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Comment. Math. Helv. 80 (2005), 771793 Commentarii Mathematici Helvetici Swiss Mathematical Society
 

Summary: Comment. Math. Helv. 80 (2005), 771793 Commentarii Mathematici Helvetici
Swiss Mathematical Society
The Weinstein conjecture for planar contact structures in dimen-
sion three
Casim Abbas, Kai Cieliebak and Helmut Hofer
Abstract. In this paper we describe a general strategy for approaching the Weinstein conjecture
in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of
contact manifolds (planar contact manifolds). We also discuss how the present approach reduces
the general Weinstein conjecture in dimension three to a compactness problem for the solution
set of a first order elliptic PDE.
Mathematics Subject Classification (2000). 53D35, 37J45.
Keywords. Contact structure, open book, periodic orbit, holomorphic curve.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
1.1 Versions of the Weinstein conjecture . . . . . . . . . . . . . . . . . 772
1.2 Generalized holomorphic curve equations . . . . . . . . . . . . . . 773
1.3 Open book decompositions and the main result . . . . . . . . . . . 774
2 Recollections on finite energy spheres . . . . . . . . . . . . . . . . . . . 775
2.1 The Reeb flow near a periodic orbit . . . . . . . . . . . . . . . . . . 777
2.2 Asymptotics near a puncture . . . . . . . . . . . . . . . . . . . . . 777

  

Source: Abbas, Casim - Department of Mathematics, Michigan State University

 

Collections: Mathematics