Ko Honda, William H. Kazez, Gordana Mati´c
Tight Contact Structures on Fibered Hyperbolic 3--Manifolds
Preprint series: ESI preprints

MSC:

57M50 Geometric structures on low-dimensional manifolds

53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)

Abstract: We take a first step towards understanding the relationship between foliations
and universally tight contact structures in hyperbolic manifolds. If a surface
bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of
``extremal'' tight contact structures. Specifically, there is exactly one
contact structure whose Euler class, when evaluated on the fiber, equals the
Euler number of the fiber. This rigidity theorem is a consequence of properties
of the action of pseudo-Anosov maps on the complex of curves of the fiber and a
remarkable flexibility property of convex surfaces in such a space. Indeed this
flexibility may be seen in surface bundles over an interval where the analogous
classification theorem is also established.
Keywords: tight, contact structure