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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
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References:

[1] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[2] Y Eliashberg, Invariants in contact topology (1998) 327 · Zbl 0913.53010
[3] Y Eliashberg, E Hofer, in preparation,
[4] A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513 · Zbl 0674.57027
[5] P Griffiths, J Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons] (1978) · Zbl 0408.14001
[6] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 · Zbl 0592.53025 · doi:10.1007/BF01388806
[7] R Gompf, private communication
[8] H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515 · Zbl 0797.58023 · doi:10.1007/BF01232679
[9] H H W Hofer, Dynamics, topology, and holomorphic curves (1998) 255 · Zbl 0908.58020
[10] H Hofer, Holomorphic curves and dynamics in dimension three, IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 35 · Zbl 1004.53062
[11] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 337 · Zbl 0861.58018
[12] H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995) 270 · Zbl 0845.57027 · doi:10.1007/BF01895669
[13] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser (1999) 381 · Zbl 0924.58003
[14] R B Lockhart, R C McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. \((4)\) 12 (1985) 409 · Zbl 0615.58048
[15] C Luttinger, unpublished
[16] D McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds, J. Differential Geom. 34 (1991) 143 · Zbl 0736.53038
[17] C B Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften 130, Springer New York, New York (1966) · Zbl 0142.38701
[18] D McDuff, D Salamon, \(J\)-holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994) · Zbl 0809.53002
[19] C H Taubes, Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms, Geom. Topol. 3 (1999) 167 · Zbl 1027.53111 · doi:10.2140/gt.1999.3.167
[20] C H Taubes, Seiberg-Witten invariants, self-dual harmonic 2-forms and the Hofer-Wysocki-Zehnder formalism, Surv. Differ. Geom., VII, Int. Press, Somerville, MA (2000) 625 · Zbl 1061.53063
[21] C H Taubes, The geometry of the Seiberg-Witten invariants (1998) 493 · Zbl 0956.57019
[22] C H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on \(S^1\times B^3\), Geom. Topol. 2 (1998) 221 · Zbl 0908.53013 · doi:10.2140/gt.1998.2.221
[23] C H Taubes, \(L^2\) moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology, I, International Press (1993) · Zbl 0830.58004
[24] C H Taubes, \(\mathrm{Gr}\Longrightarrow\mathrm{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999) 203 · Zbl 1036.53066
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