It is proved that on any bounded domain in the complex Euclidean space C(n) the Bergman metric is always greater than or equal to the Carathéodory distance. This leads to a number of interesting consequences. Here two such consequences are given. (i) The Bergman metric is complete whenever the Carathéodory distance is complete on a bounded domain. (ii) The Weil-Petersson metric is not uniformly...

In this paper, we shall study differential geometric properties of bounded domains in Cn. Here is the summary of our results. We consider an w-dimensional complex manifold M and the Hilbert space of square integrable holomorphic re-forms on M. After Bergman [3; 4; 5], we define the kernel form on M (instead of the kernel function) and, under certain assumptions, we define the invariant metric o...

We obtain a conceptually new differential geometric proof of P.F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature kg(z) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ C approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂Ω.

and Applied Analysis 3 Let K z, z be the Bergman kernel function. Then the Bergman metric Hz β, β can be defined as

We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the bisectional curvatures of the Bergman metric of a smooth bounded pseudoconvex domain near the boundary with constant Levi rank.

Given a bounded strongly pseudoconvex domain D in C with smooth boundary, we characterize (p, q, α)-Bergman Carleson measures for 0 < p < ∞, 0 < q < ∞, and α > −1. As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric.

In this paper we study the complete invariant metrics on CartanHartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, w...

Using Fefferman’s classical result on the boundary singularity of the Bergman kernel, we give an analogous description of the boundary behaviour of various related quantities like the Bergman invariant, the coefficients of the Bergman metric, of the associated Laplace-Beltrami operator, of its curvature tensor, Ricci curvature and scalar curvature. The main point is that even though one would e...