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February  2004, 10(1&2): 21-30. doi: 10.3934/dcds.2004.10.21

Recent results in contact form geometry

Abbas Bahri 1,

1. 

Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, United States

Received  February 2002 Revised  December 2002 Published  October 2003

After briefly recalling the pseudo-holomorphic approach in contact form geometry and after sketching the ways with which this approach defines invariants, we introduce another approach, of more technical type, which starts with a variational problem on Legendrian curves. We show how this approach leads also to the definition of a homology.
Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].
Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.
This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].
Keywords: critical points at infinity., Reeb vector-fields, Contact structures, Legendrian curves.
Mathematics Subject Classification: Primary: 58E15, 58F05, 58F2.
Citation: Abbas Bahri. Recent results in contact form geometry. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21
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