Colloquium | Institute of Mathematical Sciences
Time：16:00-17:00, Mar. 26, Tuesday
Location：Room 302, Library
Speaker: Bin Xu, USTC/ShanghaiTech University
Abstract: Singular spherical, flat and hyperbolic metrics are conformal metrics with constant curvature +1, 0 and -1, respectively, and with isolated singularities on Riemann surfaces. The Gauss-Bonnet formula gives a necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces, and it is also sufficent for both cone flat and cone hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over twenty years on compact Riemann surfaces. I will introduce the respectful audience some progress on this problem and some recent results on singular flat metrics and hyperbolic metrics.
1. We obtained on compact Riemann surfaces a correspondence between meromorphic one-forms with simple poles and real periods and cone spherical metrics with monodromy in U(1), called reducible metrics. As an application, we found a necessary and sufficient condition for cone angles of reducible metrics on the Riemann sphere and showed that the Friedrichs Laplacians of reducible metrics have eigenvalue 2. We obtained on compact Riemann surfaces with positive genera a correspondence between irreducible metrics with cone angiles in 2\pi N and line subbundles of rank two stable vector bundles. As an application, we found a new existence result about cone spherical metrics on compact Riemann surfaces with genera greater than one.
2. We classified the isolated singularities of flat metrics whose areas satisfy the polynomial growth condition into three classes and proposed an open question about the existence of flat metrics with such singularities prescribed on compact Riemann surfaces.
3. We showed that the singular hyperbolic metrics have Zariski desne monodromy groups in PSL(2,R) on compact Riemann surfaces and constructed a new class of hyperbolic metrics with infinitely many isolated singularities on the unit disk.
The talk is based on my joint works with Qing Chen, Xuemian Chen, Yiran Cheng, Yu Feng, Si'en Gong, Bo Li, Jin Li, Lingguang Li, Hongyi Liu, Santai Qu, Jijian Song, Yingyi Wu, Xuwen Zhu.