§ 2.40 Products of closed shrinking curves44DD8966BC9A

Let $m\geq 2$ and let each $\gamma _ j:S^1\to \mathbb R^2$ be a normalized closed Abresch–Langer shrinker, including the round circle if desired. Then

satisfies $\vec H=-X^\perp /2$, and $X_ t=\sqrt{-t}\, X$ for $t<0$. The product formula is exact, but any noncircular factor still requires the curvature ODE and closing condition.

• The domain is the compact $m$-torus. These are complete proper immersions of dimension and codimension $m$, Lagrangian in $\mathbb C^ m$, with flat normal bundle and nowhere vanishing mean-curvature vector. The product metric is intrinsically flat.

• A product is embedded when every factor is an embedded circle. A noncircular factor has self-intersections, and so does the product immersion. Multiple covers must not be treated as embeddings.

• Unless all factor images are circles, the image is not contained in a sphere. The all-circle product in $\mathbb R^4$ is the Clifford torus described separately above.

• These compact products have no ends and finite entropy. The centered Gaussian integral factors, so $\lambda (\gamma _1\times \cdots \times \gamma _ m)=\prod _ j\lambda (\gamma _ j)$, with the immersion multiplicity counted. They have a Type I homothetic extinction at the origin. No general stability or uniqueness claim for this family is taken from the cited product construction.

The figure displays the planar factors and identifies the ambient dimension of their product.