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 Lempert function of the symmetrized polydisc in higher dimensions is not a distance

Authors: Nikolai Nikolov, Peter Pflug and Wlodzimierz Zwonek
Journal: Proc. Amer. Math. Soc. 135 (2007), 2921-2928
MSC (2000): Primary 32F45
Published electronically: May 10, 2007
MathSciNet review: 2317970
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.

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

  • 1. J. Agler, N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375-403.MR 2077158 (2005e:32022)
  • 2. C. Costara, Dissertation, Université Laval (2004).
  • 3. C. Costara, The symmetrized bidisc and Lempert's theorem, Bull. London Math. Soc. 36 (2004), 656-662. MR 2070442 (2005d:32019)
  • 4. C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), 23-55. MR 2142182 (2006d:30054)
  • 5. A. Edigarian, A note on Costara's paper, Ann. Polon. Math. 83 (2004), 189-191.MR 2111408 (2005h:32028)
  • 6. A. Edigarian, W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), 364-374. MR 2135687 (2006b:32020)
  • 7. M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis-revisited, Diss. Math. 430 (2005), 1-192.MR 2167637 (2006h:32010)
  • 8. M. Kobayashi, On the convexity of the Kobayashi metric on a taut complex manifold, Pacific J. Math. 194 (2000), 117-128.MR 1756629 (2001b:32018)
  • 9. L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474.MR 0660145 (84d:32036)
  • 10. N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Polon. Math., 88 (2006), 279-283.
  • 11. N. Nikolov, P. Pflug, On the definition of the Kobayashi-Buseman pseudometric, Internat. J. Math., 17 (2006), 1145-1149.
  • 12. M.-Y. Pang, On infinitesimal behavior of the Kobayashi distance, Pacific J. Math. 162 (1994), 121-141. MR 1247146 (94i:32034)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32F45

Retrieve articles in all journals with MSC (2000): 32F45

Additional Information

Nikolai Nikolov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Peter Pflug
Affiliation: Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany

Wlodzimierz Zwonek
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland

Keywords: Symmetrized polydisc, Carath\'eodory distance and metric, Kobayashi distance and metric, Lempert function
Received by editor(s): January 31, 2006
Received by editor(s) in revised form: June 9, 2006
Published electronically: May 10, 2007
Additional Notes: This paper was written during the stays of the first and third named authors at Universität Oldenburg supported by grants from the DFG (January – March 2006 and November 2005 (DFG Projekt 227/8-1/2)). They would like to thank both institutions for their support. The third author was also supported by the Research Grant No. 1 PO3A 005 28, which is financed by public means in the programme promoting science in Poland in the years 2005-2008.
The authors thank the referee for his remarks which essentially improved this paper.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society