Existence questions — Question 2.3

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. For metric spaces, a quasi-isometry has constants $a\ge1,b\ge0$ with $a^{-1}d(x,y)-b\le d(f(x),f(y))\le a d(x,y)+b$, and every target point lies within a bounded distance of its image. Leaf geometry uses the intrinsic path metric from a metric on $M$.

Statement recorded in the supplied source.

Given a collection $\mathscr C$ of surface homeomorphism types, characterize the $3$-manifolds carrying a taut foliation every one of whose leaves is homeomorphic to a member of $\mathscr C$. For a collection of geometric surface types, ask the corresponding question with each leaf coarsely quasi-isometric to some member of $\mathscr C$. The requirement concerns every leaf, not an almost-everywhere condition. [S2]

Short English statement

Determine which manifolds can be foliated tautly with every leaf of a prescribed topological or coarse geometric type.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300003 (AMR-102-0003). 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 2.3; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.