Taut foliation

Relation to Reebless foliations

Taut foliations are closely related to the concept of Reebless foliation. A taut foliation cannot have a Reeb component, since the component would act like a "dead-end" from which a transverse curve could never escape; consequently, the boundary torus of the Reeb component has no transverse circle puncturing it. A Reebless foliation can fail to be taut but the only leaves of the foliation with no puncturing transverse circle must be compact, and in particular, homeomorphic to a torus.

Properties

The existence of a taut foliation implies various useful properties about a closed 3-manifold. For example, a closed, orientable 3-manifold, which admits a taut foliation with no sphere leaf, must be irreducible, covered by \mathbb R^3, and have negatively curved fundamental group.

Rummler–Sullivan theorem

By a theorem of Hansklaus Rummler and Dennis Sullivan, the following conditions are equivalent for transversely orientable codimension one foliations \left(M,{\mathcal{F}}\right) of closed, orientable, smooth manifolds M:

• \mathcal{F} is taut;
• there is a flow transverse to \mathcal{F} which preserves some volume form on M;
• there is a Riemannian metric on M for which the leaves of \mathcal{F} are least area surfaces.

References

References

1. Alvarez Lopez, Jesús A.. (1990). "On riemannian foliations with minimal leaves". [[Annales de l'Institut Fourier]].
2. Calegari, Danny. (2007). "Foliations and the Geometry of 3-Manifolds". [[Clarendon Press]].