Drawing cone spherical metrics via Strebel differentials

TL;DR

This paper introduces a novel method for constructing cone spherical metrics on compact Riemann surfaces by leveraging Strebel differentials and connected metric ribbon graphs, enriching the geometric toolkit.

Contribution

It presents a new approach that connects Strebel differentials with cone spherical metrics through the use of metric ribbon graphs, expanding the methods for constructing such metrics.

Findings

Constructed new cone spherical metrics using Strebel differentials.

Established a correspondence between ribbon graphs and cone metrics.

Provided explicit examples of the constructed metrics.

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.