Semi-Bloch Functions in Several Complex Variables

Abstract

Let \(M\) be an \(n\)-dimensional complex manifold. A holomorphic function \(f:M\rightarrow {\mathbb {C}}\) is said to be semi-Bloch if for every \(\lambda \in {\mathbb {C}}\) the function \(g_{\lambda }=\text {exp}(\lambda f(z))\) is normal on \(M\). We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate \(M\) in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.