Jenkins-Strebel Differentials on the Riemann Sphere with Four Simple Poles

TL;DR

This paper provides explicit formulas and an algorithm for Jenkins-Strebel differentials on the Riemann sphere with four simple poles, enhancing understanding of their structure and classification.

Contribution

It offers the first explicit expressions for these differentials using the Weierstrass  function and introduces a simple algorithm for classifying them via simple closed curves.

Findings

Explicit formulas for Jenkins-Strebel differentials using Weierstrass  function

An algorithm for classifying differentials based on simple closed curves

Enhanced understanding of differentials on the Riemann sphere with four poles

Abstract

A celebrated and deep theorem in the theory of Riemann surfaces states the existence and uniqueness of the Jenkins-Strebel differentials on a Riemann surface under some conditions, but the proof is non-constructive and examples are difficult to find. This paper deals with an example of a simple case, namely Jenkins-Strebel differentials on the Riemann sphere with four fixed simple poles. We will give explicit expressions of these Jenkins-Strebel differentials by means of the Weierstrass $\wp$ function and expose a simple algorithm determining the correspondence between these differentials and some classes of simple closed curves on the Riemann sphere with four points removed.