Kirby Problem 3.1

Definitions and mathematical statement

A hyperbolic three-manifold here is a connected complete Riemannian three-manifold of constant sectional curvature $-1$ and finite volume. Its cusps are ends isometric outside compact sets to quotients of horoballs. A cusp cross-section is a closed flat surface: a torus for an orientable cusp and a Klein bottle for a nonorientable cusp. An orientable manifold has only torus cusps, but a nonorientable manifold can also have torus cusps.

For each of the following classes $\mathcal A$, determine $v(\mathcal A)=\inf\{\operatorname{vol}(M):M\in\mathcal A\}$ and identify all manifolds attaining it, up to isometry:

1. closed nonorientable hyperbolic three-manifolds;
2. manifolds with exactly $n$ cusps, without an orientability restriction, for each $n\geq3$;
3. manifolds with exactly $n$ orientable cusps, allowing nonorientable ambient manifolds and any additional nonorientable cusps;
4. orientable manifolds with exactly $n$ cusps, for every $n\geq1$.

For (c), this reads the source's "$n$ orientable cusps" as counting torus cusps only. The alternative reading requires exactly $n$ cusps in total, all tori; its minimum should be distinguished if additional cusps are excluded. Neither reading requires ambient orientability. In each family the curvature normalization is fixed, since otherwise rescaling the metric would make the volume comparison meaningless.

Short English statement

Find the least hyperbolic volume and every minimizing manifold in the specified closed, cusped, and orientability classes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2799 (KP-3.1), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.1. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.