Title One : Conformal hyperbolic metrics and bounded projective functions with singularities
Presenter: Prof. Bin Xu (University of Science and Technology of China)
Time&Date: 9:30 A.M11:00 A.M, Thursday, January 11, 2018
Venue: Room H113, Building H, Qingyuan Campus of Anhui University
Abstract: More than half an century ago, J. Nitsche showed that isolated singularities of conformal hyperbolic metrics are either cusp singularities or cone ones, and M. Heins proved that the GaussBonnet formula actually forms a necessary and sufficient condition for the existence of conformal hyperbolic metrics with isolated singularities on compact Riemann surfaces. Projective functions are multivalued locally univalent meromorphic functions on Riemann surfaces such that their monodromy lies in the group PGL(2,C) consisting of all Möbius transformations.From the viewpoint of Complex Analysis, by using the analysis of PDE, we characterized conformal hyperbolic metrics with finitely many singularities on compact Riemann surfaces by bounded projective functions on the punctured surface by the singularities such that the Schwarzian derivative of the projective functions have double poles with coefficients prescribed by the angles of the singularities.
Title Two: Cone spherical metrics on compact Riemann surfaces
Presenter: Prof. Bin Xu (University of Science and Technology of China)
Time&Date: 9:30 A.M11:00 A.M, Friday, January 12, 2018
Venue: Room H113, Building H, Qingyuan Campus of Anhui University
Abstract: Cone spherical, flat and hyperbolic metrics are conformal metrics with constant curvature +1; 0 and 1, espectively, and with finitely many conical singularities on compact Riemann surfaces. The GaussBonnet formula gives a natural necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces. The condition is also sufficient for both flat and hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over thirty years. Projective functions are multivalued locally univalent meromorphic functions on Riemann surfaces such that their monodromy lies in the roup PGL(2,C) consisting of all Möbius transformations. We observed that the developing maps of cone spherical metrics are projective functions on the surfaces punctured by the conical ngularities whose monodromy lie in PSU(2), and whose Schwarzian derivatives have double poles at the conical singularities with coefficients prescribed by the cone angles. Starting from this observation, we made some progresses on cone spherical metrics by using Complex Algebraic Geometry, which consist of the joint works with Qing Chen, Yiran Cheng, Bo Li, Lingguang Li, Santai Qu, Jijian Song and Yingyi Wu.
