Extension and Finiteness Theorems

Abstract

The classical little Picard theorem states that every entire function f missing two values must be constant. In (3.10.2) we stated E. Borel’s generalization to a system of entire functions. One of its geometric consequences is that given n + p hyperplanes H ₁,..., H _{n +p} in P _n C in general position, every $f \in Hol(C,{P_n}C - \cup _{i = 1}^{n + p}{H_i})$ has its image in a linear subspace of dimension ≤ [n/p], see (3.10.7). For p = n + 1, this means that $f \in Hol(C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i})$ must be constant, (see (3.10.8)). This has been further strengthened to the statement that ${P_n}C - \cup _{i = 1}^{2n + 1}{H_i}$ is complete hyperbolic and is hyperbolically imbedded in P _n C, (see (3.10.9)), which is a statement on Hol($D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}$) rather than on Hol($C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}$) since the hyperbolicity and hyperbolic imbeddedness of $X = {P_2}C - \cup _{i = 1}^{2n + 1}{H_i}$ is defined in terms of d _X which, in turn, is constructed by Hol($D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}$). Thus, (3.10.9) is a statement en termes finis in the sense of Bloch [1].