题目：Conformal metrics with constant curvature one and finite conical singularities on compact Riemann surfaces

报告人：中科大学院 许斌副教授

时间：2014年5月14日下午4:00-5:00

地点：管理科研楼1518

摘要: It is an open problem about the existence and uniqueness of the metrics in the title. Troyanov and Luo-Tian proved some classical results via the method of PDEs in the 1980s.  Afterwards, Umehara-Yamada, Furuta-Hattori and Eremenko et al completely solved the problem on the two-sphere and with three singularities via the method of Complex Analysis.  Applying the second method, we firstly divide the metrics into two classes: irreducible and reducible ones. We then prove that reducible metrics exist on the surface if and only if so do some Abelian differentials of the 3rd kind,  which have real residues and whose real parts are exact outside their poles.  As a by-product,  we find an explicit necessary condition for the existence of reducible metrics on the two-sphere.  This is a joint work with Qing Chen, Yifei Chen, Mao Sheng, Wei Wang and Yingyi Wu.