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Branch structure of $ J$-holomorphic curves near periodic orbits of a contact manifold

Authors: Adam Harris and Krzysztof Wysocki
Journal: Trans. Amer. Math. Soc. 360 (2008), 2131-2152
MSC (2000): Primary 32Q65, 53D10
Published electronically: October 30, 2007
MathSciNet review: 2366977
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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.

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Additional Information

Adam Harris
Affiliation: School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia

Krzysztof Wysocki
Affiliation: School of Mathematics and Statistics, Melbourne University, Parkville, VIC 3010, Australia

Received by editor(s): July 18, 2005
Received by editor(s) in revised form: June 1, 2006
Published electronically: October 30, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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