THE WEINSTEIN CONJECTURE FOR PLANAR CONTACT STRUCTURES IN DIMENSION THREE

By Casim Abbas, Kai Cieliebak, Helmut Hofer. 2004

77 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c8f57c1b6b01c92733cbcce1d124654900365a3

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Seiberg–Witten equations and the Weinstein conjecture (2006, background): “The theorem of Hofer [H] noted above asserts that the Reeb vector field for a contact form whose kernel is overtwisted has contractible, closed orbits.”
Planar open books and Floer homology (2005, background): “In [28], Thurston and Winkelnkemper showed that an open book decomposition of a 3–manifold Y gives rise in a natural way to a contact structure over Y , and hence that every 3–manifold admits some contact structure.”
Taubes’s proof of the Weinstein conjecture in dimension three (2009, unknown): “7 (Abbas-Cieliebak-Hofer [2]).”
Reeb vector fields and open book decompositions (2008, background): “es an inﬁnite number of simple periodic orbits. Prior to the work of Taubes, the Weinstein conjecture in dimension three was veriﬁed for contact structures which admit planar open book decompositions [ACH] (also see related work of Etnyre [Et2]), for certain Stein ﬁllable contact structure s [Ch, Ze”
Rounding corners of polygons and the embedded contact homology of T 3 (2004, background): “For example, one version of the Weinstein conjecture asserts that any contact 1–form on a closed oriented 3–manifold Y has a Reeb orbit, see eg Abbas–Cieliebak–Hofer [1].”
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, background): “n10c] and generalized in [Abb11], and lies in the background of various contact topological results on planar contact manifolds, such as the proof of the Weinstein conjecture by Abbas-Cieliebak-Hofer [ACH05] and the author’s proof that strong and Stein ﬁllability are equivalent [Wen10b]. Given exist”
Strongly fillable contact manifolds and $J$-holomorphic foliations (2008, methodology): “ve Theorem 2 contains no nodal curves and thus deﬁnes an honest symplectic ﬁbration. In fact, it is easy to construct two such ﬁbrations, whose ﬁbers intersect each other 2The compactness argument in [ACH05] contains a minor gap, as it ignores the possibility of nodal degenerations. Our argument ﬁll”