Foliated Teichmüller theory — Question 11.2

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. For an integral $2$-homology class $u$, the Thurston norm $x(u)$ is the minimum of $\sum_j\max\{0,-\chi(S_j)\}$ over embedded oriented representatives $S=\bigcup_j S_j$; extend it by homogeneity and continuity to real homology. Its unit ball has a polyhedral structure. A fibered face is a face whose cone contains classes represented by fibers of fibrations over $S^1$. A pseudo-Anosov flow has stable and unstable hyperbolic directions off finitely many closed prong-singular orbits. A transverse flow has its direction outside $T\mathcal F$; it preserves a volume form $\nu$ when its generating field $X$ satisfies $\mathcal L_X\nu=0$. Bers' construction for surface automorphisms uses extremal Teichmüller geometry.

Statement recorded in the supplied source.

Find an analytic construction of transverse pseudo-Anosov flows in the taut-foliation setting, analogous to the extremal approach underlying Bers' proof of the classification of surface automorphisms. Study optimization over leafwise conformal structures and transverse flows or dilatations, with appropriate natural parameterization. The question is a construction principle where such flows can occur, not the assertion that every taut foliation has one. [S2]

Short English statement

Seek an analytic, Teichmüller-type construction of pseudo-Anosov transverse flows rather than only topological constructions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300048 (AMR-102-0048). 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 11.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.