On the boundedness of invariant hyperbolic domains

Jiafu Ning; Xiangyu Zhou

doi:10.5802/crmath.42

Analyse complexe, Géométrie analytique

On the boundedness of invariant hyperbolic domains
[Sur le caractère borné des domaines hyperboliques invariants]

Jiafu Ning ¹ ; Xiangyu Zhou ²

¹ Department of Mathematics, Central South University, Changsha, Hunan 410083, China.
² Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China

Comptes Rendus. Mathématique, Volume 358 (2020) no. 3, pp. 321-326.

Résumé
Abstract

Dans cet article, nous généralisons un théorème de A. Kodama sur le caractère borné des domaines circulaires hyperboliques. Nous démontrons que si $K$ est un groupe de Lie compact qui agit linéairement sur $ℂ^{n}$ et vérifie $𝒪 {(ℂ^{n})}^{K} = ℂ$ , et si $Ω$ est un domaine $K$ -invariant orbitalement convexe de $ℂ^{n}$ qui contient $0$ , alors $Ω$ est borné si et seulement s’il est hyperbolique au sens de Kobayashi.

In this paper, we generalize a theorem of A. Kodama about boundedness of hyperbolic circular domains. We will prove that if $K$ is a compact Lie group which acts linearly on $ℂ^{n}$ with $𝒪 {(ℂ^{n})}^{K} = ℂ$ , and $Ω$ is a $K$ -invariant orbit convex domain in $ℂ^{n}$ which contains $0$ , then $Ω$ is bounded if and only if $Ω$ is Kobayashi hyperbolic.

Reçu le : 2019-10-28
Accepté le : 2020-03-24
Publié le : 2020-07-10

DOI : 10.5802/crmath.42

Classification : 32M05, 32T99, 32A07

Affiliations des auteurs :

Jiafu Ning ¹ ; Xiangyu Zhou ²

¹ Department of Mathematics, Central South University, Changsha, Hunan 410083, China.
² Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jiafu Ning and Xiangyu Zhou},
     title = {On the boundedness of invariant hyperbolic domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {321--326},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.42},
     language = {en},
}

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Jiafu Ning; Xiangyu Zhou. On the boundedness of invariant hyperbolic domains. Comptes Rendus. Mathématique, Volume 358 (2020) no. 3, pp. 321-326. doi : 10.5802/crmath.42. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.42/

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