The surface of circumferences for Jenkins-Strebel differentials

TL;DR

This paper investigates the existence of Jenkins-Strebel differentials with prescribed circumferences on Riemann surfaces, using the surface of circumferences derived from an extremal problem, and studies their degenerations and behavior under surface variations.

Contribution

It introduces the surface of circumferences as a tool to analyze the existence and degenerations of Jenkins-Strebel differentials with given circumferences.

Findings

Surface of circumferences characterizes degenerations of Jenkins-Strebel differentials.

Existence depends on the underlying Riemann surface and prescribed circumferences.

Behavior of the surface varies with changes in the Riemann surface.

Abstract

There are some existence problems of Jenkins-Strebel differentials on a Riemann surface. The 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. However, the solution may not exist for some given underlying surface, core curves, and positive numbers. In this paper, we investigate the existence of such the solutions. Our method is to use the surface of circumferences, which is determined by the extremal problem for Jenkins-Strebel differentials. We can see degenerations of the characteristic ring domains of Jenkins-Strebel differentials by the surface. Moreover, we also consider the behavior of the surface when the underlying Riemann surface varies.