具有正则奇点的 Schwarzian 方程的单值性
Monodromy of Schwarzian equations with regular singularities
AI总结：
本文刻画有限型负 Euler 示性数穿孔曲面上带正则奇点的 Schwarzian 方程的全部单值性表示，并通过构造指定完整群的复仿射结构处理表观奇点，进而推出带锥点球面度量的完整群。
摘要：
设 $S$ 是一个有限型且 Euler 示性数为负的穿孔曲面。我们确定了所有可能的表示 $ρ:π_1(S) \to \text{PSL}_2(\mathbb{C})$，这些表示可作为 $S$ 上在穿孔处具有正则奇点的 Schwarzian 方程的单值性而出现。等价地，我们确定了 $S$ 上复射影结构的完整群表示，这些结构的 Schwarzian 导数（相对于某个一致化结构）在穿孔处具有至多二阶的极点。在处理不存在表观奇点情形的早期工作之后，我们的证明归结为实现具有表观奇点的退化表示这一情形。这主要涉及在穿孔曲面上显式构造具有指定完整群的复仿射结构。作为推论，我们确定了可作为 $S$ 上在穿孔处具有锥点的球面度量之完整群而出现的表示。
英文摘要：
Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $ρ:π_1(S) \to \text{PSL}_2(\mathbb{C})$ that arise as the monodromy of the Schwarzian equation on $S$ with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on $S$, whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on $S$ with cone-points at the punctures.
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