Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

TL;DR

This paper classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds, establishing an h-principle and analyzing the space of Legendrian unknots and their contact mapping class groups.

Contribution

It provides a complete classification of such disks up to contact isotopy, including cases with tight neighborhoods and boundary violations of the Bennequin-Eliashberg inequality.

Findings

Classification coincides with the formal one, confirming the h-principle.

The space of Legendrian unknots satisfies the h-principle at the fundamental group level.

Determination of the contact mapping class group for Legendrian unknots with non-positive tb in overtwisted 3-spheres.

Abstract

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood of such a disk is tight or where the boundary violates the Bennequin-Eliashberg inequality. We show that this classification coincides with the formal one, establishing an h-principle for these disks. As a corollary, we deduce that the space of Legendrian unknots that lie in some Darboux ball in a closed overtwisted contact 3-manifold satisfies the h-principle at the level of fundamental groups. Finally, we determine the contact mapping class group of the complement of each Legendrian unknot with non-positive tb invariant in an overtwisted 3-sphere.