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Register a new userComparison of the Bergman kernel and the Caratheodory--Eisenman volume
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It is proved that for any domain in $mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the multidimensional Suita conjecture theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel.
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The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tians partial $C^0$-estimate.
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $mathcal C^1$ 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 $mathcal C^1$ 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 $mathcal C^2$ case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.
The Schur (resp. Caratheodory) class consists of all the analytic functions $f$ on the unit disk with $|f|le 1$ (resp. $Re f>0$ and $f(0)=1$). The Schur parameters $gamma_0,gamma_1,dots (|gamma_j|le 1)$ are known to parametrize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the $n$-th coefficient of a Caratheodory function in terms of $n$ independent variables $gamma_1,dots, gamma_n$ with $|gamma_j|le 1.$ The mapping properties of those correspondences are also studied.
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