Comptes Rendus
Analyse complexe, Géométrie analytique
On the boundedness of invariant hyperbolic domains
[Sur le caractère borné des domaines hyperboliques invariants]
Comptes Rendus. Mathématique, Volume 358 (2020) no. 3, pp. 321-326.

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 n et vérifie 𝒪( n ) K =, et si Ω est un domaine K-invariant orbitalement convexe de 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 n with 𝒪( n ) K =, and Ω is a K-invariant orbit convex domain in n which contains 0, then Ω is bounded if and only if Ω is Kobayashi hyperbolic.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.42
Classification : 32M05, 32T99, 32A07
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
@article{CRMATH_2020__358_3_321_0,
     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},
}
TY  - JOUR
AU  - Jiafu Ning
AU  - Xiangyu Zhou
TI  - On the boundedness of invariant hyperbolic domains
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 321
EP  - 326
VL  - 358
IS  - 3
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.42
LA  - en
ID  - CRMATH_2020__358_3_321_0
ER  - 
%0 Journal Article
%A Jiafu Ning
%A Xiangyu Zhou
%T On the boundedness of invariant hyperbolic domains
%J Comptes Rendus. Mathématique
%D 2020
%P 321-326
%V 358
%N 3
%I Académie des sciences, Paris
%R 10.5802/crmath.42
%G en
%F CRMATH_2020__358_3_321_0
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/

[1] Kazuo Azukawa Hyperbolicity of circular domains, Tôhoku Math. J., Volume 35 (1983), pp. 259-265 | MR | Zbl

[2] Enrico Casadio Tarabusi; Stefano Trapani Envelopes of holomorphy of Hartogs and circular domains, Pac. J. Math., Volume 149 (1991) no. 2, pp. 231-249 | DOI | MR | Zbl

[3] Peter Heinzner Geometric Invariant Theory on Stein spaces, Math. Ann., Volume 289 (1991) no. 4, pp. 631-662 | DOI | MR | Zbl

[4] Sigurdur Helgason Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press Inc., 1978 | MR | Zbl

[5] Wilhelm Kaup Über das Randverhalten von Holomorphen Automorphismen beschränkter Gebiete, Manuscr. Math., Volume 3 (1970), pp. 257-270 | DOI | Zbl

[6] Shoshichi Kobayashi Hyperbolic Manifolds and Holomorphic Mappings: an introduction, World Scientific, 2005 | DOI | Zbl

[7] Akio Kodama Boundedness of circular domains, Proc. Japan Acad., Ser. A, Volume 58 (1982), pp. 227-230 | DOI | MR | Zbl

[8] Steven Krantz Function Theory of Several Complex Variables, American Mathematical Society, 2001 | Zbl

[9] David Mumford; John Fogarty; Frances Kirwan Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 34, Springer, 1993 | Zbl

[10] Jiafu Ning; Hui-Ping Zhang; Xiang-Yu Zhou Proper holomorphic mappings between invariant domains in n , Trans. Am. Math. Soc., Volume 369 (2017) no. 1, pp. 517-536 | DOI | MR | Zbl

[11] Nolan R. Wallach Hilbert-Mumford type theorems (an unpublished lecture, www.math.ucsd.edu/~nwallach/lectures-3-math207.pdf)

[12] Xiao-Wen Wu; Fu-Sheng Deng; Xiang-Yu Zhou Rigidity and regularity in group actions, Sci. China, Ser. A, Volume 51 (2008) no. 4, pp. 819-826 | MR | Zbl

[13] Xiang-Yu Zhou On orbit connectedness, orbit convexity, and envelopes of holomorphy, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 58 (1994) no. 2, pp. 196-205 | MR | Zbl

[14] Xiang-Yu Zhou Some results related to group actions in several complex variables, Proceedings of the international congress of mathematicians (ICM 2002). Vol. II: Invited lectures, Higher Education Press, 2002, pp. 743-753 | Zbl

Cité par Sources :

Commentaires - Politique