Quantitative Darboux theorems in contact geometry
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- by John B. Etnyre, Rafal Komendarczyk and Patrick Massot PDF
- Trans. Amer. Math. Soc. 368 (2016), 7845-7881 Request permission
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.References
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Additional Information
- John B. Etnyre
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 619395
- Email: etnyre@math.gatech.edu
- Rafal Komendarczyk
- Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
- Email: rako@tulane.edu
- 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
- MR Author ID: 844086
- Email: patrick.massot@polytechnique.edu
- Received by editor(s): September 17, 2012
- Received by editor(s) in revised form: January 9, 2015
- Published electronically: March 1, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7845-7881
- MSC (2010): Primary 53D10, 53D35; Secondary 57R17
- DOI: https://doi.org/10.1090/tran/6821
- MathSciNet review: 3546786