Commentarii Mathematici Helvetici
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The Weinstein conjecture for planar contact structures in dimension three
Casim Abbas (1), Kai Cieliebak (2) and David Geraghty (3)(1) Department of Mathematics, Michigan State University, MI 48824, EAST LANSING, UNITED STATES
(2) Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14, 86159, AUGSBURG, GERMANY
(3) School of Mathematics, Institute for Advanced Study, Einstein Drive, NJ 08540, PRINCETON, UNITED STATES
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.
Keywords: Contact structure, open book, periodic orbit, holomorphic curve