Constructing Strebel Differentials via Belyi Maps on the Riemann Sphere

In this paper, by using Belyi maps and dessin d'enfants, we construct some concrete examples of Strebel differentials with four double poles of residues 1, 1, 1, 1 on the Riemann sphere. We also prove that they have either two double zeroes or four simple zeroes. In particular, we show that they have two double zeroes if and only if their poles are coaxial; in such cases we also obtain their explicit expressions. On the other hand, for those differentials with four non-coaxial poles and whose metric ribbon graphs have edges of rational lengths, we characterize them optimally in terms of Belyi maps in the sense that the Belyi maps used here have minimal degree, and work out the explicit expressions of the five simplest ones among them. As applications, we obtain some explicit cone spherical metrics on the Riemann sphere.