Families of bloch functions on higher dimensional complex spaces

A holomorphic function f for which (1+|z|f(z)| is bounded on the complex disk D{zεC.|z|<} is classically called Bloch. We introduce the notion of uniformly Bloch families on complex spaces, which notion includes that of Bloch function as defined and studied by authors in various settings, show that several known characterizations and properties of Bloch functions have generalizations to uniformly Bloch families on higher dimensional complex spaces and establish some new properties of Bloch functions. For example we show that if X is a complex space. A⊂C and C-A does not contain arbitrarily large disks, then F⊂H(X)is uniformly Bloch iff for some . As is the case for spaces of Bloch functions defined on the unit disk and on homogeneous bounded domains in the space of Bloch functions (mod constants) defined on a hyperbolic space is a complex Banach space. We also show that members of uniformly Bloch families on complex manifolds satisfy extension theorems of big Picard type if the nature of the singularies is that of normal crossings; moreover, the function which maps a member of a uniformly Bloch family to its extension is a topological imbedding. The family of continuous extensions of the members of a uniformly Bloch family from the punctured disk D * or more generally from ( D *)n is shown to be uniformly Bloch.