Revêtements et isométries pour la métrique infinitésimale de Kobayashi

Vigué, Jean-Pierre

doi:10.1090/S0002-9939-01-05977-9

Revêtements et isométries pour la métrique infinitésimale de Kobayashi
HTML articles powered by AMS MathViewer

by Jean-Pierre Vigué

Proc. Amer. Math. Soc. 129 (2001), 3279-3284

DOI: https://doi.org/10.1090/S0002-9939-01-05977-9

Published electronically: March 29, 2001

PDF | Request permission

Abstract:

In this paper, we prove that, under some hypothesis on the domains, if a holomorphic mapping $f:D_{1}\longrightarrow D_{2}$ is an isometry for the Kobayashi infinitesimal metric at a point, it is a covering map. In the case $D_{1} = D_{2}$, we prove, in certain cases, that $f$ is an analytic isomorphism.

References

Larbi Belkhchicha, Caractérisation des isomorphismes analytiques sur la boule-unité de $\textbf {C}^n$ pour une norme, Math. Z. 215 (1994), no. 1, 129–141 (French). MR 1254816, DOI 10.1007/BF02571702
H. Cartan. Sur les fonctions de plusieurs variables complexes. L’itération des transformations intérieures d’un domaine borné. Math. Z. 35 (1932), p. 760-773.
Tullio Franzoni and Edoardo Vesentini, Holomorphic maps and invariant distances, Notas de Matemática [Mathematical Notes], vol. 69, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 563329
Ian Graham, Holomorphic mappings into strictly convex domains which are Kobayashi isometries at one point, Proc. Amer. Math. Soc. 105 (1989), no. 4, 917–921. MR 961406, DOI 10.1090/S0002-9939-1989-0961406-0
Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
Shoshichi Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), no. 3, 357–416. MR 414940, DOI 10.1090/S0002-9904-1976-14018-9
L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), no. 4, 257–261 (English, with Russian summary). MR 690838, DOI 10.1007/BF02201775
H. Royden and P. Wong. Carathéodory and Kobayashi metrics on convex domains. Preprint (1983).
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
Jean-Pierre Vigué, Caractérisation des automorphismes analytiques d’un domaine convexe borné, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 4, 101–104 (French, with English summary). MR 756530
Jean-Pierre Vigué, Sur la caractérisation des automorphismes analytiques d’un domaine borné, Portugal. Math. 43 (1985/86), no. 4, 439–453 (1987) (French). MR 911450
H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. MR 224869, DOI 10.1007/BF02392083