Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach

TL;DR

This paper develops quantitative estimates for the tightness radius around Reeb orbits in 3D contact manifolds using sub-Riemannian metrics, introducing contact Jacobi curves and comparing bounds with classical models.

Contribution

It introduces the concept of contact Jacobi curves and provides sharp lower bounds for tightness radius in 3D contact manifolds, extending previous results with new sub-Riemannian techniques.

Findings

Established sharp bounds for tightness radius in classical models.

Derived lower bounds in terms of Schwarzian derivative and curvature.

Proved a contact analogue of the Cartan--Hadamard theorem for K-contact manifolds.

Abstract

Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem.