Speaker: 许斌 (中国科大)

Time: Friday Mar. 7, 9:30-11:30am.

Venue: Room 29 at Quan Zhai, BICMR

Abstract: It is an open problem about the existence and uniqueness of the metrics in the title. More than twenty years ago, Troyanov and Luo-Tian obtained some classical results via the method of PDEs. 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 third kind, which have real residues and are exact outside their poles. As a byproduct, we show that such a metric on the two-sphere with conical angles of 2pi times integers is the pull-back of the standard metric on the two-sphere by some rational function. This is a joint work with Qing Chen, Wei Wang and Yingyi Wu.