Kirby Problem 3.12

Definitions and mathematical statement

Let $M$ be a hyperbolic three-manifold with curvature $-1$. For a smooth immersion $f:S\to M$ of a closed connected orientable surface of genus at least two, define its area using the pullback metric: \[ A(f)=\int_S dA_{f^*g_M}. \] Self-overlaps are counted with multiplicity. Require $f_*:\pi_1(S)\to\pi_1(M)$ to be injective. Taking the infimum over immersions in a homotopy class is allowed; one is not required to choose a totally geodesic representative.

Determine the following universal infima: (a) $A(f)$ over all such immersions in all closed hyperbolic three-manifolds; (b) the same infimum with the genus fixed at a specified $g\geq2$; (c) the infima obtained by requiring $f$ to be an embedding; and (d) the analogous question for closed orientable essential surfaces when the ambient hyperbolic manifold is allowed to be nonclosed. For (d), essential excludes spheres and peripheral surfaces, including surfaces homotopic into a cusp, and retains $\pi_1$-injectivity.

The ambient manifold varies in these optimizations; a lower bound depending on one fixed $M$ is not a universal answer. The question asks for the actual infimum, so one must pair a proposed universal bound with examples approaching it, without assuming that a minimum is attained.

Short English statement

Find the smallest possible areas of essential orientable surfaces in hyperbolic three-manifolds, with variants fixing genus or requiring an embedding.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2810 (KP-3.12), 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 3.12. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.