Quantitative Darboux theorems in contact geometry

TL;DR

This paper explores quantitative bounds relating Riemannian geometry and contact topology, providing estimates for Darboux ball sizes and neighborhoods of Reeb orbits using classical comparison theorems and holomorphic curves.

Contribution

It introduces new quantitative bounds for Darboux balls and Reeb orbit neighborhoods in contact manifolds, linking Riemannian and contact geometric properties.

Findings

Lower bound for the radius of Darboux balls in contact manifolds

Estimate for the size of neighborhoods of closed Reeb orbits in dimension three

Use of classical comparison theorems and holomorphic curves techniques

Abstract

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.