On Bounded Positive (m, p)-Circle Domains

Abstract

Let D be a bounded positive (m, p)-circle domain in ℂ². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ℂ². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ℂ² by the Lie group technique.

Keywords

(m, p)-Circular domain / Reinhardt domain / Holomorphically equivalent

References

[1] Cartan H.. Sur les groupes de transformations analytiques, 1935

[2] Thullen P.. Zur den Abbildungen durch analytische Funktionen mehrerer komplexer Veränder-lichen. Math. Ann., 1931, 104: 244-259

[3] Cartan H.. Les fonctions de deux variables complexes et le problème de la représentation analytique. J. de Math., 1931, 10: 1-114

[4] Cartan H.. Sur les transformations analytiques des domaines cerclés et semi-cerclés bornés. Math. Ann., 1932, 106: 540-573

[5] Xu Y. C.. On the holomorphic automorphism group of the bounded domain with positive (m. p) circle type, Acta Math. Sinica, 1963, 13: 419-432

[6] Xu Y. C., Wang D. L.. On the classification of the bounded domains with semi-circle type in C2. Acta Math. Sinica, 1980, 23: 372-384

[7] Yamamori A.. Automorphisms of normal quasi-circular domains. Bull. Sci. Math., 2014, 138: 406-415

[8] Yamamori A.. On the linearity of origin-preserving automorphisms of quasi-circular domains in Cn. J. Math. Anal. Appl., 2015, 426: 612-623

[9] Rong F.. On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math., 2016, 140: 92-98

[10] Xu Y. C.. On the classification of symetric schlicht domains in several complex variable. Adv. Math., 1965, 8: 109-144