Drawing cone spherical metrics via Strebel differentials

Abstract

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on compact Riemann surfaces by drawing on the surfaces some class of connected metric ribbon graphs.Comment: 25 pages, 8 figures. Version 2: minor typo corrections; revised according to referee's comments. We substantially revised the proof of the second theorem to make its exposition easier to understand. We added a new section, where we discuss on the Riemann sphere the consistence of metrics generated by Strebel differentials with the two angle conditions by Mondello-Panov and Eremenko, respectivel