Leaf topology of minimal hyperbolic foliations with non simply-connected generic leaf

TL;DR

This paper constructs minimal hyperbolic foliations on closed 3-manifolds with non-simply connected generic leaves, demonstrating that surfaces satisfying a specific end genus condition can be realized as leaves.

Contribution

It proves that surfaces meeting condition $(igstar)$ can be realized as leaves in minimal hyperbolic foliations on closed 3-manifolds, including countably many such surfaces simultaneously.

Findings

Surfaces satisfying $(igstar)$ are homeomorphic to leaves of minimal hyperbolic foliations.

Multiple noncompact surfaces satisfying $(igstar)$ can coexist as leaves.

All constructed examples are hyperbolic foliations with leafwise negative curvature.

Abstract

A noncompact (oriented) surface satisfies the condition $(\star)$ if their isolated ends are ''accumulated by genus''. We show that every surface satisfying this condition is homeomorfic to the leaf of a minimal codimension one foliation on a closed $3$-manifold whose generic leaf is not simply connected. Moreover, the above result is also true for any prescription of a countable family of noncompact surfaces (satisfying $(\star)$): they can coexist in the same minimal codimension one foliation as above. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature.