Sublaminations and superlaminations — Question 6.6

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. A manifold is atoroidal if it has no incompressible embedded torus other than a boundary-parallel one. Write $\widetilde M$ for its universal cover, and put a tilde on a foliation or lamination for its lift. A loosesse lamination may have compressing monogons but satisfies the other essentiality requirements. Realization by minimal surfaces means that all leaves have zero mean curvature for one ambient Riemannian metric.

Statement recorded in the supplied source.

Construct an atoroidal $3$-manifold $M$ with a lamination $\Lambda$, possibly loosesse, for which no Riemannian metric on $M$ makes every leaf minimal, but whose existence proves an identifiable useful topological property of $M$. Supply both the obstruction to such minimal realization and the certified topological consequence. [S2]

Short English statement

Find an atoroidal manifold with a lamination that cannot be minimal for any metric but still certifies useful topology.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300015 (AMR-102-0015). 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 6.6; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.