Abstract
We shall study smooth maps ũ: S → ℝ x M of finite energy defined on the punctured Riemann surface S = S\Γ and satisfying a Cauchy-Riemann type equation Tũ ∘ j = Jũ ∘ Tũ for special almost complex structures J, related to contact forms A on the compact three manifold M. Neither the domain nor the target space are compact. This difficulty leads to an asymptotic analysis near the punctures. A Fredholm theory determines the dimension of the solution space in terms of the asymptotic data defined by non-degenerate periodic solutions of the Reeb vector field associated with λ on M, the Euler characteristic of S, and the number of punctures. Furthermore, some transversality results are established.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C. Abbas and H. Hofer. Holomorphic curves and global questions in contact geometry. To appear in Birkhäuser.
L. Ahlfors and L. Bers. “Riemann’s mapping theorem for variable metrics.” Ann. of Math. 72 (1960), 385–404.
D. Bennequin. “Entrelacements et équations de Pfaff.” Astérisque, 107-108:83–161, 1983.
S.S. Chern. “An elementary proof of the existence of isothermal parameters on a surface.” Proc. Amer. Math. Soc. 6 (1955), 771–782.
Y. Eliashberg. “Classification of overtwisted contact structures on three manifolds.” Inv. Math., pages 623–637, 1989.
Y. Eliashberg. “Filling by holomorphic discs and its applications.” London Math. Society Lecture Notes, pages 45–67, 1991. Series 151.
Y. Eliashberg. “Contact 3-manifolds, twenty year since J. Martinet’s work.” Ann. Inst. Fourier 42 (1992), 165–192.
Y. Eliashberg. “Legendrian and transversal knots in tight contact manifolds”. In: Topological methods in modern mathematics. Publish or Perish, 1993.
A. Floer. “The unregularised gradient flow of the symplectic action.” Comm. Pure Appl. Math. 41 (1988), 775–813.
A. Floer. “Symplectic fixed points and holomorphic spheres.” Comm. Math. Phys. 120 (1989), 575–611.
A. Floer, H. Hofer, and D. Salamon. “Transversality results in the elliptic Morse theory for the action functional.” Duke Math. Journal 30 (1995), No. 1, 251–292.
E. Giroux. “Convexité en topologie de contact.” Comm. Math. Helvetici 66 (1991), 637–677.
J.W. Gray. “Some global properties of contact structures.” Ann. of Math. 2(69) (1959), 421–450.
M. Gromov. “Pseudoholomorphic curves in symplectic manifolds.” Invent. Math. 82 (1985), 307–347.
H. Hofer. “Pseudoholomorphic curves in symplectizations with application to the Weinstein conjecture in dimension three.” Invent. Math. 114 (1993), 515–563.
H. Hofer, V. Lizan, and J.-C. Sikorav. On generecity for holomorphic curves in 4~ dimensional almost complex manifolds. Preprint.
H. Hofer, K. Wysocki, and E. Zehnder. “A characterisation of the tight three-sphere.” Duke Math. Journal 81 (1995), 159–226.
H. Hofer, K. Wysocki, and E. Zehnder. “Properties of pseudoholomorphic curves in symplectizations I: Asymptotics.” Ann. I. H. P. Analyse Non Linéaire 13 (1996), 337–379.
H. Hofer, K. Wysocki, and E. Zehnder. “Properties of pseudoholomorphic curves in symplectizations II: Embedding controls and algebraic invariants.” Geometrical and Functional Analysis 5, No. 2 (1995), 270–328.
H. Hofer, K. Wysocki, and E. Zehnder. “Unknotted periodic orbits for Reeb flows on the 3-sphere.” Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center 7 (1996), 219–244.
H. Hofer, K. Wysocki, and E. Zehnder. “The dynamics on a strictly convex energy surface in R 4”. To appear in Ann. of Math.
H. Hofer, K. Wysocki, and E. Zehnder. “A characterization of the tight three-sphere II”. Preprint.
H. Hofer, K. Wysocki, and E. Zehnder. “Properties of pseudoholomorphic curves in symplectizations V: Fredholm theory with singularities”. In preparation.
H. Hofer and E. Zehnder. Hamiltonian Dynamics and Symplectic Invariants. Birkhäuser, 1994.
J. Martinet. “Formes de contact sur les variétés de dimension 3.” Springer LNM 209 (1971), 142–163.
D. McDuff. “The local behavior of J-holomorphic curves in almost complex 4-manifolds.” J. Diff. Geom. 34 (1991), 143–164.
M. Micallef and B. White. “The structure of branch points in minimal surfaces and in pseudoholomorphic curves.” Ann. of Math. 141 (1994), 35–85.
A. Nijenhuis and W. Woolf. “Some integration problems in almost-complex and complex manifolds.” Annals of Mathematics 77, No. 3 (1963), 424–489.
M. Schwarz. Cohomology Operations from S 1-Cobordisms in Floer Homology. Ph.D thesis. ETH-Zürich, 1996.
J.-C. Sikorav. “Singularities of J-holomorphic curves.” Math. Zeit. 226 (1997), 359–373.
R. Ye. “Filling by holomorphic disks in symplectic 4-manifolds.” Trans. Amer. Math. Soc. 350, 1 (1998), 213–250.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this chapter
Cite this chapter
Hofer, H., Wysocki, K., Zehnder, E. (1999). Properties of Pseudoholomorphic Curves in Symplectizations III: Fredholm Theory. In: Escher, J., Simonett, G. (eds) Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 35. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8765-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8765-6_18
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9764-8
Online ISBN: 978-3-0348-8765-6
eBook Packages: Springer Book Archive