Weighted Bergman Kernels on Planar Domains

Abstract

Boundary Behaviour of Weighted Bergman Kernels: For a planar domain $D \subset \mathbb{C}$ and an admissible weight function $\mu$ on it, some aspects of the boundary behaviour of the corresponding weighted Bergman kernel $K_{D, \mu}$ are studied. First, under the assumption that $\mu$ extends continuously to a smooth boundary point $p$ of $D$ and is non-vanishing there, we obtain a precise relation between $K_{D, \mu}$ and the classical Bergman kernel $K_D$ near $p$. Second, when viewed as functions of such weights, the weighted Bergman kernel is shown to have a suitable additive and multiplicative property near such boundary points. A Study on Holomorphic Isometries of Weighted Bergman Metrics: For a domain $D \subset \mathbb{C}^n$ and an admissible weight $\mu$ on it, we consider the weighted Bergman kernel $K_{D, \mu}$ and the corresponding weighted Bergman metric on $D$. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study the nature of holomorphic isometries from the disc $\mathbb{D} \subset \mathbb{C}$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb{D}}^{-d}$ for some integer $d \geq 0$. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in $\mathbb{D}^n$ and $\mathbb{D} \times \mathbb{B}^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.