Holomorphic Maps into Hyperbolic Spaces

Abstract

Let X and Y be complex spaces. Let C(X, Y) denote the family of continuous maps from X into Y with compact-open topology. Let D(X, Y) be the subfamily of distance-decreasing maps from X into Y with respect to their intrinsic pseudo-distances d _X and d _Y . Then D(X, Y) is closed in C(X, Y). The family Hol(X, Y) of holomorphic maps from X into Y is a closed subset of D(X, Y).