The Lempert function of the symmetrized polydisc in higher dimensions is not a distance
HTML articles powered by AMS MathViewer
- by Nikolai Nikolov, Peter Pflug and Wlodzimierz Zwonek PDF
- Proc. Amer. Math. Soc. 135 (2007), 2921-2928 Request permission
Abstract:
We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.References
- J. Agler and N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), no. 3, 375–403. MR 2077158, DOI 10.1007/BF02922097
- C. Costara, Dissertation, Université Laval (2004).
- C. Costara, The symmetrized bidisc and Lempert’s theorem, Bull. London Math. Soc. 36 (2004), no. 5, 656–662. MR 2070442, DOI 10.1112/S0024609304003200
- Constantin Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), no. 1, 23–55. MR 2142182, DOI 10.4064/sm170-1-2
- Armen Edigarian, A note on C. Costara’s paper: “The symmetrized bidisc and Lempert’s theorem” [Bull. London Math. Soc. 36 (2004), no. 5, 656–662; MR2070442], Ann. Polon. Math. 83 (2004), no. 2, 189–191. MR 2111408, DOI 10.4064/ap83-2-9
- Armen Edigarian and Włodzimierz Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), no. 4, 364–374. MR 2135687, DOI 10.1007/s00013-004-1183-z
- Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis—revisited, Dissertationes Math. 430 (2005), 192. MR 2167637, DOI 10.4064/dm430-0-1
- Masashi Kobayashi, On the convexity of the Kobayashi metric on a taut complex manifold, Pacific J. Math. 194 (2000), no. 1, 117–128. MR 1756629, DOI 10.2140/pjm.2000.194.117
- László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427–474 (French, with English summary). MR 660145, DOI 10.24033/bsmf.1948
- N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Polon. Math., 88 (2006), 279–283.
- N. Nikolov, P. Pflug, On the definition of the Kobayashi-Buseman pseudometric, Internat. J. Math., 17 (2006), 1145–1149.
- Myung-Yull Pang, On infinitesimal behavior of the Kobayashi distance, Pacific J. Math. 162 (1994), no. 1, 121–141. MR 1247146, DOI 10.2140/pjm.1994.162.121
Additional Information
- Nikolai Nikolov
- Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
- MR Author ID: 332842
- Email: nik@math.bas.bg
- Peter Pflug
- Affiliation: Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany
- MR Author ID: 139035
- Email: pflug@mathematik.uni-oldenburg.de
- Wlodzimierz Zwonek
- Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland
- Email: Wlodzimierz.Zwonek@im.uj.edu.pl
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2921-2928
- MSC (2000): Primary 32F45
- DOI: https://doi.org/10.1090/S0002-9939-07-08817-X
- MathSciNet review: 2317970