Tight contact structures on fibered hyperbolic 3-manifolds

TL;DR

This paper explores the relationship between foliations and tight contact structures on hyperbolic 3-manifolds, providing a classification of extremal structures for fibered manifolds with pseudo-Anosov holonomy.

Contribution

It introduces a classification of extremal tight contact structures on fibered hyperbolic 3-manifolds with pseudo-Anosov monodromy, revealing a rigidity property linked to the Euler class.

Findings

Exactly one extremal tight contact structure per fibered hyperbolic 3-manifold with pseudo-Anosov holonomy.

The classification relies on properties of the action of pseudo-Anosov maps on the complex of curves.

Flexibility of convex surfaces plays a key role in the analysis.

Abstract

We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-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.