Generalization of the big Picard theorem

Abstract

The big Picard theorem states that a holomorphic mapping from a punctured disk D* = {z e C: 0 < I z I < 1} into the Riemann sphere P1(C) minus three points can be extended to a holomorphic mapping from the disk {z E C: I z I < 1} into P1(C). H. Huber extended this theorem to the case where the image space is a domain G of hyperbolic type (in the classical sense) of a Riemann surface R such that the closure of G in R is compact [9]. The purpose of this paper is to prove the following. A holomorphic mapping from the punctured disk into a compact hyperbolic manifold M can be extended to a holomorphic mapping from the whole disk into M. Part of our proof uses some of the arguments H. Grauert and H. Reckziegel introduced in their paper [7, p. 120]. Generalizing further, we shall consider the extension problem of a holomorphic mapping defined on a complex manifold minus an analytic subset.