Meromorphic quadratic differentials with half-plane structures

TL;DR

This paper proves the existence of meromorphic quadratic differentials with prescribed local data on any Riemann surface, resulting in a singular flat metric composed of glued half-planes, generalizing Strebel's classical results.

Contribution

It introduces 'half-plane differentials' with higher-order poles and prescribed local data, extending Strebel's theorem to more general meromorphic quadratic differentials.

Findings

Existence of half-plane differentials with specified local data on any Riemann surface

Construction of singular flat metrics from glued half-planes

Generalization of Strebel's theorem to higher-order poles

Abstract

We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their boundaries. The local data is associated with the poles and consists of the integer order, a non-negative real residue, and a positive real leading order term. This generalizes a result of Strebel for differentials with double-order poles, and associates metric spines with the Riemann surface.