Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity
Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity
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DOI：
10.2140/gt.2020.24.309
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
R. Mazzeo;Xuwen Zhu
中科院分区：
文献类型：
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作者：
R. Mazzeo;Xuwen Zhu
我们引入了黎曼曲面上简单正因数空间的紧化，以及该空间之上的穿孔曲面的通用族的紧化。这些是带有角的实流形。然后，我们研究这个黎曼曲面上的常曲率度量空间，在这些除数处具有规定的圆锥奇点。我们感兴趣的是这些指标的局部变形，特别是圆锥点合并时的行为。我们在已知存在这些指标的情况下证明了这种现象的尖锐规律性定理。该设置将在后续论文中用于研究具有大锥角的球锥度量空间，其中存在理论仍然不完整。
We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete.