We derive conditions under which a holomorphic mapping of a taut Riemann surface must be an automorphism. This is an analogue involving invariant distances of a result of H. Cartan. Using similar methods we prove an existence result for 1-dimensional holomorphic retracts in a taut complex manifold.
We introduce a new invariant Kähler metric on relatively compact domains in a complex manifold, which is the Bergman metric of the L² space of holomorphic sections of the pluricanonical bundle equipped with the Hermitian metric introduced by Narasimhan-Simha.
We show that the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂):|λ₁|,|λ₂| < 1} ⊂ ℂ² cannot be exhausted by domains biholomorphic to convex domains.
We show a relation between the Kobayashi pseudodistance of a holomorphic fiber bundle and the Kobayashi pseudodistance of its base. Moreover, we prove that a holomorphic fiber bundle is taut iff both the fiber and the base are taut.
Let be the open unit ball of a Banach space , and let be a holomorphic map with . In this paper, we discuss a condition whereby is a linear isometry on .
We give a simple proof of almost properness of any extremal mapping in the sense of Lempert function or in the sense of Kobayashi-Royden pseudometric.
Let a and m be positive integers such that 2a < m. We show that in the domain the holomorphic sectional curvature of the Bergman metric at z in direction X tends to -∞ when z tends to 0 non-tangentially, and the direction X is suitably chosen. It seems that an example with this feature has not been known so far.
We provide a structure theorem for Carnot-Carathéodory balls defined by a family of Lipschitz continuous vector fields. From this result a proof of Poincaré inequality follows.
It is shown that given a bounded strictly convex domain in with real analitic boundary and a point in , there exists a larger bounded strictly convex domain with real analitic boundary, close as wished to , such that is a ball for the Kobayashi distance of with center . The result is applied to prove that if is not biholomorphic to the ball then any automorphism of extends to an automorphism of .
The behaviour of the Carathéodory, Kobayashi and Azukawa metrics near convex boundary points of domains in is studied.