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A compendium of pseudoholomorphic beasts in \({\mathbf R}\times (S^1\times S^2)\). (English) Zbl 1021.32008

This article describes various moduli spaces of pseudoholomorphic curves on the symplectization of a particular overtwisted contact structure on \(S^{1}\times S^{2}\). This contact structure appears when one considers a closed self-dual form of a \(4\)-manifold as a symplectic form on the complement of a disjoint union of circles which are its zero locus (see [C. H. Taubes, Surveys in Diff. Geom. VII, 625-672 (2002; Zbl 1044.53002)]).
An HWZ subvariety, \(C\) of the symplectization \(R \times (S^{1}\times S^{2})\), is a non-empty closed subset which is a pseudoholomorphic surface of a countable nowhere accumulating subset, with \(\int_{C\cap K} \omega\) finite for any compact subset \(K\) and with \(\int_{C} d\alpha\) finite (where \(\omega\) and \(\alpha\) are the symplectic and contact forms, respectively).
The HWZ subvarieties \(C\) converge for \(|s|\) very large (\(s\) in the coordinate in the \(R\) factor) to multiple covers of embedded Reeb orbits of \(S^{1}\times S^{2}\), and these limits are described in combinatoric terms. Let \(X_{C}\) be the number of ends in the negative side of \(C\). Let \(I_{C}\) be the (formal) dimension of the moduli space of HWZ subvarieties containing \(C\). The paper describes in detail the structure of the moduli spaces for which \(X_{C}\leq I_{C}\leq X_{C}+1\). All the HWZ subvarieties in these cases are cylinders, discs or three-punctured spheres. The natural action of \(R\times S^{1} \times S^{1}\) on \(R\times (S^{1} \times S^{2})\) is very relevant in order to describe the HWZ subvarieties and the moduli spaces which appear.

MSC:

32Q65 Pseudoholomorphic curves
57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds

Citations:

Zbl 1044.53002

References:

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