Minimal surfaces — Question 4.1

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.

Statement recorded in the supplied source.

Given a taut foliation $\mathcal F$ on $M$, describe the set of Riemannian metrics $g$ for which some ambient isotopy $h_t$ with $h_0=\operatorname{id}_M$ makes every leaf of $(h_1)_*\mathcal F$ a minimal surface in $(M,g)$. The metric is fixed while the isotopy is chosen; the question asks for the full set of such metrics, not only existence of one. [S2]

Short English statement

Characterize all metrics for which a given taut foliation can be isotoped so that every leaf is minimal.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300007 (AMR-102-0007). 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 4.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.