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

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 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 et al (2012 Invent. Math. 188 621-57; 2016 Trans. Am. Math. Soc. 368 7845-81), 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.