Taut and Tight Complex Manifolds

Abstract

Taut and tight manifolds, introduced recently by H. Wu, are characterized as follows. Let $D$ denote the open unit disk in $C$. The complex manifold $N$ is taut iff the set $A(D,N)$ of holomorphic maps from $D$ into $N$ is a normal family. If $d$ is a metric inducing the topology on $N,(N,d)$ is tight iff $A(D,N)$ is equicontinuous. It is also shown that every taut manifold is tight in a suitable metric.