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Conical Metrics on Riemann Surfaces, II: Spherical Metrics
International Mathematics Research Notices ( IF 1 ) Pub Date : 2021-01-15 , DOI: 10.1093/imrn/rnab011
Rafe Mazzeo 1 , Xuwen Zhu 2
Affiliation  

We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi $. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38].

中文翻译:

黎曼曲面上的圆锥度量,II:球面度量

我们继续我们在 [34] 中开始的研究,研究具有恒定曲率和孤立圆锥奇点的黎曼曲面。使用在简单除数的扩展配置系列的早期论文中开发的机制,我们研究了部分或全部锥角大于 $2\pi $ 的球面圆锥度量的存在和变形理论。当数字 $2$ 位于拉普拉斯算子的弗里德里希扩展谱中时,变形就被阻碍了。我们的主要结果是,在这种情况下,可以通过允许锥点分裂来找到解的平滑局部模空间。这一分析事实反映了 [37, 38] 中的几何结构。
更新日期:2021-01-15
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