# Taut foliation

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In mathematics, **tautness** is a rigidity property of foliations. A **taut foliation** is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle.^{[1]}^{: 155 } By transverse circle, is meant a closed loop that is always transverse to the leaves of the foliation.

If the foliated manifold has non-empty tangential boundary, then a codimension 1 foliation is taut if every leaf meets a transverse circle or a transverse arc with endpoints on the tangential boundary. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Furthermore, for compact manifolds the existence, for every leaf , of a transverse circle meeting , implies the existence of a single transverse circle meeting every leaf.

Taut foliations were brought to prominence by the work of William Thurston and David Gabai.

## Relation to Reebless foliations

[edit]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

[edit]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 , and have negatively curved fundamental group.

## Rummler–Sullivan theorem

[edit]By a theorem of Hansklaus Rummler and Dennis Sullivan, the following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifolds M:^{[2]}^{[1]}^{: 158 }

- is taut;
- there is a flow transverse to which preserves some volume form on M;
- there is a Riemannian metric on M for which the leaves of are least area surfaces.

## References

[edit]- ^
^{a}^{b}Calegari, Danny (2007).*Foliations and the Geometry of 3-Manifolds*. Clarendon Press. **^**Alvarez Lopez, Jesús A. (1990). "On riemannian foliations with minimal leaves".*Annales de l'Institut Fourier*.**40**(1): 163–176.