Branch structure of $J$–holomorphic curves near periodic orbits of a contact manifold
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- by Adam Harris and Krzysztof Wysocki PDF
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Abstract:
Let $M$ be a three–dimensional contact manifold, and $\tilde {\psi }:D\setminus \{0\}\to M\times {\mathbb R}$ a finite–energy pseudoholomorphic map from the punctured disc in ${\mathbb C}$ that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that $\tilde {\psi }$ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into ${\mathbb E}_{p,q}\times {\mathbb R}$, where ${\mathbb E}_{p,q}$ denotes a rational ellipsoid (contact structure induced by the standard complex structure on ${\mathbb C}^{2}$), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.References
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Additional Information
- Adam Harris
- Affiliation: School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia
- MR Author ID: 607698
- Email: adamh@turing.une.edu.au
- Krzysztof Wysocki
- Affiliation: School of Mathematics and Statistics, Melbourne University, Parkville, VIC 3010, Australia
- MR Author ID: 184985
- Email: wysocki@ms.unimelb.edu.au
- Received by editor(s): July 18, 2005
- Received by editor(s) in revised form: June 1, 2006
- Published electronically: October 30, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 2131-2152
- MSC (2000): Primary 32Q65, 53D10
- DOI: https://doi.org/10.1090/S0002-9947-07-04350-4
- MathSciNet review: 2366977