Boundedly holomorphic convex domains

Abstract

shown in this paper: In a Riemann domain, a boundedly holomorphic convex domain is a domain of bounded holomorphy. With some restrictions, the converse is true. The spectrum of the algebra B of bounded holomorphic functions is an envelope of bounded holomorphy provided that the completion of B with the topology of uniform convergence on compact subsets is stable under differentiation. Finally, Stein manifolds of bounded type are introduced. Let (Xl9 Aj) and (Xi9 A2) be complex (analytic) manifolds. A map oi Xx —> X2 said to be biholomorphίc if a: is a homeomorphism of Xλ onto X2 and both a and or1 are holomorphic. a is called a spread map if a is a locally biholomorphic. We denote a complex manifold (X, A; a) with a spread map a. A Riemann domain is a complex manifold which spreads in (C*, <£?). We denote B{X) for the algebra of all bounded holomorphic functions on X. DEFINITION l Let (X, A) be a complex manifold and D be open in X. Let B = B(D). D is said to be boundedly holomorphic convex if Ίcm\\BK= KB = {xeD; \f(x) ^ \\f\\κ for all feB] is compact provided if is a compact subset of D. An open set D of X is called a region of bounded holomorphy if there is an / e B(D) for which every boundary point of D is a boundary singularity in the sense that / has no bounded analytic continuation to any open neighborhood of any boundary point (see [5]). The following natural questions arise; if boundedly holomorphic convex domains are domains of bounded holomorphy, and vice versa. The answer for the first is affirmative.