Foliated Teichmüller theory — Question 11.4

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. Equip leaves with a conformal structure. Let $(\gamma_i)_{i\in I}$ be a collection of transverse circles whose union meets every leaf and $n_i\ge0$ integers. The divisor notation $D=\sum_i n_i\gamma_i$ specifies poles of order at most $n_i$ at each leaf intersection with $\gamma_i$. Write $\mathscr O(D)$ for functions leafwise holomorphic off these circles and leafwise meromorphic with these pole bounds.

Statement recorded in the supplied source.

Determine how $\mathscr O(\sum_i n_i\gamma_i)$ varies with the transverse circles $\gamma_i$. In particular determine effects of topological modifications of these circles, including crossing changes and cabling, while retaining valid transverse data. Include the function spaces themselves, not only dimensions in a particular example. [S2]

Short English statement

Describe how spaces of leafwise holomorphic functions with prescribed transverse poles change when the transverse circles move, cross or are cabled.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300050 (AMR-102-0050). 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 11.4; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.