AI 中文总结
本文研究起源于Levi问题的并合问题的拟共形变体，利用Kiernan拟共形Schwarz引理与Ferrand共形容量，分类一类可被一致有界伸缩商的拟共形像耗尽的n-黎曼流形，还拓展了Gehring区域的相关性质。
AI 中文摘要
并合问题起源于经典Levi问题，旨在对复流形M进行分类，这类复流形可被子流形M_j的递增并集耗尽，其中所有M_j均双全纯等价于ℂⁿ中的某个固定区域。本文探讨该问题的拟共形变体，旨在对n-黎曼流形M进行分类，要求每个M_j拟共形等价于ℝⁿ中的一个有界区域。研究发现，当这些拟共形等价具有一致有界的伸缩商时，该分类可行。利用n=2时的Kiernan拟共形Schwarz引理，以及n≥3时的Ferrand共形容量，本文对一类n-黎曼流形M进行分类，其中每个M_j为K_j-拟共形等价于Ω\backslash A，且满足sup K_j < ∞，Ω⊂ℝⁿ为C²-光滑有界区域，A⊂Ω至多有限。由此可得，Gehring构造的ℝⁿ中有界区域（除一点外均为C¹-光滑边界，且已知其拟共形不等价于ℝⁿ中的单位球）还具有额外性质：它甚至无法被具有一致有界伸缩商的单位球的拟共形像耗尽。
英文摘要
The Union Problem, which has its genesis in the classical Levi problem, asks for a classification of complex manifolds $M$ that can be exhausted by an increasing union of submanifolds $M_j \subset M$ which are all biholomorphic to a fixed domain in $\mathbb C^n$. We explore a quasiconformal variant of this question and seek to classify $n$-Riemannian manifolds $M$ such that each $M_j$ is quasiconformally equivalent to a bounded domain in $\mathbb R^n$. It turns out that this is possible when these quasiconformal equivalences have uniformly bounded dilatations. Using Kiernan's quasiconformal Schwarz lemma when $n=2$ and Ferrand's conformal capacity when $n \geq 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $Ω\setminus A$, where $\sup K_j < \infty$ and $Ω\subset \mathbb R^n$ is a $C^2$-smoothly bounded domain and $A \subset Ω$ is at most finite. As a consequence, we obtain that Gehring's example of a bounded domain in $\mathbb R^n$ which has $C^1$-smooth boundary everywhere except at a point and is known to be quasiconformally inequivalent to the unit ball in $\mathbb R^n$, possesses the additional property that it cannot even be exhausted by quasiconformal images of the unit ball with uniformly bounded dilatations.
Comments17 pages
