Kirby Problem 3.14

Definitions and mathematical statement

Let $K\subset S^3$ be a smooth knot and suppose $M=S^3\setminus K$ carries its complete finite-volume hyperbolic metric, normalized to sectional curvature $-1$. A surface $S\subset M$ is closed if it is compact without boundary, and embedded if its inclusion is injective and locally a smooth submanifold. It is totally geodesic if its second fundamental form vanishes identically; equivalently, every geodesic of its induced metric is locally an ambient hyperbolic geodesic.

Can there exist such a pair $(K,S)$ with $S$ a closed embedded totally geodesic surface? The Menasco--Reid conjecture gives the negative answer: \[ \text{No hyperbolic knot complement in }S^3\text{ contains a closed embedded totally geodesic surface.} \] All three adjectives matter. A punctured or spanning totally geodesic surface is not closed; an immersed one need not be embedded; a surface with arbitrarily small but nonzero principal curvatures is not totally geodesic. The knot must have one component and lie in $S^3$, so examples in multicomponent link complements or in other ambient three-manifolds do not decide this question.

Short English statement

Can the complement of a hyperbolic knot in the three-sphere contain a compact embedded surface that is exactly totally geodesic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2812 (KP-3.14), 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 3.14. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.