Lectures of Bin Xu on conical spherical metrics
Freiburg, January 30 - February 2, 2018Abstract
Projective functions are multi-valued locally univalent meromorphic functions on Riemann surfaces such that their monodromy lies in the group PGL(2, C) consisting of all Mobius transformations. We observed that the developing maps of cone spherical metrics are projective functions on the surfaces punctured by the conical singularities whose monodromy lie in PSU(2), and whose Schwarzian derivatives have double poles at the conical singularities with coecients determined by the cone angles. Starting from this observation, we made the following progresses on cone spherical metrics by using Complex Algebraic Geometry.
- We obtained on compact Riemann surfaces a correspondence between meromorphic one-forms with simple poles and real periods and cone spherical metrics whose developing maps have monodromy in U(1), called reducible metrics. As an application, we found a necessary and sucient condition for cone angles of reducible metrics on the Riemann sphere.
- We obtained on compact Riemann surfaces a correspondence between meromorphic Jenkins-Strebel differentials with real periods and cone spherical metrics with monodromy in U(1)oZ2, called quasi-reducible metrics. Moreover, by using the Mumford-Thurston correspondence, we could construct new quasi-reducible metrics by drawing certain connected metric ribbon graphs.
- We obtained on compact Riemann surfaces with positive genera a correspondence between irreducible metrics with cone angles in 2πZ_{>1} and line sub-bundles of rank two stable vector bundles. As an application of it and a theorem of Lange-type proved by us, we found a new existence result about cone spherical metrics on compact Riemann surfaces with genera greater than one.
Schedule
Nr | Date | Time | Room |
---|---|---|---|
1 | 30.01 | 13:00-14:00 | SR-119 |
2 | 31.01 | 10:15-11:15 | SR-119 |
3 | 01.02 | 13:00-14:00 | SR-403 |
4 | 02.02 | 10:15-11:15 | SR-404 |