The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalen...

The pseudohyperbolic metric is developed for the unit ball of Cn and is applied to a study of uniformly discrete sequences and Bergman spaces of holomorphic functions on the ball. The pseudohyperbolic metric plays an important role in the study of Bergman spaces over the unit disk. It is defined in terms of Möbius self-mappings of the disk. As is well known, these conformal automorphisms genera...

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

The spectral theory of a metric of Bergman type on a strictly pseudoconvex manifold is described and the scattering matrix is shown to be a pseudodifferential operator of Heisenberg type.

Let Ω be a non-compact complex manifold of dimension n, ω = ∂∂Ψ a Kähler form on Ω, and Kα(x, y) the reproducing kernel for the Bergman space Aα of all analytic functions on Ω square-integrable against the measure e−αΨ|ωn|. Under the condition Kα(x, x) = λαe αΨ(x) F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on (Ω, ω) which has recently...