Bergman Kernel For Domains Research Articles

Growth of Sibony metric and Bergman kernel for domains with low regularity

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with C1 boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general C1 boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the C2 case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.

The Bergman kernels on Cartan-Hartogs domain

The main point is the calculation of the Bergman kernel for the so-called Cartan-Hartogs domains. The Bergman kernels on four types of Cartan-Hartogs domains are given in explicit formulas. First by introducing the idea of semi-Reinhardt domain is given, of which the Cartan-Hartogs domains are a special case. Following the ideas developed in the classic monograph of Hua, the Bergman kernel for these domains is calculated. Along this way, the method of “inflation”, is made use of due to Boas, Fu and Straube.