Every noncompact surface is a leaf of a minimal foliation

Paulo Gusmão, Carlos Meniño Cotón

Published 2019-10-30Version 1

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. Some of these foliations are suspensions of continuous minimal actions of surface groups. 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, trivial euler class and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely $C^2$-smoothable.