§ 2.37 Flow-generated sphere–plane shrinkers036F44595178

(Hoffman–Martin–White sphere–plane family) Write $d_1=\sqrt{2\pi /e}$ and $d_2=4/e$. For every integer $g\geq 1$ and every fixed

Hoffman–Martin–White construct a smooth embedded shrinker $\Sigma _ g\subset \mathbb R^3$ of genus $g$ with $\lambda (\Sigma _ g)<\delta $. They produce it as a tangent shrinker of an intrinsically flowing compact surface in $S^2\times \mathbb R$; the tangent shrinker itself is Euclidean. It satisfies $\vec H=-X^\perp /2$, with homothetic flow $\sqrt{-t}\, \Sigma _ g$ for $t<0$. No explicit parametrization is known from this construction.

• This is a connected, orientable, complete, properly embedded surface of dimension two and codimension one. It contains the union $Q_ g$ of $g+1$ horizontal lines through the origin at angles $j\pi /(g+1)$, and $\Sigma _ g\cap \{ z=0\} =Q_ g$. In particular it is noncompact. Its complement of the horizontal plane has genus zero.

• The symmetry group $G_ g$ used here is generated by reflection in vertical planes at angles $(2j+1)\pi /(2(g+1))$, rotations about the vertical axis through $2\pi /(g+1)$, and half-turns about the lines in $Q_ g$. This is the Costa–Hoffman–Meeks symmetry group specified in Section 5.

• For all sufficiently large $g$, $\Sigma _ g$ has exactly one asymptotically conical end. For small $g$, the theorem does not specify an exact number of ends. For a fixed admissible $\delta $, as $g\to \infty $ the surfaces converge to $\{ z=0\} \cup S^2(2)$, smoothly with multiplicity one away from the equatorial intersection circle.

• The normalized Gaussian area equals entropy and obeys

  \[ F_{0,1}(\Sigma _ g)=\lambda (\Sigma _ g)<\delta <1+\sqrt{2\pi /e}<3. \]

  The choice of $\delta $ is part of the existence statement; it does not assign an exact entropy to a fixed uniquely determined member.

• No identification with the Buzano–Nguyen–Schulz min-max examples or the Kapouleas–Kleene–Moller and Nguyen gluing examples is proved. The shared topology and limiting configuration do not establish that they coincide or are distinct.

See [HMW2026], version 3: Section 5 (PDF pp. 16–17), Theorems 40–42 (pp. 26–31), Section 9 (p. 40), and Lemma 64 (pp. 45–46). The source uses $\leq \delta $ in Theorem 40; the strict bound above is its refined choice in Theorem 41. Figures 1 and 5 (PDF pp. 3 and 26) show numerical simulations credited to Francisco Torralbo and Mario Schulz. Our interactive picture is an original topology schematic informed by those figures and the sphere–disk surgery in Figure 6 (p. 27): a connected surface, a cutaway exposing its central sheet and lower cap, the exact horizontal trace $Q_ g$, and the separate sphere–plane limit. Its one-ended finite-genus shape and cone slope are illustrative; they are not computed from the shrinker equation and do not determine the small-genus end count.