§ 2.46 Cylinders over embedded rotational shrinking tori (Angenent, 1992; Berchenko-Kogan, 2018; Chen–Lee–Sun–Zhao, 2026)99055D91C188

Let $T^2\subset \mathbb R^3$ be a normalized embedded Angenent shrinking torus as in Example 2.13. For every integer $k\geq 1$, its Euclidean product

is a smooth embedded shrinker of dimension $k+2$ and codimension one. If $X_ T(p)$ parametrizes the torus, the exact product and flow formulas are

The torus factor itself is specified by its closed weighted geodesic or rotational ODE, not by a closed elementary formula. Since the Euclidean factor is tangent to the product,

This verifies the product construction directly.

• Each product is connected, complete, proper, noncompact, and without boundary, with topology $T^2\times \mathbb R^ k$. It has two topological ends when $k=1$ and one when $k\geq 2$. Its intrinsic dimension is at least three, so a surface genus is not assigned; genus one describes the torus factor only.

• The product inherits the torus's axial rotational symmetry and is invariant under translations and orthogonal transformations in the Euclidean factor. The two ends of $\Sigma _1\subset \mathbb R^4$ have the fixed torus as cross-section.

• Gaussian integration in the Euclidean factor gives, for $a\in \mathbb R^3$, $b\in \mathbb R^ k$, and $\tau >0$,

  \[ F_{(a,b),\tau }(\Sigma _ k)=F_{a,\tau }(T^2),\qquad \lambda (\Sigma _ k)=\lambda (T^2)=F_{0,1}(\Sigma _ k). \]

  Indeed, the extra factor is $(4\pi \tau )^{-k/2}\int _{\mathbb R^ k}e^{-|z-b|^2/(4\tau )}\, dz=1$. Thus the numerical estimate $1.85122$ for the torus also estimates these products, with the same lack of a certified error enclosure; the product identity itself is exact.

• As Euclidean products of a closed nonflat embedded shrinker, these examples fall within the realization theorem of [CLSZ2026], Theorem 1.2, PDF p. 2: each can occur with multiplicity one as a first-singularity tangent flow of a closed embedded hypersurface in the corresponding Euclidean space. The theorem does not assert that the initial closed hypersurface is this noncompact product.

The factor is constructed in [Angenent1992]; the numerical entropy estimate is [BerchenkoKogan2021], Section 5, with its error-bound limitation in Section 6. The interactive picture shows only three $z$-slices of $\Sigma _1$ after projection from $\mathbb R^4$ to $\mathbb R^3$. It uses the catalogue's numerical torus mesh. Apparent intersections are projection artifacts; the three slices do not represent three components or the entire three-dimensional hypersurface.