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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

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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

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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

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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

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