Journal d’Analyse Mathematique

, Volume 72, Issue 1, pp 261–278 | Cite as

Regularity of discs attached to a submanifold ofC n

Article

Abstract

Letp be an analytic disc attached to a generating CR-submanifoldM of C n . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC α holomorphic perturbations ofp alongM is a Banach submanifold of (Aα(D))n are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A α (D))n.

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References

  1. [1]
    H. Alexander,Hulls of deformations in C n, Trans. Amer. Math. Soc.266 (1981), 243–257.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. S. Baouendi, L. P. Rothschild and J.-M. Trépreau,On the geometry of analytic discs attached to real manifolds, J. Differential Geom.39 (1994), 379–405.MATHMathSciNetGoogle Scholar
  3. [3]
    E. Bedford,Stability of the polynomial hull of T 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)8 (1981), 311–315.MATHMathSciNetGoogle Scholar
  4. [4]
    E. Bedford,Levi flat hypersurfaces in C2 with prescribed boundary: Stability, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)9 (1982), 529–570.MATHMathSciNetGoogle Scholar
  5. [5]
    E. Bedford and B. Gaveau,Envelopes of holomorphy of certain two-spheres in C2, Amer. J. Math.105(1983), 975–1009.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. černe,Analytic discs attached to a generating CR-manifold, Ark. Mat.33 (1995), 217–248.CrossRefMathSciNetGoogle Scholar
  7. [7]
    F. Forstnerič,Analytic discs with boundaries in a maximal real submanifolds of C 2, Ann. Inst. Fourier37 (1987), 1–44.Google Scholar
  8. [8]
    F. Forstneric,Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J.37 (1988), 869–889.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Globevnik,Perturbation by analytic discs along maximal real submanifold of C n, Math. Z.217 (1994), 287–316.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Globevnik,Perturbing analytic discs attached to maximal real submanifolds of C n, Indag. Math.7(1996), 37–46.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Gromov,Pseudo-holomorphic curves in symplectic manifolds, Invent. Math.81 (1985), 307–347.CrossRefMathSciNetGoogle Scholar
  12. [12]
    L. Lempert,La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France109 (1981), 427–474.MATHMathSciNetGoogle Scholar
  13. [13]
    Y.-G. Oh,The Fredholm-regularity and realization of the Riemann—Hilbert problem and application to the perturbation theory of analytic discs, preprint.Google Scholar
  14. [14]
    Y.-G. Oh,Fredholm theory of holomorphic discs with Lagrangian or totally real boundary conditions under the perturbation of boundary conditions, Math. Z.222 (1996), 505–520.MATHMathSciNetGoogle Scholar
  15. [15]
    S. Trapani,Defect and evaluations, preprint.Google Scholar
  16. [16]
    J-M. Trépreau,On the global Bishop equation, preprint.Google Scholar
  17. [17]
    A. E. Tumanov,Extension of CR-functions into a wedge from a manifold of finite type (Russian), Mat. Sbornik136 (1988), 128–139; English transi.: Math. USSR Sbornik64 (1989), 129–140.Google Scholar
  18. [18]
    N. P. Vekua,Systems of Singular Integral Equations, Nordhoff, Groningen, 1967.Google Scholar
  19. [19]
    N. P. Vekua,Systems of Singular Integral Equations, 2nd edition (Russian), Nauka, Moscow, 1970.Google Scholar

Copyright information

© Hebrew University 1997

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia

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