Hyperbolic geometry — Question 10.5

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct.

Statement recorded in the supplied source.

Is there $\epsilon>0$ independent of all data such that for every hyperbolic $3$-manifold $M$, every taut foliation $\mathcal F$ of $M$, and every closed geodesic $\gamma$ of length less than $\epsilon$, the curve $\gamma$ is isotopic either to a curve contained in one leaf or to a curve transverse to $\mathcal F$? Ask the same universal assertion with isotopy replaced by homotopy. [S2]

Short English statement

Is every sufficiently short geodesic, uniformly across manifolds and taut foliations, isotopic into a leaf or transverse to the foliation; and does the homotopy version hold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300043 (AMR-102-0043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 10.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.