Every noncompact surface is a leaf of a minimal foliation

Paulo Gusmão; Carlos Meniño Cotón

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arXiv · 1910.13839

Every noncompact surface is a leaf of a minimal foliation

Paulo Gusmão·Carlos Meniño Cotón

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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.

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Paulo Gusmão, Carlos Meniño Cotón. 2019-10-30. Every noncompact surface is a leaf of a minimal foliation. https://arxiv.org/abs/1910.13839

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