Quantitative Darboux theorems in contact geometry

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.

33 pages, corrects several inaccuracies in earlier version

References in corpus (2)

• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• Tightness in contact metric 3-manifolds

Cited by in corpus (4)

• Tightness in contact metric 3-manifolds
• Dynamics and Topology of Conformally Anosov Contact 3-Manifolds
• On the existence of critical compatible metrics on contact $3$-manifolds
• Ricci Curvature, Reeb Flows and Contact 3-Manifolds