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)



Isometries for the Carathéodory metric

Authors: Marco Abate and Jean-Pierre Vigué
Journal: Proc. Amer. Math. Soc. 136 (2008), 3905-3909
MSC (2000): Primary 32H02
Published electronically: May 20, 2008
MathSciNet review: 2425730
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given two open unit balls $ B_1$ and $ B_2$ in complex Banach spaces, we consider a holomorphic mapping $ f\colon B_1\to B_2$ such that $ f(0)=0$ and $ f'(0)$ is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that $ f(B_1)$ is a complex closed analytic submanifold of $ B_2$.

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

  • 1. L. Belkhchicha. Caractérisation des isomorphismes analytiques sur la boule-unité de $ \mathbb{C}^{n}$ pour une norme. Math. Z. 215 (1994), 129-141. MR 1254816 (94m:32037)
  • 2. T. Franzoni and E. Vesentini. Holomorphic maps and invariant distances. Notas de Matematica [Mathematical Notes], 69. North-Holland Publishing Co., Amsterdam-New York, 1980. MR 563329 (82a:32032)
  • 3. I. Graham. Holomorphic mappings into strictly convex domains which are Kobayashi isometries at one point. Proc. Amer. Math. Soc. 105 (1989), 917-921. MR 961406 (89k:32048)
  • 4. M. Jarnicki and P. Pflug. Invariant distances and metrics in complex analysis. de Gruyter Expositions in Mathematics, 9. Walter de Gruyter and Co., Berlin, 1993. MR 1242120 (94k:32039)
  • 5. L. Lempert. La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109 (1981), 427-474. MR 660145 (84d:32036)
  • 6. L. Lempert. Holomorphic retracts and intrinsic metrics in convex domains. Anal. Math. 8 (1982), 257-261. MR 690838 (84f:32026)
  • 7. P. Mazet. Principe du maximum et lemme de Schwarz à valeurs vectorielles. Canad. Math. Bull. 40 (1997), 356-363. MR 1464844 (99b:30063)
  • 8. J.-P. Vigué. Sur les points fixes d'applications holomorphes. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 927-930. MR 873396 (88a:32034)
  • 9. J.-P. Vigué. Sur la caractérisation des isomorphismes analytiques entre domaines bornés d'un espace de Banach complexe. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 145-155. MR 1276766 (95c:58016)
  • 10. J.-P. Vigué. Un lemme de Schwarz pour les boules-unités ouvertes. Canad. Math. Bull. 40 (1997), no. 1, 117-128. MR 1443733 (98c:32028)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32H02

Retrieve articles in all journals with MSC (2000): 32H02

Additional Information

Marco Abate
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy

Jean-Pierre Vigué
Affiliation: LMA, Université de Poitiers, CNRS, Mathématiques, SP2MI, BP 30179, 86962 Futuroscope cedex, France

Received by editor(s): July 17, 2007
Received by editor(s) in revised form: September 19, 2007
Published electronically: May 20, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society