Cone spherical, flat and hyperbolic metrics are conformal metrics with constant curvature $+1,\,0$ and $-1$, respectively, and with finitely many conical singularities on compact Riemann surfaces. The Gauss-Bonnet formula gives a natural necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces. The condition is also sufficient for both flat and hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over twenty years. Projective functions are multi-valued locally univalent meromorphic functions on Riemann surfaces such that their monodromy lies in the group ${\rm PGL(2,\,{\Bbb C})}$ consisting of all M{\" o}bius 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 ${\rm PSU(2)}$, and whose Schwarzian derivatives have double poles at the conical singularities with coefficients 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 ${\rm U(1)}$, called reducible metrics. As an application, we found a necessary and sufficient condition for cone angles of reducible metrics on the Riemann sphere.
We obtained on compact Riemann surfaces a correspondence between meromorphic Jenkins-Strebel differentials with real periods and cone spherical metrics with monodromy in
${\rm U}(1)\rtimes {\Bbb Z}_2$, 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 $2\pi\,{\Bbb Z}_{>1}$ 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.
This is my joint works with Qing Chen, Xuemiao Chen, Yiran Cheng, Yu Feng, Bo Li, Lingguang Li, Santai Qu, Jijian Song and Yingyi Wu.
