关于短$\mathbb{C}^k$的注记
Notes on Short $\mathbb{C}^k$'s
摘要：
在复分析中，不可避免地会出现球的递增并（直到双全纯）和Kobayashi度量为零的域。在这篇文章中，我们表明，在高维这些域有无穷大的体积和Bergman空间的这些域是平凡的。因此，它们在每个边界点不是严格伪凸的，尽管这些域根据定义是伪凸的。这些域可以是不同的类型，其中之一是短$\mathbb{C}^k$。为了确定Runge Short $\mathbb{C}^k$'s（直到双全纯），我们引入了一类特殊的Short $\mathbb{C}^k$'s，称为Loewner Short $\mathbb{C}^k$'s.这些是那些短$\mathbb{C}^k$的可以用尽在一个连续的方式由严格增加的参数化家庭的开放集，其中每一个是biholomphically相当于单位球，因此，他们是龙格高达biholomorphism。虽然，是否所有的短$\mathbb{C}^k$的是龙格（直到biholomorphism），或是否所有的短$\mathbb{C}^k$的是Loewner的问题仍然悬而未决，我们表明，典型的短$\mathbb{C}^k$的是Loewner。在最后一节中，我们构建了一堆非自治的吸引力盆地，它们作为Short $\mathbb{C}^2$的有趣例子。
英文摘要：
Domains that are increasing union of balls (up to biholomorphism) and on which the Kobayashi metric vanishes identically arise inexorably in complex analysis. In this article we show that in higher dimensions these domains have infinite volume and the Bergman spaces of these domains are trivial. As a consequence they fail to be strictly pseudo-convex at each of their boundary points although these domains are pseudo-convex by definition. These domains can be of different types and one of them is Short $\mathbb{C}^k$'s. In pursuit of identifying the Runge Short $\mathbb{C}^k$'s (up to biholomorphism), we introduce a special class of Short $\mathbb{C}^k$'s, called Loewner Short $\mathbb{C}^k$'s. These are those Short $\mathbb{C}^k$'s which can be exhausted in a continuous manner by a strictly increasing parametrized family of open sets, each of which is biholomrphically equivalent to the unit ball and therefore, they are Runge up to biholomorphism. Although, the question of whether all Short $\mathbb{C}^k$'s are Runge (up to biholomorphism), or whether all Short $\mathbb{C}^k$'s are Loewner remains unsettled, we show that the typical Short $\mathbb{C}^k$'s are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short $\mathbb{C}^2$'s.
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