Taut foliations and contact pairs in dimension three

TL;DR

This paper introduces a novel method to construct taut foliations from contact structures in three dimensions, offering new insights into contact geometry and its relation to taut foliations, with implications for the L-space conjecture.

Contribution

It provides a new construction of taut foliations from contact pairs, complementing Eliashberg and Thurston's approximation theorem, and generalizes key technical results on branching foliations.

Findings

Constructs taut foliations from contact pairs under certain conditions

Reveals flexible phenomena in taut foliations via contact geometry

Provides new insights into the L-space conjecture

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 taut foliations entirely in terms of contact geometry. This viewpoint reveals some surprising flexible phenomena for taut foliations, and provides new insight into the $L$-space conjecture. The first part of the proof builds upon the work on 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.