Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Recent results in contact form geometry

Pages: 21 - 30, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.21

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

Abstract: 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:  Contact structures, Reeb vector-fields, Legendrian curves, critical points at infinity.
Mathematics Subject Classification:  Primary: 58E15, 58F05, 58F22.

Received: February 2002;      Revised: December 2002;      Available Online: October 2003.