论文标题
平面域上的加权伯格曼内核
Weighted Bergman Kernels on Planar Domains
论文作者
论文摘要
加权伯格曼内核的边界行为：对于平面域$ d \ subset \ mathbb {c} $和可允许的权重函数$μ$，研究了相应加权的伯格曼内核$ k_ {d，μ} $的边界行为的某些方面。首先，假设$μ$连续扩展到平稳的边界点$ p $ $ d $，并且在那里不存在，我们获得了$ k_ {d，μ} $和古典伯格曼内核$ k_d $近$ p $之间的精确关系。其次，当将其视为此类权重的功能时，加权的伯格曼内核被证明在此类边界点附近具有合适的添加性和乘法性能。 一项关于加权伯格曼指标的骨膜异构的研究：对于域$ d \ subset \ mathbb {c}^n $和可允许的权重$μ$，我们认为加权的伯格曼内核$ k_ {d，μ} $和相应的加权bergman un $ d $ d $。特别是，由Mok，Ng，Chan-Yuan和Chan-Xiao-Xiao-Yuan等动机，我们研究了来自碟片$ \ Mathbb {D} \ subset \ Mathbb {C Mathbb {C C} $的圆锥形同构的性质k _ {\ mathbb {d}}}^{ - d} $对于某些整数$ d \ geq 0 $。这些指标提供了一类自然的例子，这些示例产生了阳性的共形常数，这些常数已在最近的异构体中考虑过。详细研究的异构体的具体示例包括$ \ Mathbb {d}^n $和$ \ Mathbb {d} \ times \ times \ Mathbb {b}^n $中的值，其中每个因素都接纳上述可能不同的非式伯格曼公制。最后，还介绍了在可能不同的维度上的多磁性之间的异构体的情况，其中每个因素的加权伯格曼度量都如上所述。
Boundary Behaviour of Weighted Bergman Kernels: For a planar domain $D \subset \mathbb{C}$ and an admissible weight function $μ$ on it, some aspects of the boundary behaviour of the corresponding weighted Bergman kernel $K_{D, μ}$ are studied. First, under the assumption that $μ$ extends continuously to a smooth boundary point $p$ of $D$ and is non-vanishing there, we obtain a precise relation between $K_{D, μ}$ 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 $μ$ on it, we consider the weighted Bergman kernel $K_{D, μ}$ 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 $μ= K_{\mathbb{D}}^{-d}$ 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.
