Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces
Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces
复制标题
DOI：
10.2140/pjm.2015.273.75
复制
发表时间：
2013-02
影响因子：
0.6
通讯作者：
Qing Chen;Wei Wang;Yingyi Wu;Bin Xu
中科院分区：
文献类型：
--
作者：
Qing Chen;Wei Wang;Yingyi Wu;Bin Xu
一个正形g常曲率度量和有限许多锥形奇异点在一个紧凑的黎曼surface6可以被认为是标准的回调2度量的球体由多值局部单价的亚纯函数f on6nfsingularitiesg,叫做度规的发展图g。当发展中映射f的度量g紧凑的黎曼曲面上6可约单值,我们表明,f上一些莫比乌斯变换,对数微分d.l ogf / (f)是6上的第三类阿贝尔微分，它满足某些性质，称为g的字符1形式。反过来，给定这样一个阿贝尔微分！对于满足上述性质的第三类，我们证明了在6上存在一个唯一的1参数共形度量族，使得所有这些度量都具有常曲率1，相同的圆锥奇点，并且具有！作为他们性格的一种形式。这提供了常曲率为1且具有奇点的紧致黎曼曲面上的共形度量的新例子。此外，我们还证明了该展开映射是双球上曲率为1的共形公规g和角为2z> 1的有限多个圆锥奇点的有理函数。
A conformal metric g with constant curvature one and finitely many conical singularities on a compact Riemann surface6 can be thought of as the pullback of the standard metric on the 2-sphere by a multivalued locally univalent meromorphic function f on6nfsingularitiesg, called the developing map of the metric g. When the developing map f of such a metric g on the compact Riemann surface 6 has reducible monodromy, we show that, up to some Mobius transformation on f , the logarithmic differential d.log f/ of f turns out to be an abelian differential of the third kind on 6, which satisfies some properties and is called a character 1-form of g. Conversely given such an abelian differential ! of the third kind satisfying the above properties, we prove that there exists a unique 1-parameter family of conformal metrics on6 such that all these metrics have constant curvature one, the same conical singularities, and have! as one of their character 1-forms. This provides new examples of conformal metrics on compact Riemann surfaces of constant curvature one and with singularities. Moreover we prove that the developing map is a rational function for a conformal metric g with constant curvature one and finitely many conical singularities with angles in 2 Z>1 on the two-sphere.