2016/06/15 by Xujia Chen, Bin Xu, Chen, Xujia +1
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods #math.CV #math.GT #msc:30F10
paper · pdf · doi:10.48550/arxiv.1606.04655
8 pages, 5 figures. Second version: We have made the paper more readable by fixing some gaps and typos. Any comments are welcome
arxiv created 2017/03/15 · arxiv updated 2017/03/16
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.