normalized Gaussian area satisfies the proved bound

  \[ 1<F_{0,1}(\Theta _ g)\leq 1.02+\frac4e<3. \]

  This is Lemmas 2.5–2.6, PDF p. 7: the multiplicity is one and $F_{0,1}(\Theta _ g)$ equals the min-max width. For these normalized complete shrinkers, Gaussian area equals entropy.

• The numerical simulation for $g=1$ gives Gaussian area approximately $1.7143$ (Section 4, Table 1, PDF p. 15). It is not a certified value for a uniquely identified min-max surface. The source conjectures that $\Theta _1$ has lower entropy than the Angenent torus and Gaussian Morse index five; neither statement is its existence theorem.

The relation between the large-genus members and the Kapouleas–Kleene–Møller construction is not proved to be equality. The interactive illustration is an original topology schematic informed by Section 4, Figures 3–4, PDF pp. 14–15, not a numerical solution of the shrinker equation. A vertical cutaway reveals the central sheet and lower cap. The one-ended shape at low genus reflects the paper's numerical evidence; it is not a proved end count.