题目： Reducible metrics and 1-forms on compact Riemann surfaces
报告人：许斌 （中国科学技术大学 ）
报告人：许斌 （中国科学技术大学 ）
时间: 2015年12月25日, 周五 下午15:00-16:30
地点：管研楼1611
摘要： Does there there exist a conformal metric of positive constant curvature with finitely many conical singularities prescribed on a compact Riemann surface? This is a problem with a long history traced to H. Poincare and E. Picard. It is still widely open nowadays besides a lot of mathematicians obtained many classical results. Such a metric is called reducible if its developing map has diagonalized monodromy representation. The speaker will explain an observation in detail that all the information of a reducible metric can be encapsulated into some meromorphic 1-form with simple poles and with purely imaginary periods. Moreover, on the other hand, any such a 1-form gives naturally a 1-parameter family of reducible metrics, which form new examples of conical metrics of positive constant curvature. The speaker shall announce some new existence theorems about 1-forms with simple poles (and with purely imaginary periods) on compact Riemann surfaces and their two consequences as follows:
1. An explicit necessary and sufficient condition on the existence of reducible metrics on the Riemann sphere,
2. An explicity necessary and sufficient condition of prescribed conical angles under which there exists a reducible metric with those conical angles on some compact Riemann surface with genus g for each positive integer g.
Be sketched will the proof of the above existence theorem on the Riemann sphere in case that all the conical angles are pi times rationals. This special case also gives a new sufficient condition for the Hurwitz problem, which asks whether a admissible branch data is realized by some rational function. If time permitted, some problems interesting to the speaker will be proposed.
This is a joint work with Professor Qing Chen and my students Bo Li, Santai Qu and Ji-Jian Song. Since the beginning of this work, the speaker has been in a great debt to Professors Mao Sheng, Yifei Chen and Yingyi Wu at AMSS, to whom he express his deep gratitude.
摘要： Does there there exist a conformal metric of positive constant curvature with finitely many conical singularities prescribed on a compact Riemann surface? This is a problem with a long history traced to H. Poincare and E. Picard. It is still widely open nowadays besides a lot of mathematicians obtained many classical results. Such a metric is called reducible if its developing map has diagonalized monodromy representation. The speaker will explain an observation in detail that all the information of a reducible metric can be encapsulated into some meromorphic 1-form with simple poles and with purely imaginary periods. Moreover, on the other hand, any such a 1-form gives naturally a 1-parameter family of reducible metrics, which form new examples of conical metrics of positive constant curvature. The speaker shall announce some new existence theorems about 1-forms with simple poles (and with purely imaginary periods) on compact Riemann surfaces and their two consequences as follows:
1. An explicit necessary and sufficient condition on the existence of reducible metrics on the Riemann sphere,
2. An explicity necessary and sufficient condition of prescribed conical angles under which there exists a reducible metric with those conical angles on some compact Riemann surface with genus g for each positive integer g.
Be sketched will the proof of the above existence theorem on the Riemann sphere in case that all the conical angles are pi times rationals. This special case also gives a new sufficient condition for the Hurwitz problem, which asks whether a admissible branch data is realized by some rational function. If time permitted, some problems interesting to the speaker will be proposed.
This is a joint work with Professor Qing Chen and my students Bo Li, Santai Qu and Ji-Jian Song. Since the beginning of this work, the speaker has been in a great debt to Professors Mao Sheng, Yifei Chen and Yingyi Wu at AMSS, to whom he express his deep gratitude.
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