Quantitative Darboux theorems in contact geometry

Abstract:

This paper begins the study of relations between Riemannian geometry and contact topology on $(2n+1)$–manifolds and continues this study on 3–manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact $(2n+1)$–manifold $(M,\xi )$ that can be embedded in the standard contact structure on $\mathbb {R}^{2n+1}$, 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 $\alpha$ for $\xi$. In dimension 3, this 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 curve techniques to provide a lower bound for the radius of a PS-tight ball.