# zbMATH — the first resource for mathematics

Localization principle of automorphisms on generalized pseudoellipsoids. (English) Zbl 0943.32006
The authors prove the following result. Let $$\Sigma$$ and $$\Sigma'$$ be generalized pseudoellipsoids in $${\mathbb C}^N$$, $$N\geq 2$$, such that the sets of weak pseudoconvexity are contained in analytic sets of codimension at least $$2$$. Let $$F$$ be a biholomorphic map from $$U\cap \Sigma$$ in $$\Sigma'$$, such that $$F(U\cap \partial \Sigma) \subset \partial \Sigma'$$, then $$F$$ extends to a biholomorphism of $$\Sigma$$ onto $$\Sigma'$$.
This result generalizes some previous works, for example, H. Alexander [Math. Ann. 209, 249-256 (1974; Zbl 0281.32019)] for balls, S. I. Pinchuk [Math. USSR, Sb. 27(1975), 375-392 (1977; Zbl 0366.32010), ibid. 34, 503-519 (1978; Zbl 0438.32009)] for simply connected strongly pseudoconvex $$\partial D$$, and R. Greene and S. G. Krantz [Lect. Notes Math. 1268, 121-157 (1987; Zbl 0626.32023)] for pseudoellipsoids. Some related results on proper holomorphic mappings are also discussed.

##### MSC:
 32D15 Continuation of analytic objects in several complex variables 32H35 Proper holomorphic mappings, finiteness theorems 32E35 Global boundary behavior of holomorphic functions of several complex variables
##### Citations:
Zbl 0281.32019; Zbl 0626.32023; Zbl 0366.32010; Zbl 0438.32009
Full Text:
##### References:
 [1] Alexander, H. Holomorphic mappings from the ball and polydisc,Math. Ann.,209, 249–257, (1974). · Zbl 0281.32019 [2] Bedford, E. and Bell, S. Boundary continuity of proper holomorphic correspondences,Seminaire Dolbeaut-Lelong-Skoda, 1983. [3] Dini, G. and Selvaggi Primicerio, A. Proper holomorphic mappings between Reinhardt domains and strictly pseudoconvex domains,Rend. Circ. Mat. di Palermo, Serie II, Tomo XXXVIII 60–64, (1989). · Zbl 0699.32013 [4] Dini, G. and Selvaggi Primicerio, A. Proper holomorphic mappings between generalized pseudoellipsoids,Annali di matematica pura e applicata, CLVIII, 219–229, (1991). · Zbl 0736.32002 [5] Dini, G. and Selvaggi Primicerio, A. On the factorization of proper holomorphic mappings from pseudoellipsoids,Boll. U.M.I., (7)6-A, 379–386, (1992). · Zbl 0777.32013 [6] Fornæss, J. Biholomorphic mappings between weakly pseudoconvex domains.,Pac. J. Math.,74, 63–65, (1978). · Zbl 0371.32015 [7] Greene, R.E. and Krantz, S.G. Characterization of certain weakly pseudoconvex domains with non-compact automorphism groups,Lecture Notes in Math.,1268, (1987). [8] Greene, R.E. and Krantz, S.G. Techniques for studying automorphism of weakly pseudoconvex domains.Math. Notes,38, Princeton University Press, Princeton, NJ. · Zbl 0779.32017 [9] Kim, K.T. Complete localization of domains with non compact automorphism groups,Trans. Amer. Math. Soc.,319, 139–153,(1990). · Zbl 0705.32008 [10] Kodama, A. A characterization of certain domains with good boundary points in the sense of Greene-Krantz,Kodai Math. J.,12, 257–269, (1989). · Zbl 0693.32008 [11] Kodama, A., Krantz, S.G., and Ma, D. A characterization of generalized pseudoellipsoids in CN and related results,Indiana Univ. Math. J.,41, (1992) [12] Naruki, I. The holomorphic equivalence problem for a class of Reinhardt domains,Pubbl. RIMS, Kyoto Univ. Ser.,A4, 527–543,(1968). · Zbl 0199.41101 [13] Pinchuk, S. On the analytic continuation of holomorphic mappings,Math. USSR Sb.,27, 375–519, (1975). · Zbl 0366.32010 [14] Pinchuk, S. On holomorphic mappings of real analytic hypersurfaces,Math. USSR Sb.,34, 503–519, (1978). · Zbl 0438.32009 [15] Sunada, T. Holomorphic equivalence problem for bounded Reinhardt domains,Math. Ann.,235, 111–128, (1978). · Zbl 0371.32001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.