Numerical invariants — Question 13.1

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 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. For a cooriented codimension-one foliation choose a nowhere-zero $1$-form $\alpha$ with $T\mathcal F=\ker\alpha$. Integrability gives $d\alpha=\alpha\wedge\omega$. The Godbillon--Vey class is $\operatorname{gv}(\mathcal F)=[\omega\wedge d\omega]\in H^3(M;\mathbb R)$, independent of these choices, and its value on an oriented closed $M$ is the integral of this $3$-form.

Statement recorded in the supplied source.

Let $\mathcal F$ be a minimal taut $C^2$ foliation of an atoroidal oriented closed $3$-manifold $M$, with $\operatorname{gv}(\mathcal F)[M]\ne0$. Can its defining form $\alpha$ and a corresponding $\omega$ satisfying $d\alpha=\alpha\wedge\omega$ be chosen so that the $3$-form $\omega\wedge d\omega$ is everywhere nonnegative or everywhere nonpositive relative to the orientation? This property is called monotone wobble. Nonnegative permits zero loci. [S2]

Short English statement

Can a minimal taut foliation with nonzero Godbillon--Vey number be described by a Godbillon--Vey density of one sign everywhere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300054 (AMR-102-0054). 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 13.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.