Every noncompact surface is a leaf of a minimal foliation

TL;DR

This paper proves that every noncompact oriented surface can be realized as a leaf in a minimal foliation of a closed 3-manifold, using suspensions of minimal circle actions, with examples being hyperbolic and not transversely smoothable.

Contribution

It establishes that all noncompact surfaces can be embedded as leaves in minimal foliations of 3-manifolds, extending to multiple topologies and providing hyperbolic examples.

Findings

Every noncompact oriented surface is a leaf of a minimal foliation.

Constructs include suspensions of minimal circle actions.

Examples are hyperbolic and not transversely smoothable.

Abstract

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of open surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely $C^2$-smoothable.