迈向经典 Schottky 单值化猜想的一个证明
Towards a proof of the Classical Schottky Uniformization Conjecture
AI总结：
本文证明每个 Belyi 曲线可被经典 Schottky 群单值化，从而推出经典 Schottky 单值化曲面在模空间中开且稠密，为猜想提供关键证据。
摘要：
根据 Koebe 的逆剖分定理，每个亏格 $g \geq 2$ 的闭黎曼曲面都可以被一个 Schottky 群单值化。Marden 观察到存在非经典的 Schottky 群，即它们不能由适当的圆族来定义。这引出了一个问题：是否每个闭黎曼曲面都能被经典 Schottky 群单值化。在本文中，我们观察到每个 Belyi 曲线都可以被经典 Schottky 群单值化。由于 Belyi 曲线在模空间 ${\mathcal M}_{g}$ 中形成稠密子集，且由经典 Schottky 群单值化的黎曼曲面所构成的集合 ${\mathcal M}_{g}^{cs} \subset {\mathcal M}_{g}$ 是非空开集，这确保了 ${\mathcal M}_{g}^{cs}$ 在 ${\mathcal M}_{g}$ 中是既开又稠密的。
英文摘要：
By Koebe's retrosection theorem, every closed Riemann surface of genus $g \geq 2$ is uniformized by a Schottky group. Marden observed that there are Schottky groups that are not classical ones, that is, they cannot be defined by a suitable collection of circles. This opened the question of whether every closed Riemann surface can be uniformized by a classical Schottky group. In this paper, we observe that every Belyi curve can be uniformized by a classical Schottky group. Since Belyi curves form a dense locus in the moduli space ${\mathcal M}_{g}$ and the locus ${\mathcal M}_{g}^{cs} \subset {\mathcal M}_{g}$ of those Riemann surfaces uniformized by classical Schottky groups is a non-empty open set, this ensures that ${\mathcal M}_{g}^{cs}$ is open and dense in ${\mathcal M}_{g}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
