Banach hyperbolicity and existence of holomorphic maps in infinite dimension (Q1373035)

Finite-dimensional hyperbolic analysis has been investigated by several authors, in particular by Kobayashi, Kwack and recently by Noguchi, Zaidenberg. The obtained results have been used to study some important problems of complex analysis and number theory. In recent years Bart, Lempert, Vesentini have obtained important results concerning the hyperbolicity of convex domains in Banach spaces. The aim of the present paper is to study some questions concerning the extension of holomorphic maps in Banach hyperbolic analysis. In Section 1 we will solve the Kobayashi problem for proper holomorphic maps between Banach analytic spaces. Applying Brody's characterization of compact hyperbolic spaces, Urata and independently Zaidenberg have solved this problem in the finite-dimensional case. In Section 2 it is shown that for a convex domain in a Banach space the Kwack extension theorem holds if and only if this domain contains no complex lines. Moreover, in the finite-dimensional case, we show that the above statement is equivalent to the \(H^\infty\)-extendability. Finally, in Section 3 we extend results of Hirschowitz and Sibony on the extension of holomorphic maps with values in complete \(C\)-spaces. We prove that every holomorphic map from a Riemann domain \(\Omega\) over a topological vector space into a Banach manifold, modelled by open sets in an injective Banach space which is complete for the distance of Caratheodory, can be extended holomorphically to \(\widehat\Omega_\infty\), the envelope of holomorphy of \(\Omega\) for the set \(H^\infty(\Omega)\) of bounded holomorphic functions on \(\Omega\).