Immersed objects — Question 14.2

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. An immersed incompressible surface $\phi:\Sigma\to M$ has injective induced map on fundamental groups. Use the pullback of a fixed ambient Riemannian metric for intrinsic surface balls. Pointed convergence on compact sets means that arbitrarily large local surface pieces based at the chosen points converge to the corresponding pieces of the specified leaf.

Statement recorded in the supplied source.

For every leaf $\lambda$ of a taut foliation $\mathcal F$ of $M$ and every $p\in\lambda$, do there exist compact immersed incompressible surfaces $\phi_i:\Sigma_i\to M$, points $p_i\in\Sigma_i$, and radii $r_i\to\infty$ such that the pointed images $\phi_i(B_{\Sigma_i}(p_i,r_i))$ converge on compact sets to $(\lambda,p)$? The approximants must be incompressible; arbitrary doubled leaf pieces do not suffice. [S2]

Short English statement

Can each pointed leaf of a taut foliation be approximated on larger and larger regions by compact immersed incompressible surfaces?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300062 (AMR-102-0062). 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 14.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.