Advertisement

Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 185–197 | Cite as

Totally geodesic discs in strongly convex domains

  • Hervé GaussierEmail author
  • Harish Seshadri
Article

Abstract

We prove that every isometry from the unit disk Δ in \({\mathbb{C}}\) , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class \({\mathcal{C}^3}\) in \({\mathbb{C}^n}\) , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class \({\mathcal{C}^3}\) in \({\mathbb{C}^n}\) to a strongly convex bounded domain of class \({\mathcal{C}^3}\) in \({\mathbb{C}^m}\) , is either holomorphic or anti-holomorphic.

Keywords

Complex Manifold Convex Domain Pseudoconvex Domain Distributional Derivative Complex Geodesic 
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. 1.
    Balogh Z.M., Bonk M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75, 504–533 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beltrami E.J.: Another proof of Weyl’s lemma. SIAM Rev. 10, 212–213 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chang C.H., Hu M.C., Lee H.P.: Extremal analytic discs with prescribed boundary data. Trans. Am. Math. Soc. 310, 355–369 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Graham I.: Boundary behavior of the Carathéodory, Kobayashi, and Bergman metrics on strongly pseudoconvex domains in C n with smooth boundary. Bull. Am. Math. Soc. 79, 749–751 (1973)zbMATHCrossRefGoogle Scholar
  5. 5.
    Kim K.T., Krantz S.G.: A Kobayashi metric version of Bun Wong’s theorem. Complex Var. Elliptic Equ. 54, 355–369 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kobayashi S.: Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Jpn. 19, 460–480 (1967)zbMATHCrossRefGoogle Scholar
  7. 7.
    Kobayashi, S.: Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 318. Springer, Berlin (1998)Google Scholar
  8. 8.
    Lempert L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109(4), 427–474 (1981)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lempert L.: Holomorphic retracts and intrinsic metrics in convex domains. Anal. Math. 8, 257–261 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Patrizio, G.: On the convexity of the Kobayashi indicatrix. In: Deformations of Mathematical Structures (Lódź/Lublin, 1985/87), pp. 171–176. Kluwer, Dordrecht (1989)Google Scholar
  11. 11.
    Royden, H.L.: Remarks on the Kobayashi metric. In: Several Complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), pp. 125–137. Lecture Notes in Math., vol. 185. Springer, Berlin (1971)Google Scholar
  12. 12.
    Seshadri H., Verma K.: On isometries of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 393–417 (2006)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Seshadri H., Verma K.: On the compactness of isometry groups in complex analysis. Complex Var. Elliptic Equ. 54, 387–399 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Venturini S.: Pseudodistances and pseudometrics on real and complex manifolds. Ann. Mat. Pura Appl. 154, 385402 (1989)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.UJF-Grenoble 1, Institut Fourier, CNRS UMR 5582GrenobleFrance
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations