Taut Foliation

A codimension one foliation  of a 3-manifold  is said to be taut if for every leaf  in the leaf space  of , there is a circle  transverse to  (i.e., a closed loop transverse to the tangent field of ) which intersects .

Taut foliations play a significant role in various aspects of topology and are credited as being one of two major tools (along with incompressible surfaces) responsible for revealing significant topological and geometric information about 3-manifolds (Gabai and Oertel 1989). As such, a considerable amount of research has gone into the study of taut foliations on 3-manifolds. One well-known result is that every taut foliation is necessarily Reebless and that, for any non-taut Reebless foliation, the leaves which don't admit a closed transversal are necessarily tori. Additionally, the closed leaves of a taut foliation are homologically nontrivial.

Some classification results for taut foliations are also known. One such result, attributed by Eliashberg and Thurston to Novikov and Sullivan, says that a foliation  on a closed 3-manifold  is taut if it is different from the foliation  on  and satisfies any one of the following:

1. Each leaf  of  is intersected by a transversal closed curve.

2. There exists a vector field  on  which is transversal to  and preserves a volume form  on .

3.  admits a Riemannian metric for which all leaves are minimal surfaces.

Moreover, a necessary and sufficient condition for tautness of a foliation  is that  contain no generalized Reeb components (Goodman 1975).

REFERENCES:

Calegari, D. Foliations and the Geometry of 3-Manifolds. Oxford, England: Clarendon Press, 2007.

Eliashberg, Y. M. and Thurston, W. P. Confoliations. Providence, RI: Amer. Math. Soc., 1998.

Gabai, D. and Oertel, U. "Essential Laminations in 3-Manifolds." Ann. Math. 130, 41-73, 1989.

Goodman, S. "Closed Leaves in Foliations of Codimension One." Comm. Math. Helv. 50, 383-388, 1975.