Mappings into Hyperbolic Spaces

In this note we state some results on extensions of holomorphic mapings into hyperbolic spaces. A theorem involves extending holomorphic mappings to a domain of holomorphy. An extension problem of holomorphic mappings into a taut complex space was considered by Fujimoto [1]. Another result is that the space of all meromorphic mappings from a complex space X into a hyperbolically imbedded space in Y is relatively compact in the space of all meromorphic mappings from X into Y. A relatively compact complex space M is said to be hyperbolically imbedded in a complex space Y if for all sequences {pn} and {qn} in M such that pn-+ peM and gw -> qeM and such that dM(pn,qn) -» 0, we have p = q. Here dM denotes the pseudo-distance defined by Kobayashi [5]. A relatively compact complex space M in Y is strictly Levi pseudoconvex if for every point p e dM there are a neighborhood Up of p and a biholomorphic map Op of Up onto a subvariety of a domain Dp in some C and a function cp defined in Up such that cp o ^ " 1 is the restriction to ®p(Up) of a strictly pluri-subharmonic function p(x) < 0}. THEOREM 1. Let X be a complex manifold and A be an analytic subset of X of codimension at least 1. Let M be a strictly Levi pseudoconvex hyperbolic space in Y. Then a holomorphic mapping f of X — A into M can be extended holomorphically to a mapping fofX into M.