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.


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


Zbl 1044.53002
Full Text: DOI arXiv EuDML EMIS


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