The surface of circumferences for Jenkins-Strebel Differentials

Abstract

There are some existence problems of Jenkins-Strebel differentials on a Riemann surface. One of them is to find a Jenkins-Strebel differential whose characteristic ring domains have given positive numbers as their circumferences, for any fixed underlying Riemann surface and core curves of the ring domains. In this paper, we investigate the existence of such solutions. Our method is to use a surface of circumferences. This is an analogue of the surface of the squares of the heights introduced by Strebel to provide an existence proof for a Jenkins-Strebel differential with given ratio of the moduli. We can see some degenerations of the characteristic ring domains of Jenkins-Strebel differentials by using the surface. We also consider the behavior of the surface when the underlying Riemann surface varies.