Geometry & Topology

A compendium of pseudoholomorphic beasts in $\mathbb{R}{\times}(S^1{\times}S^2)$

Clifford Henry Taubes

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Abstract

This article describes various moduli spaces of pseudoholomorphic curves on the symplectization of a particular overtwisted contact structure on S1×S2. This contact structure appears when one considers a closed self dual form on a 4–manifold as a symplectic form on the complement of its zero locus. The article is focussed mainly on disks, cylinders and three-holed spheres, but it also supplies groundwork for a description of moduli spaces of curves with more punctures and non-zero genus.

Article information

Source
Geom. Topol., Volume 6, Number 2 (2002), 657-814.

Dates
Received: 21 November 2002
Accepted: 18 December 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882938

Digital Object Identifier
doi:10.2140/gt.2002.6.657

Mathematical Reviews number (MathSciNet)
MR1943381

Zentralblatt MATH identifier
1021.32008

Subjects
Primary: 32Q65: Pseudoholomorphic curves
Secondary: 57R17: Symplectic and contact topology 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Keywords
pseudoholomorphic curves moduli spaces contact structures

Citation

Taubes, Clifford Henry. A compendium of pseudoholomorphic beasts in $\mathbb{R}{\times}(S^1{\times}S^2)$. Geom. Topol. 6 (2002), no. 2, 657--814. doi:10.2140/gt.2002.6.657. https://projecteuclid.org/euclid.gt/1513882938


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References

  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry in Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975a) 43–69
  • Y Eliashberg, Invariants in contact topology, from: “Proceedings of the International Congress of Mathematicians, Berlin (1998)”, Vol II, Documenta Mathematica, Extra Volume ICM (1998) 327–338
  • Y Eliashberg, E Hofer, in preparation
  • A Floer, Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988) 513–547
  • P Griffiths, J Harris, Principles of Algebraic Geometry, J Wiley and Sons, New York (1978)
  • M Gromov, Pseudoholomorphic spheres in symplectic manifolds, Invent. Math 82 (1985) 307–347
  • R Gompf, private communication.
  • H Hofer, Pseudoholmorphic curves in symplectizations with applications to the Weinstein conjecture in dimension 3, Invent Math 114 (1993) 515–563
  • H Hofer, Dynamics, topology and holomorphic curves, from: “Proceedings of the International Congress of Mathematicians, Berlin 1998”, Vol I, Documenta Mathematica, Extra Volume ICM (1998) 255–280
  • H Hofer, Holomorphic curves and dynamics in dimension 3, from: “Symplectic Geometry and Topology”, (Eliashberg and Traynor, editors) IAS/Park City Mathematics Series 7, American Mathematical Society, Providence RI (1999) 37–101
  • H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations I: Asymptotics, Ann. Inst. Henri Poincaré 13 (1996) 337–379
  • H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations II: Embedding controls and algebraic invariants, Geom. and Funct. Anal. 5 (1995) 270–328
  • H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations III: Fredholm theory, preprint.
  • R B Lockhart, R C McOwen, Elliptic differential operators on non-compact manifolds, Ann. Sci. Norm. Sup. Pisa IV-12 (1985) 409–446
  • C Luttinger, unpublished
  • D McDuff, The local behavior of J–holomorphic curves in almost complex manifolds, J. Diff. Geom. 34 (1990) 679–712
  • C B Morrey, Multiple Integrals in the Calculus of Variations, Springer–Verlag, Berlin (1966)
  • D McDuff, D Salamon, J–Holomorphic Curves and Quantum Cohomology, University Lecture Series 6, American Mathematical Society, Providence RI (1994)
  • C H Taubes, Seiberg–Witten invariants and pseudoholomorphic subvarieties for self-dual, harmonic 2–forms, Geom. Topol. 3 (1999) 167–210
  • C H Taubes, Seiberg–Witten invariants, self-dual harmonic 2–forms and the Hofer–Wysocki–Zehnder formalism, from: “Surveys in Differential Geometry, VII”, International Press (2000) 625–672
  • C H Taubes, The geometry of the Seiberg–Witten invariants, from: “Proceedings of the International Congress of Mathematicians, Berlin 1998”, Vol II, Documenta Mathematica, Extra Volume ICM (1998) 493–504
  • C H Taubes, The structure of pseudoholomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^1 \times B^3$, Geom. Topol. 2 (1998) 221–332
  • C H Taubes, $L^2$ Moduli Spaces on 4–Manifolds with Cylindrical End, International Press, Cambridge MA (1993)
  • C H Taubes, Gr$\rightarrow$SW; from pseudoholomorphic curves to Seiberg–Witten solutions, J. Diff. Geom. 51 (1999) 203–334.