Generalization of the big Picard theorem
Bulletin of the American Mathematical Society
Communicated by S. Smale, Feburary 29, 1968 S. Kobayashi defined a pseudodistance d on a complex manifold in such a manner that it depends only on the complex structure of the complex manifold in question . The definition of the pseudodistance can be extended word for word to a complex space (see  for definition of a complex space). Let D be the open unit disk in the complex plane C and p the Poincaré-Bergman metric of Z>. Given two points p and q of a complex space X, choose the
... choose the following objects: (1) points p = po, pi, • • • , pk = q of X, and (2) points ai, For each choice of points and mappings satisfying (1) and (2) , consider the number p(ai, bi)+ • • • +p(a*, 6*). Let d(£, <?) be the infimum of the numbers obtained in this manner for all possible choices. It is easy to verify that d is a pseudodistance on X. We shall call a complex space hyperbolic if the pseudodistance dx is a distance. The concept of a hyperbolic space is a generalization of a Riemann surface of hyperbolic type in the sense that a Riemann surface of hyperbolic type is a hyperbolic space. A hyperbolic space (X, dx) is said to be complete if for any point p of X and any positive number r, the closed ball of radius r around p is compact. The purpose of this paper is to generalize the big Picard theorem which states that a holomorphic mapping from the punctured disk into the Riemann sphere Pi(C) minus three points can be extended to a holomorphic mapping from the whole disk into Pi(C). H. Huber extended this theorem to the case where the image space is a domain G of hyperbolic type in a Riemann surface R such that the closure of G in R is compact . THEOREM 1. Let ƒ be a holomorphic mapping from the punctured disk D* into a hyperbolic space X. Moreover, assume that the complex space X is compact. Then ƒ can be extended to a holomorphic mapping from the whole disk into X. 1 This note is an abstract of the author's Ph.D dissertation written under the guidance of Professor S. Kobayashi. 759 760 M. H. KWACK LM7 THEOREM 2. Let M he a complex manifold and A an analytic subset of M of codimension at least 1. Also let X be a complete hyperbolic space which is compact X. Then a holomorphic mapping f : M-A-+X can be extended to a unique holomorphic mapping from M into Y. If, moreover, A is of codimension at least 2, the assumption that X is compact can be dropped. THEOREM 3. Let N be a bounded symmetric domain and T an arithmetic group of transformations acting properly discontinuously on N. (It is known that the quotient space N/T can be provided with a structure of complex space such that the projection w: N-+N/T is holomorphic) Let Y be the compactification of N/T by Borel and Baily [l]. If ƒ is a holomorphic mapping from the punctured disk D* into N/T which can be lifted locally, then ƒ can be extended to a holomorphic mapping from the whole disk into Y. (We say that a holomorphic mapping g from a complex space X into N/T can be lifted locally if given any point p of X there exist an open neighborhood U of p in X and a holomorphic mapping Jfr: U-+N such that ir o gu -g on U.) THEOREM 4. Let N, T and Y he as in the previous theorem and moreover assume that N is complete. If A is an analytic subset of a complex manifold M of codimension at least 1, then a holomorphic mapping f: M-A-+N/T which can be lifted locally can he extended to a unique holomorphic mapping from M into Y. If, in addition, A is of codimension at least 2, the image of the extension off lies in N/T.