The image of a proper holomorphic map from a pseudo-convex domain into a strongly pseudo-convex domain

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  • image of a proper holomorphic map from
  • 擬凸領域から強擬凸領域への固有正則写像の像

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It is known that if f is a proper holomorphic map from a strongly pseudo-convex domain D_1 ⊂ C^n to a strongly pseudo-convex domain D_2 ⊂ C^n , then every cone with vertex at the boundary of D_1 is mapped into an approach region in D_2. This was proved by Henkin [ 3 ]. But the case in which the domain D_1 and D_2 are different dimentional, were not investigated. In this paper we first prove that for the pseudo-convex domain D_1 = {|z|^2 + |w|^<2m> < 1 }, m > 1 and for the strongly pseudo-convex domain D_2 = B_2, f = (z, w^m) : D_1→B_2 maps every cone with vertex p ∈ ∂D_1 into an approach region of B_2 [Theorem 3]. Secondary we prove that if f is a proper holomorphic map from B_2 ⊂ C^2 into B_3 ⊂ C^3 and if f extends to <B_2>^^^- in a C^2 -way, then f maps every approach region of B_2 into an approach region of B_3 [Theorem 4] .

source:Bulletin of the Faculty of Education, Chiba University. Part II

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