Immersed objects — Question 14.3

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. A total foliation of a $3$-manifold is a triple $(\mathcal F_1,\mathcal F_2,\mathcal F_3)$ of mutually transverse codimension-one foliations which in small charts are simultaneously topologically conjugate to the three families of coordinate planes in $\mathbb R^3$. It is total taut when each of the three foliations is taut.

Statement recorded in the supplied source.

Characterize the $3$-manifolds that admit a total taut foliation. Both simultaneous local coordinate form and tautness of all three members are required. [S2]

Short English statement

Classify manifolds with three simultaneously transverse taut foliations locally modeled on the coordinate-plane foliations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300063 (AMR-102-0063). 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 14.3; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.