The classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic the upper half-plane. We characterize local order isomorphisms domains biholomorphic automorphisms of generalized half-plane, which is collection all operators with positive invertible imaginary part. describe such maps in an explicit manner, and examine properties maximal isom...

We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and Carathéodory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex dom...

A biholomorphically invariant real function hx is defined for a hyperbolic manifold X. Properties of such functions are studied. These properties are applied to prove the following theorem. If a hyperbolic manifold X can be exhausted by biholomorphic images of a strictly pseudoconvex domain D C C" with dD G C3, then X is biholomorphically equivalent either to D or to the unit ball in C". The pr...

In an important development of several complex variables, Poincaré [26] discovered that any biholomorphic map between two open pieces of the unit sphere in C2 is the restriction of a certain automorphism of B2, the unit two-ball in C2. This phenomenon fails obviously in one complex variable and reveals a strong rigidity property of holomorphic mappings in several variables. Later, Tanaka, etc (...