# Taut foliation

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In mathematics, a taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. 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.

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

It is 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.

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.

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

a) $\mathcal{F}$ is taut;

b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on M;

c) there is a Riemannian metric on M for which the leaves of $\mathcal{F}$ are least area surfaces.