Reeb components — Question 5.2

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.

Formulate useful extensions of tautness for arbitrary open $3$-manifolds, addressing both topological and geometric settings rather than fixing a single candidate in advance. Source suggestions include generalized taut structures on interiors of compact manifolds; foliations transverse, possibly uniformly transverse, to the lift of a volume-preserving flow from a compact base; eventual or uniform positive transversality along sufficiently long flow segments; and lifted positive transverse loops that collectively meet every leaf. Develop and compare suitable definitions and their consequences. [S2]

Short English statement

Develop appropriate notions of tautness on noncompact manifolds, including topological definitions and versions controlled by transverse flows or lifted loops.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300010 (AMR-102-0010). 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 5.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.