Minimal surfaces — Question 4.2

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.

Statement recorded in the supplied source.

For a specified collection $(\mathcal F_i)_{i\in I}$ of taut foliations on the same manifold $M$, determine obstructions to existence of one Riemannian metric $g$ and isotopies $h_{i,t}$ of $M$, one for each $i$, such that all leaves of $(h_{i,1})_*\mathcal F_i$ are $g$-minimal, simultaneously for every $i\in I$. [S2]

Short English statement

Determine when a collection of taut foliations can all be made minimal for one metric, allowing an isotopy of each foliation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300008 (AMR-102-0008). 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 4.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.