Asymptotic Behavior of Normal Mappings of Several Complex Variables

Abstract

Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N).In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions.