Advertisement

Mathematische Zeitschrift

, Volume 261, Issue 1, pp 39–63 | Cite as

Plurisubharmonic polynomials and bumping

  • Gautam BharaliEmail author
  • Berit Stensønes
Article

Abstract

We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain \({\Omega \subset \mathbb{C}^{n}}\) in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with ∂Ω, at the site of the bumping, are explicitly realised. Generally, when \({\Omega \subset \mathbb{C}^{n}, n \geq 3}\) , the known methods lead to bumpings with high orders of contact—which are not explicitly known either—at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in \({\mathbb{C}^3}\) . These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.

Keywords

Bumping Finite-type domain Plurisubharmonic function Weighted-homogeneous function 

Mathematics Subject Classification (2000)

Primary 32F05 32T25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bedford, E., Fornaess, J.E.: A construction of peak functions on weakly pseudoconvex domains. Ann. Math. 107, 555–568 (1978)CrossRefMathSciNetGoogle Scholar
  2. 2.
    D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115, 615–637 (1982)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Catlin, D.: Subelliptic estimates for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Diederich, K., Fornaess, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39, 129–141 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Diederich, K., Fornaess, J.E.: Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary. Ann. Math. 110, 575–592 (1979)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Diederich, K., Fornaess, J.E.: Support functions for convex domains of finite type. Math. Z. 230, 145–164 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Diederich, K., Fornaess, J.E.: Lineally convex domains of finite type: holomorphic support functions. Manuscripta Math. 112, 403–431 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Diederich, K., Herbort, G.: Pseudoconvex Domains of Semiregular Type, Contributions to Complex Analysis and Analytic Geometry. Aspects Math., pp. 127–161. Vieweg, Braunschweig (1994)Google Scholar
  9. 9.
    Fornaess, J.E.: Sup-norm estimates for \({\overline{\partial}}\) in \({\mathbb{C}^{2}}\) . Ann. of Math. 123, 335–345 (1986)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fornaess, J.E., Sibony, N.: Construction of P.S.H. functions on weakly pseudoconvex domains. Duke Math. J. 58, 633–655 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Noell, A.: Peak functions for pseudoconvex domains, Several Complex Variables—Proceedings of the Mittag-Leffler Institute, 1987–1988, pp. 529–541. Princeton University Press, Princeton (1993)Google Scholar
  12. 12.
    Range, R.M.: Integral kernels and Hölder estimates for \({\overline{\partial}}\) on pseudoconvex domains of finite type in \({\mathbb{C}^{2}}\) . Math. Ann. 288, 63–74 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shafarevich, I.R.: Basic Algebraic Geometry 1, Varieties in Projective Space (2nd edn., translated from the Russian & with notes by Miles Reid). Springer, Berlin (1994)Google Scholar
  14. 14.
    van der Waerden, B.L.: Algebra vol. 1 (translated from the German by F. Blum & J.R. Schulenberger). Frederick Ungar Publishing Co. (1970)Google Scholar
  15. 15.
    Yu, J.Y.: Peak functions on weakly pseudoconvex domains. Indiana Univ. Math. J. 43, 1271–1295 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations