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Transactions of the American Mathematical Society

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Quantitative Darboux theorems in contact geometry

Authors: John B. Etnyre, Rafal Komendarczyk and Patrick Massot
Journal: Trans. Amer. Math. Soc. 368 (2016), 7845-7881
MSC (2010): Primary 53D10, 53D35; Secondary 57R17
Published electronically: March 1, 2016
MathSciNet review: 3546786
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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.

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Additional Information

John B. Etnyre
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Rafal Komendarczyk
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118

Patrick Massot
Affiliation: Département de Mathématiques, Université Paris Sud, 91405 Orsay Cedex, France
Address at time of publication: Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France

Received by editor(s): September 17, 2012
Received by editor(s) in revised form: January 9, 2015
Published electronically: March 1, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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