Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Kobayashi indicatrix at the center of a circular domain

Author: Theodore J. Barth
Journal: Proc. Amer. Math. Soc. 88 (1983), 527-530
MSC: Primary 32F15; Secondary 32A07, 32H15
MathSciNet review: 699426
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The indicatrix of the Kobayashi infinitesimal metric at the center of a pseudoconvex complete circular domain coincides with this domain. It follows that a nonconvex complete circular domain cannot be biholomorphic to any convex domain. An example shows that a bounded pseudoconvex complete circular domain in $ {{\mathbf{C}}^2}$ need not be taut.

References [Enhancements On Off] (What's this?)

  • [1] T. J. Barth, Taut and tight complex manifolds, Proc. Amer. Math. Soc. 24 (1970), 429-431. MR 0252679 (40:5897)
  • [2] C. Carathéodory, Über die Geometrie der analytischen Abbildungen, die durch analytische Funktionen von zwei Veränderlichen vermittelt werden, Abh. Math. Sem. Univ. Hamburg 6 (1928), 96-145.
  • [3] T. Franzoni and E. Vesentini, Holomorphic maps and invariant distances, North-Holland Math. Studies, 40, Notas Mat., 69, North-Holland, Amsterdam and New York, 1980. MR 563329 (82a:32032)
  • [4] L. A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy, North-Holland Math. Studies, 34, Notas Mat., 65, North-Holland, Amsterdam and New York, 1979, pp. 345-406. MR 520667 (80j:32043)
  • [5] F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. MR 1511365
  • [6] N. Kerzman and J. P. Rosay, Fonctions plurisousharmoniques d'exhaustion bornées et domaines taut, Math. Ann. 257 (1981), 171-184. MR 634460 (83g:32019)
  • [7] S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), 357-416. MR 0414940 (54:3032)
  • [8] P. Noverraz, Pseudo-convexité, convexité polynomiale et domaines d'holomorphie en dimension infinie, North-Holland Math. Studies, 3, Notas Mat., 48, North-Holland, Amsterdam, 1973.
  • [9] P. Pflug, About the Carathéodory completeness of all Reinhardt domains, preprint. MR 771335 (86b:32026)
  • [10] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193-233. MR 0224869 (37:468)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32F15, 32A07, 32H15

Retrieve articles in all journals with MSC: 32F15, 32A07, 32H15

Additional Information

Keywords: Kobayashi metric, complete circular domain, taut domain, pseudoconvex domain, convex domain
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society