Taut foliations and contact pairs in dimension three

Abstract

We present a new construction of codimension-one foliations from pairs of contact structures in dimension three. This constitutes a converse result to a celebrated theorem of Eliashberg and Thurston on approximations of foliations by contact structures. Under suitable hypotheses on the initial contact pairs, the foliations we construct are taut, allowing us to characterize the existence of taut foliations entirely in terms of contact geometry. This viewpoint reveals some surprising flexibility phenomena for taut foliations, and provides new insight into the $L$-space conjecture. The first part of the proof builds upon the work of Colin and Firmo on positive contact pairs. The second part involves a wide generalization of a technical result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, and might be of independent interest.

arXiv commentV2: 114 pages, 20 figures. Corrected various typos and minor mistakes, added clarifications, and slightly revised terminology. Comments welcome!