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 tangent field 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 ${\displaystyle L}$, of a transverse circle meeting ${\displaystyle L}$, 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

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 ${\displaystyle \mathbb {R} ^{3}}$, 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 ${\displaystyle \left(M,{\mathcal {F}}\right)}$ of closed, orientable, smooth manifolds M:^{[2][1]: 158}

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

References