Families of holomorphic maps into Riemann surfaces

Abstract

In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence { f j } \{ {f_j}\} of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence { f j − 1 } \{ f_j^{ - 1}\} of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).