Advertisement

Inventiones mathematicae

, Volume 103, Issue 1, pp 651–671 | Cite as

Symplectic manifolds with contact type boundaries

  • Dusa McDuff
Article

Summary

An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ω) is symplectomorphic to a neighbourhood ofS2n−1 in standard Euclidean space, and if ω vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB2n.

Keywords

Manifold Euclidean Space Symplectic Manifold Type Boundary Contact Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [E 1] Eliashberg, Ya.: On Symplectic Manifolds which are bounded by standard Contact Spheres and Exotic Contact Structures on Spheres of Dimension >3. J. Differ. Geom. (to appear)Google Scholar
  2. [E 2] Eliashberg, Ya.: Topological Characterization of Stein Manifolds of Dimension >2. Int. J. Math.1, 19–46 (1990)Google Scholar
  3. [E 3] Eliashberg, Ya.: Filling by Holomorphic Discs and Its Applications. (preprint 1989)Google Scholar
  4. [EG] Eliashberg, Ya., Gromov, M. Convex Symplectic Manifolds, preprint (1990)Google Scholar
  5. [F] Floer, A.: Symplectic Fixed Points and Holomorphic Spheres. Commun. Math. Phys.120, 575–611 (1989)Google Scholar
  6. [G 1] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math.82, 307–347 (1985)Google Scholar
  7. [G 2] Gromov, M.: Partial Differential Relations. Ergeb. Math. Vol. 3. Folgeband 9. Berlin Heidelberg New York, Springer 1986Google Scholar
  8. [H] Hofer, H. (private communication)Google Scholar
  9. [McD 1] McDuff, D.: Symplectic Diffeomorphisms and the Flux Homomorphism. Invent. Math. 77, 353–366 (1984)Google Scholar
  10. [McD 2] McDuff, D.: Examples of simply-connected non-Kählerian manifolds. J. Differ. Geom.20, 267–277 (1984)Google Scholar
  11. [McD 3] McDuff, D.: Examples of symplectic structures. Invent. Math.89, 13–36 (1987)Google Scholar
  12. [McD 4] McDuff, D.: The Structure of Rational and Ruled symplectic 4-manifolds. J. Am. Math. Soc.3, 679–712 (1990)Google Scholar
  13. [McD 5] McDuff, D.: Elliptic Methods in Symplectic Geometry. Bull. Am. Math. Soc.23, 311–358 (1990)Google Scholar
  14. [McD 6] McDuff, D.: The Local Behaviour of Holomorphic Curves in Almost Complex 4-manifolds. J. Differ. Geom. (to appear)Google Scholar
  15. [V] Viterbo, C. (private communication)Google Scholar
  16. [W] Wolfson, J.: Gromov's compactness of pseudo-holomorphic curves and symplectic geometry. J. Differ. Geom.28, 383–405 (1988)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Dusa McDuff
    • 1
  1. 1.Mathemathical DepartmentSUNYStony BrookUSA

Personalised recommendations