Condition R and proper holomorphic maps between equidimensional product domains

Abstract

Abstract. We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.

Analysis of Orbit Accumulation Points and The Greene–Krantz Conjecture

“…We will prove Theorem 0.1 in Section 2 and we will add some remarks about Greene-Krantz conjecture in Section 3. In Section 3 we also use the recent result of [2] to prove the bidisc is not biholomorphic to any bounded domain with smooth boundary and finally affirm the Greene-Krantz conjecture for a special case (see Corollary 3.1).…”

“…We observe D is pseudoconvex and satisfies condition R. Suppose there is a biholomorphism f which maps the bidisc D × D onto a bounded domain Ω with a (globally) smooth boundary. Then f extends smoothly up to boundary of the bidisc and Ω by Theorem 1.1 in [2]. However, the bidisc does not have a smooth boundary, which contradicts the extension of f .…”

“…Although D ⋊ θ H + is very possible not to biholomorphic to any bounded domain with a smooth boundary, we can not prove it here now. One possible method is to generalize the theorem in [2] from product domains to "⋊ θ " domains.…”

Bergman Kernel in Complex Analysis

“…The relationship between the Bergman kernel, condition (R), geometry of domains, and their group of automorphisms has been intensively studied since then (e.g., [17,25]). …”

Comparison of invariant functions and metrics

“…Actually, the local uniform convergence of the functions r tμ follows simply from (3). Similarly, making use of (4), one may deduce the local uniform convergence of partial derivatives of the first order.…”