Geodesic
On~orbit connectedness, orbit convexity and envelopes of holomorphy
Izvestiya. Mathematics , Tome 44 (1995) no. 2, pp. 403-413.

Voir la notice de l'article provenant de la source Math-Net.Ru

We are concerned with the univalence and discription of the envelope of holomorphy $E(D)$ for a domain $D$ having a compact Lie group action. Our main result is the following: Let $X$ be a holomorphic Stein $K^C$-manifold, $D\subset X$ a $K$-invariant orbit connected domain. Then $E(D)$ is schlicht and orbit convex if and only if $E(K^C\cdot D)$ is schlicht. Moreover, in this case, $E(K^C\cdot D)=K^C\cdot e(d)$. 
@article{IM2_1995_44_2_a10,
     author = {Xiang-Yu Zhou},
     title = {On~orbit connectedness, orbit convexity and envelopes of holomorphy},
     journal = {Izvestiya. Mathematics },
     pages = {403--413},
     publisher = {mathdoc},
     volume = {44},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a10/}
}
TY  - JOUR
AU  - Xiang-Yu Zhou
TI  - On~orbit connectedness, orbit convexity and envelopes of holomorphy
JO  - Izvestiya. Mathematics 
PY  - 1995
SP  - 403
EP  - 413
VL  - 44
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a10/
LA  - en
ID  - IM2_1995_44_2_a10
ER  - 
%0 Journal Article
%A Xiang-Yu Zhou
%T On~orbit connectedness, orbit convexity and envelopes of holomorphy
%J Izvestiya. Mathematics 
%D 1995
%P 403-413
%V 44
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a10/
%G en
%F IM2_1995_44_2_a10
Xiang-Yu Zhou. On~orbit connectedness, orbit convexity and envelopes of holomorphy. Izvestiya. Mathematics , Tome 44 (1995) no. 2, pp. 403-413. http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a10/

[1] E. Bedford, J. Dadok, Matrix Reinhardt domains, Preprint | MR | Zbl

[2] E. Casadio Tarabusi, S. Trapani, “Envelopes of holomorphy of Hartogs domains and circular domains”, Pacific J. of Math., 149:2 (1991) | MR | Zbl

[3] G. Coeure, J.-J. Loeb, “Univalence de certaines envelopes d'holomorphie”, C.R. Acad. Sci. Paris. Serie 1, 302:2 (1989), 59–61 | MR

[4] E. Fornaess, R. Narasimhan, “The Levi problem on complex spaces with singularities”, Math. Ann., 248 (1980), 47–72 | DOI | MR | Zbl

[5] H. Grauert, R. Remmert, Theory of Stein spaces, Grundlehren der Mathematischen Wissenschaften, 236, Springer-Verlag, 1979 | MR | Zbl

[6] P. Heinzner, “Geometric invariant theory on Stein spaces”, Math. Ann., 289:4 (1991), 631–662 | DOI | MR | Zbl

[7] A. G. Sergeev, P. Khaintsner, “Rasshirennyi matrichnyi disk yavlyaetsya oblastyu golomorfnosti”, Izv. RAN. Ser. matem., 55:3 (1991), 647–657 | MR

[8] A. T. Huckleberry, “Actions of holomorphical transformation groups”, Itogi Nayki i Tehniki. Current Math. Problem. Fundamental direction, 69, VINITI, 1991, 166–221 (Russian)

[9] A. T. Hucklebery, G. Fels, A characterization of $K$-invariant Stein domains in symmetric embeddings, Preprint

[10] G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco–London–Amsterdam, 1965 | MR | Zbl

[11] M. Lassale, “Deux generalisations du “theoreme des trois cercles” de Hadamard”, Math. Ann., 249 (1980), 17–26 | DOI | MR

[12] M. Lassale, “Series de Laurent des fonctions holomorphes dans la complexification d'un expace symmetrique compact”, Ann. Sci. Ecole Norm. Sup., 11 (1978), 167–210 | MR

[13] J.-J. Loeb, “Plurisousharmonicite et convexite sur les groupes reductifs complexes”, Publ. IRMA-Lille, 2:8 (1986)

[14] J.-J. Loeb, “Pseudo-convexite des ouverts invariants et convexite geodesique dans certains expaces symmetriques”, Seminaire d'Analyse, Lect. Notes in Math., 1198, eds. P. Lelong, P. Dolbeault, H. Skoda, Springer-Verlag, 1983–1984 | MR

[15] D. Luna, Th. Vust, “Plongements d'espaces homogenes”, Comment Math. Helvetici, 58:2 (1983), 186–245 | DOI | MR | Zbl

[16] O. Rothaus, “Envelopes of holomorphy of domains in complex Lie groups”, Problems in analysis, Princeton university press, 1970, 309–317 | MR | Zbl

[17] A. G. Sergeev, On matrix Reinhardt domains, Preprint, Mittag-Leffler Institute, Stockholm | Zbl

[18] B. V. Shabat, Introduction to complex analysis, vol. 2, Nayka, Moskva, 1985 (Russian) | MR

[19] D. M. Snow, “Reductive group action on Stein spaces”, Math. Ann., 259 (1982), 79–97 | DOI | MR | Zbl

[20] V. S. Vladimirov, Methods of the theory of functions of many complex variables, Nayka, Moskva, 1964 (Russian)

[21] X.-Y. Zhou, “On matrix Reinhardt domains”, Math. Ann., 287 (1990), 35–46 | DOI | MR | Zbl

[22] X.-Y. Zhou, “On orbit convexity of certain torus linearly invariant domains of holomorphy”, Dokl. A.N. SSSR, 322:2 (1992), 262–267 (Russian) | Zbl