§ 2.38 Flow-generated three-ended shrinkers5245200FA4A7

(Hoffman–Martin–White three-ended family) For every sufficiently large integer $g$, Hoffman–Martin–White construct a smooth embedded shrinker $\Sigma _ g\subset \mathbb R^3$ with genus $2g$ and exactly three ends. It has the normalization and smooth evolution

The construction obtains the surface as a multiplicity-one tangent shrinker of a flow whose initial surface is compact and embedded in $S^2\times \mathbb R$ with its product metric. The shrinker lives in the Euclidean tangent space. There is no explicit parametrization.

• It is a connected, orientable, complete, proper noncompact surface of dimension two and codimension one. Its topology is a closed genus-$2g$ surface with three punctures; the three ends are annular and asymptotically conical.

• It has the group $G_ g$ and the union of lines $Q_ g$ specified in Example 2.37. In particular, $\Sigma _ g\cap \{ z=0\} =Q_ g$ and $\Sigma _ g\setminus \{ z=0\} $ has genus zero. These are the symmetries of the genus-$g$ Costa–Hoffman–Meeks minimal surface, but the shrinker has genus $2g$.

• As $g\to \infty $, the surfaces converge to the horizontal plane with multiplicity three. After passage to a subsequence, convergence is smooth away from one circle in that plane. The circle is part of the limiting analysis, not an independently prescribed radius.

• The entropy has a bound $\lambda (\Sigma _ g)\leq \Lambda <\infty $ independent of $g$, and $F_{0,1}(\Sigma _ g)=\lambda (\Sigma _ g)$. The paper does not specify a numerical value of $\Lambda $ in this theorem. This entry therefore does not assert $\lambda <3$ or an exact entropy limit.

• For each small $g$ the preceding construction produces a shrinker with genus $g$ or $2g$ and a uniform entropy bound; the assertions of genus $2g$ and three ends require $g$ sufficiently large. The paper's genus-two picture is explicitly conjectural and is not a numerical approximation. No uniqueness or identification with Shao–Zou's tripling construction is proved.

See [HMW2026], version 3: Theorem 43 (PDF p. 31), Theorem 55 and Conjecture 56 (p. 38), Theorem 59 (p. 39), Proposition 62 and Lemma 64 (pp. 45–46). The interactive illustration is an original connected topology model guided by Figure 7: three exterior sheets are joined by alternating upper and lower necks. Its cutaway exposes the neck interiors, and its plan relates the necks to $Q_ g$. The separate limit view shows a single plane with multiplicity three. Neck sizes, end slopes, and the displayed circle radius are illustrative; no shrinker equation or mean curvature flow has been computed, and the displayed small values of $g$ do not assert three-ended existence.