Abstract
Let D be a bounded domain in C 2 with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.
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References
Bedford E., Fornaess J.E.: Biholomorphic maps of weakly pseudoconvex domains. Duke Math. J. 45, 711–719 (1978)
Bedford E., Pinchuk S.: Domains in C 2 with noncompact holomorphic automorphism group (English). Math. USSR Sb. 63, 141–151 (1989)
Bedford E., Pinchuk S.: Domains in C n + 1 with noncompact automorphism group. J. Geom. Anal. 1, 165–191 (1991)
Bedford E., Pinchuk S.: Convex domains with noncompact automorphism group (English). Russ. Acad. Sci. Sb. Math. 82(1), 1–10 (1995)
Bedford E., Pinchuk S.: Domains in C 2 with noncompact automorphism group. Indiana Univ. Math. J. 47(1), 199–222 (1998)
Berteloot F.: Characterization of models in C 2 by their automorphism groups. Int. J. Math. 5, 619–634 (1994)
Berteloot F.: Sur certains domaines faiblement pseudoconvexes dont le groupe d’automorphismes analytiques est non compact. Bull. Sc. Math. 2e Sér. 114, 411–420 (1990)
Chirka E.: Complex Analytic Sets. Kluwer, Dordrecht (1990)
Chirka E.: Regularity of the boundary of analytic sets (English). Math. USSR Sb. 45(3), 291–335 (1983)
Coupet, B., Pinchuk, S.: Holomorphic equivalence problem for weighted homogeneous rigid domains in C n + 1. In: Chirka, E.M. (ed.) Collection of Papers Dedicated to B. V. Shabat, pp. 111–126. Farsis, Moscow (1997)
Coupet B., Pinchuk S., Sukhov A.: On boundary rigidity and regularity of holomorphic mappings. Int. J. Math. 7(5), 617–643 (1996)
Diederich K., Fornaess J.E.: Biholomorphic maps between certain real analytic domains in C 2 Math. Ann. 245, 255–272 (1979)
Diederich K., Fornaess J.E.: Proper holomorphic mappings between real analytic pseudoconvex domains in C n. Math. Ann. 282, 681–700 (1988)
Diederich K., Pinchuk S.: Proper holomorphic maps in dimension two extend. Indiana Univ. Math. J. 44, 1089–1126 (1995)
Diederich K., Pinchuk S.: Regularity of continuous CR mappings in arbitrary dimension. Mich. Math. J. 51(1), 111–140 (2003)
Diederich K., Webster S.: A reflection principle for degenerate real hypersurfaces. Duke Math. J. 47(4), 835–843 (1980)
Efimov, A.M.: A generalization of the Wong–Rosay theorem for the unbounded case, (Russian) Math. Sb., 186, 41–50; translation in Sb. Math., 186 (1995), 967–976
Fu S., Isaev A.V., Krantz S.G.: Examples of domains with non-compact automorphism groups. Math. Res. Lett. 3(5), 609–617 (1996)
Fujita R.: Domaines sans point critique intérieur sur l’espace projectif complexe. J. Math. Soc. Jpn. 15, 443–473 (1963)
Gaussier H.: Characterization of convex domains with noncompact automorphism group. Mich. Math. J. 44, 375–388 (1997)
Greene, R., Krantz, S.: Invariants of Bergman geometry and results concerning the automorphism groups of domains in C n, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989), Sem. Conf., vol. 8. EditEl, Rende, pp. 107–136 (1991)
Henkin G.M.: An analytic polyhedron is not holomorphically equivalent to a strictly pseudoconvex domain (English). Sov. Math. Dokl. 14, 858–862 (1973)
Isaev A.V.: Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group. Geom. Topol. 12, 643–711 (2008)
Isaev A.V.: Hyperbolic n-dimensional manifolds with automorphism group of dimension n 2. Geom. Funct. Anal. (GAFA) 17, 192–219 (2007)
Isaev A.V.: Analogues of Rossi’s map and E. Cartan’s classification of homogeneous strongly pseudoconvex 3-dimensional hypersurfaces. J. Lie Theory 16, 407–426 (2006)
Isaev, A.V.: Lectures on the Automorphism Groups of Kobayashi-hyperbolic manifolds. Lecture Notes in Mathematics, vol. 1902. Springer, Berlin (2007)
Isaev A.V., Krantz S.G.: Domains with non-compact automorphism group: a survey. Adv. Math. 146(1), 1–38 (1999)
Kaup W.: Relle transformationsgruppen und invariante metriken auf komplexen räumen. Inv. Math. 3, 43–70 (1967)
Kim K.T.: Complete localization of domains with noncompact automorphism group. Trans. Am. Math. Soc. 319(1), 139–153 (1990)
Kim K.T.: Domains in C n with piecewise Levi flat boundaries which possess a noncompact automorphism group. Math. Ann. 292, 575–586 (1992)
Kim K.T., Krantz S.: The automorphism groups of domains. Am. Math. Mon. 112(7), 585–601 (2005)
Kim K.T., Verdiani L.: Complex n-dimensional manifolds with a real n 2-dimensional automorphism group. J. Geom. Anal. 14, 701–713 (2004)
Krantz, S.: Geometric Analysis and Function Spaces. CBMS and American Mathematical Society, Providence (1993)
Kruzhilin N., Soldatkin P.A.: Affine and holomorphic equivalence of tube domains in C 2. Math. Notes 75(5), 623–634 (2004)
Narasimhan R.: Several Complex Variables. Univ. of Chicago Press, Chicago, IL (1971)
Oeljeklaus K.: On the automorphism group of certain hyperbolic domains in C 2. Asterisque 217, 193–217 (1993)
Oeljeklaus K.: Une remarque sur le groupe des automorphismes holomorphes de domaines tubes dans C n. C. R. Acad. Sci. Paris Ser. I 312, 967–968 (1991)
Pinchuk S., Tsyganov S.: Smoothness of CR mappings between strictly pseudoconvex hypersurfaces. Math. USSR Izv. 35, 457–467 (1990)
Rosay J.P.: Sur une caractérisation de la boule parmi les domaines de C n par son groupe d’automorphismes. Ann. Inst. Four. Grenoble. XXIX, 91–97 (1979)
Shafikov R.: Analytic continuation of germs of holomorphic mappings between real hypersurfaces in C n. Mich. Math. J. 47, 133–149 (2000)
Shiffman B.: Separate analyticity and Hartogs theorems. Indiana Univ. Math. J. 38(4), 943–957 (1989)
Shiffman B.: On the removal of singularities of analytic sets. Mich. Math. J. 15, 111–120 (1968)
Sukhov A.: On boundary regularity of holomorphic mappings (English). Russ. Acad. Sci. Sb. Math. 83(2), 541–551 (1995)
Verma K.: A note on uniform extendability of automorphisms. Complex Var. Theory Appl. 49(3), 183–195 (2004)
Vladimirov V.S.: Methods of the theory of functions of many complex variables. MIT Press, Cambridge (1966)
Webster S.: On the mapping problem for algebraic real hypersurfaces. Inv. Math. 43, 53–68 (1977)
Wong B.: Characterization of the ball in C n by its automorphism group. Inv. Math. 41, 253–257 (1977)
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The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.
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Verma, K. A characterization of domains in C 2 with noncompact automorphism group. Math. Ann. 344, 645–701 (2009). https://doi.org/10.1007/s00208-008-0321-5
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DOI: https://doi.org/10.1007/s00208-008-0321-5