Inventiones mathematicae

, 176:461 | Cite as

A Bishop surface with a vanishing Bishop invariant

Article

Abstract

We give a solution to the equivalence problem for Bishop surfaces with the Bishop invariant λ=0. As a consequence, we answer, in the negative, a problem that Moser asked in 1985 after his work with Webster in 1983 and his own work in 1985. This will be done in two major steps: We first derive the formal normal form for such surfaces. We then show that two real analytic Bishop surfaces with λ=0 are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature. Our convergence proof is done through a new hyperbolic geometry associated with the surface.

As an immediate consequence of the work in this paper, we will see that the modular space of Bishop surfaces with the Bishop invariant vanishing and with the Moser invariant s<∞ is of infinite dimension. This phenomenon is strikingly different from the celebrated theory of Moser–Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants where the surfaces only have two and one half invariants. Notice also that there are many real analytic hyperbolic Bishop surfaces, which have the same Moser–Webster formal normal form but are not holomorphically equivalent to each other as shown by Moser–Webster and Gong. Hence, Bishop surfaces with the Bishop invariant λ=0 behave very differently from hyperbolic Bishop surfaces and elliptic Bishop surfaces with non-vanishing Bishop invariants.

References

  1. 1.
    Ahern, P., Gong, X.: Real analytic submanifolds in ℂn with parabolic complex tangents along a submanifold of codimension one. Preprint (2006)Google Scholar
  2. 2.
    Baouendi, S., Ebenfelt, P., Rothschild, L.: Local geometric properties of real submanifolds in complex space. Bull. Am. Math. Soc., New. Ser. 37(3), 309–333 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bedford, E., Gaveau, B.: Envelopes of holomorphy of certain 2-spheres in ℂ2. Am. J. Math. 105, 975–1009 (1983)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cartan, É.: Sur les variétés pseudo-conformal des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. Pura Appl. 11(4), 17–90 (1932)MATHMathSciNetGoogle Scholar
  6. 6.
    Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Eliashberg, Y.: Filling by holomorphic discs and its applications. In: Geometry of Low-Dimensional Manifolds. Lond. Math. Soc. Lect. Note Ser., vol. 151 (1997)Google Scholar
  8. 8.
    Forstneric, F.: Most real analytic Cauchy–Riemann manifolds are nonalgebraizable. Manuscr. Math. 115, 489–494 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gong, X.: On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents. Comment. Math. Helv. 69(4), 549–574 (1994)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gong, X.: Normal forms of real surfaces under unimodular transformations near elliptic complex tangents. Duke Math. J. 74(1), 145–157 (1994)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gong, X.: Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics. Indiana Univ. Math. J. 53(1), 83–95 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gromov, M.: Pseudo holomorphic curves in symplectic geometry. Invent Math. 82, 307–347 (1985)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Huang, X.: Local Equivalence Problems for Real Submanifolds in Complex Spaces. Lect. Notes Math., vol. 1848, pp. 109–161. Springer, Berlin, Heidelberg, New York (2004)Google Scholar
  14. 14.
    Huang, X.: On some problems in several complex variables and Cauchy–Riemann geometry. In: Yang, L., Yau, S.T. (eds.) Proceedings of ICCM. AMS/IP Stud. Adv. Math., vol. 20, pp. 383–396 (2001)Google Scholar
  15. 15.
    Huang, X.: On an n-manifold in ℂn near an elliptic complex tangent. J. Am. Math. Soc. 11, 669–692 (1998)MATHCrossRefGoogle Scholar
  16. 16.
    Huang, X., Krantz, S.: On a problem of Moser. Duke Math. J. 78, 213–228 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ji, S.: Algebraicity of real analytic hypersurfaces with maxium rank. Am. J. Math. 124, 255–264 (2002)CrossRefGoogle Scholar
  18. 18.
    Kenig, C., Webster, S.: The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67, 1–21 (1982)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kenig, C., Webster, S.: On the hull of holomorphy of an n-manifold in ℂn. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11(2), 261–280 (1984)MATHMathSciNetGoogle Scholar
  20. 20.
    Mir, N.: Convergence of formal embeddings between real-analytic hypersurfaces in codimension one. J. Differ. Geom. 62, 163–173 (2002)MATHMathSciNetGoogle Scholar
  21. 21.
    Moser, J.: Analytic surfaces in ℂ2 and their local hull of holomorphy. Ann. Acad. Sci. Fenn., Math. 10, 397–410 (1985)MATHGoogle Scholar
  22. 22.
    Moser, J., Webster, S.: Normal forms for real surfaces in ℂ2 near complex tangents and hyperbolic surface transformations. Acta Math. 150, 255–296 (1983)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Poincaré, H.: Les fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo, II. Ser. 23, 185–220 (1907)MATHCrossRefGoogle Scholar
  24. 24.
    Stolovitch, L.: Family of intersecting totally real manifolds of (ℂn,0) and CR-singularities. Preprint (2006)Google Scholar
  25. 25.
    Webster, S.: Pairs of intersecting real manifolds in complex space. Asian J. Math. 7(4), 449–462 (2003)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Department of Mathematics, Hill Center-Busch CampusRutgers UniversityPiscatawayUSA

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