Kirby Problem 3.18

Definitions and mathematical statement

Let $(M,g)$ be a closed connected hyperbolic three-manifold with sectional curvature $-1$. A smooth codimension-one foliation $\mathcal F$ partitions $M$ into connected immersed surfaces called leaves, with local charts in which leaves are the level sets of one coordinate. In particular it has no singularities and its tangent planes form an integrable rank-two subbundle of $TM$.

A leaf is minimal with respect to $g$ if its mean-curvature vector vanishes identically. Locally, after choosing a unit normal, this says that the sum of its two principal curvatures is zero. It means that area has zero first variation under every compactly supported normal deformation.

Does there exist at least one closed hyperbolic three-manifold with a codimension-one foliation all of whose leaves are minimal in its hyperbolic metric? The leaves may be noncompact or dense in $M$; they are not required to be fibers of a surface bundle. The question is existential over $M$ and $\mathcal F$, not an assertion for every hyperbolic manifold. Making a foliation minimal by choosing a different, nonhyperbolic Riemannian metric does not answer it, since the curvature $-1$ metric is part of the requirement.

Short English statement

Does any closed hyperbolic three-manifold admit a nonsingular foliation whose every leaf has zero mean curvature in the hyperbolic metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2816 (KP-3.18), 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 3.18. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.