报告人： 许斌 副教授

     中国科学技术大学数学系

报告题目：Differentials and conical elliptic metrics on compact Riemann surfaces

报告时间：2016年9月30日16:00点

报告地点：海韵数理楼661

学院联系人:刘轼波教授

报告摘要：A celebrated open problem asks whether there exists a conformal metric with positive constant curvature (conical elliptic metric) on a compact Riemann surface such that it has the prescribed finitely many conical singularities. To attack it, since 1980s people have used at least the two different methods --- Calculus of Variation and Complex Analysis. In the first part of the talk, besides the background of conical elliptic metrics, we shall survey some general existence theorems proved by Calculus of Variation and sketch their proof. Then we shall turn to the Complex Analysis aspect of some research works about the problem. A developing map of a conical elliptic metric has monodromy in SO(3). A conical elliptic metric is called reducible if its monodromy lies in U(1).  Otherwise, it is called irreducible. We call a conical elliptic metric quasi-reducible if its monodromy lies in the semi-direct product of U(1) and Z/2. We observe that all the information of a reducible metric on a compact Riemann surface can be encapsulated into a one-form on the surface with simple poles such that all its periods are purely imaginary. By using techniques of translation surfaces, we prove some general existence theorems of one-forms on compact Riemann surfaces and apply some of them to reducible metrics. In particular, for any genus g and n positive numbers given, we give a necessary and sufficient condition for them under which there exists a compact Riemann surface of genus g and a reducible metric on the surface with conical angles prescribed by these n positive numbers.  If time permitted, we shall also mention an reduction of quasi-reducible metrics to Jenkins-Strebel differentials whose periods are all real, give some explicit examples, and propose an existence problem of quasi-reducible metrics. This is a joint work with Qing Chen, Bo Li, Santai Qu and Ji-Jian Song.

报告人简介:  许斌，1997年本科毕业于中国科学技术大学数学系，2003年在东京大学获数学博士学位。2003年至今在中国科学技术大学数学系任教，主要从事复分析、微分几何、流形上的分析等方面的研究。现为数学系副教授。

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