The Lempert function of the symmetrized polydisc in higher dimensions is not a distance. (English) Zbl 1123.32007
The authors prove that the Lempert function of the symmetrized polydisc \(G_n\) in dimension \(n\geq 3\) is not a distance (whereas it is a distance, and coincides with the Carathéodory distance, in dimension 2). In particular, the symmetrized polydisc of any dimension \(n\geq 2\) cannot be exhausted by domains biholomorphic to convex domains.
Furthermore, the authors also show that if \(D\subset{\mathbb C}^m\) is a balanced convex bounded domain then \(G_2\times D\) is a bounded pseudoconvex domain that cannot be exhausted by domains biholomorphic to convex domains but where the Carathédory and Kobayashi distances agree.
Furthermore, the authors also show that if \(D\subset{\mathbb C}^m\) is a balanced convex bounded domain then \(G_2\times D\) is a bounded pseudoconvex domain that cannot be exhausted by domains biholomorphic to convex domains but where the Carathédory and Kobayashi distances agree.
Reviewer: Marco Abate (Pisa)
MSC:
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
Keywords:
symmetrized polydisc; Caratheodory distance and metric; Kobayashi distance and metric; Lempert functionReferences:
| [1] | J. Agler and N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), no. 3, 375 – 403. · Zbl 1055.32010 · doi:10.1007/BF02922097 |
| [2] | C. Costara, Dissertation, Université Laval (2004). |
| [3] | C. Costara, The symmetrized bidisc and Lempert’s theorem, Bull. London Math. Soc. 36 (2004), no. 5, 656 – 662. · Zbl 1065.32006 · doi:10.1112/S0024609304003200 |
| [4] | Constantin Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), no. 1, 23 – 55. · Zbl 1074.30035 · doi:10.4064/sm170-1-2 |
| [5] | 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. · Zbl 1098.32006 · doi:10.4064/ap83-2-9 |
| [6] | Armen Edigarian and Włodzimierz Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), no. 4, 364 – 374. · Zbl 1068.32012 · doi:10.1007/s00013-004-1183-z |
| [7] | Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis — revisited, Dissertationes Math. (Rozprawy Mat.) 430 (2005), 192. · Zbl 1085.32005 · doi:10.4064/dm430-0-1 |
| [8] | Masashi Kobayashi, On the convexity of the Kobayashi metric on a taut complex manifold, Pacific J. Math. 194 (2000), no. 1, 117 – 128. · Zbl 1020.32007 · doi:10.2140/pjm.2000.194.117 |
| [9] | 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). · Zbl 0492.32025 |
| [10] | N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Polon. Math., 88 (2006), 279-283. · Zbl 1111.32020 |
| [11] | N. Nikolov, P. Pflug, On the definition of the Kobayashi-Buseman pseudometric, Internat. J. Math., 17 (2006), 1145-1149. · Zbl 1125.32005 |
| [12] | Myung-Yull Pang, On infinitesimal behavior of the Kobayashi distance, Pacific J. Math. 162 (1994), no. 1, 121 – 141. · Zbl 0788.32020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
