Hyperbolic geometry — Question 10.8

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 surface lamination $\Lambda$ is a closed subset of $M$ locally homeomorphic to a product of a disk with a closed transverse set. It is essential when it has no sphere leaf or torus leaf bounding a solid torus, and its complementary regions are irreducible, have incompressible boundary, and have no compressing monogon. Such a monogon is a compressing disk with one boundary arc on a boundary leaf and the other along a transverse interval fiber. It is genuine if some complementary region is not an interval bundle. A sublamination is a closed union of leaves.

Statement recorded in the supplied source.

For an essential surface lamination $\Lambda$ in a hyperbolic $3$-manifold $M$, is there an ambient isotopy taking $\Lambda$ to a lamination whose leafwise intrinsic Gaussian curvature, for the metric induced from the fixed hyperbolic metric, is at least $-2$ at every point? Changing the ambient metric to a freely chosen leafwise hyperbolic metric is not the requested operation. [S2]

Short English statement

Can an essential lamination be isotoped within the fixed hyperbolic metric so every leaf has Gaussian curvature at least minus two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300046 (AMR-102-0046). 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 10.8; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.