The Levi problem on strongly pseudoconvex G-bundles

  • Joe J. Perez
Original Paper


Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle GMX so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.


\({\bar\partial}\)-Neumann problem Subelliptic operators Harmonic analysis 

MR Classification Numbers

32E40 32W05 43A30 


  1. 1.
    Arnal D., Ludwig J.: Q.U.P. and Paley–Wiener properties of unimodular, especially nilpotent, Lie groups. Proc. Amer. Math. Soc. 125(4), 1071–1080 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brudnyi A.: Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains. J. Funct. Anal. 231(2), 418–437 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brudnyi A.: On holomorphic L 2 functions on coverings of strongly pseudoconvex manifolds. Publ. Res. Inst. Math. Sci. 43(4), 963–976 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brudnyi A.: On holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds. J. Funct. Anal. 249(2), 354–371 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brudnyi A.: Hartogs type theorems for CRL 2 functions on coverings of strongly pseudoconvex manifolds. Nagoya Math. J. 189, 27–47 (2008)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, S.-C., Shaw, M.-C.: Partial differential equations in several complex variables. AMS-IP Stud. Adv. Math. 19, (2001)Google Scholar
  7. 7.
    Engliš, M.: Pseudolocal estimates for \({\bar\partial}\) on general pseudoconvex domains, Indiana Univ. Math. J. 50(4), 1593–1607 (2001), and Erratum, to appear in Indiana Univ. Math. J.Google Scholar
  8. 8.
    Folland G.B.: Real Analysis, Modern Techniques and Their Applications. Wiley, New York (1984)zbMATHGoogle Scholar
  9. 9.
    Folland G.B., Kohn J.J.: The Neumann Problem for the Cauchy-Riemann Complex, Annals of Mathematical Studies, No. 75. Princeton University Press, Princeton (1972)Google Scholar
  10. 10.
    Grauert H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gromov M., Henkin G., Shubin M.: Holomorphic L 2 functions on coverings of pseudoconvex manifolds. Geom. Funct. Anal. 8(3), 552–585 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Heinzner, P., Huckleberry, A.T., Kutzschebauch, F.: Abels’ theorem in the real analytic case and applications to complexifications. In: Complex Analysis and Geometry, Lecture Notes in Pure and Applied Mathematics, pp. 229–273. Marcel Dekker, New York (1995)Google Scholar
  13. 13.
    Kohn J.J.: Harmonic integrals on strongly pseudoconvex manifolds I. Ann. Math. 78, 112–148 (1963)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kohn J.J.: Harmonic integrals on strongly pseudoconvex manifolds, II. Ann. Math. 79, 450–472 (1964)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Knapp A.W.: Lie Groups Beyond an Introduction Vol. 2. Birkhäuser, Boston (2002)Google Scholar
  16. 16.
    Levi E.E.: Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse. Ann. Mat. Pura Appl. 18, 69–79 (1911)Google Scholar
  17. 17.
    Margulis G.A.: Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17. Springer-Verlag, Berlin (1991)Google Scholar
  18. 18.
    Pedersen G.K.: C*-Algebras and Their Automorphism Groups. Academic Press, London (1979)Google Scholar
  19. 19.
    Perez J.J.: The G-Fredholm property for the \({\bar\partial}\)-Neumann problem. J. Geom. Anal. 19, 87–106 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rudin W.: Functional Analysis. Vol. 2. McGraw-Hill, New York (1991)Google Scholar
  21. 21.
    Siu Y.-T.: Pseudoconvexity and the problem of Levi. Bull. Amer. Math. Soc. 84(4), 481–512 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Takesaki M.: Theory of Operator Algebras. Vol. I. Springer-Verlag, Berlin (1979)Google Scholar
  23. 23.
    Todor R., Chiose I., Marinescu G.: Morse inequalities for covering manifolds. Nagoya Math. J. 163, 145–165 (2001)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Todor R., Chiose I., Marinescu G.: L 2 holomorphic sections of bundles over weakly pseudoconvex coverings. Geom. Dedicata 91, 23–43 (2002)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Universität WienViennaAustria

Personalised recommendations