Meromorphic quadratic differentials with half-plane structures

Abstract

We prove the existence of half-plane with prescribed local data on any Riemann surface. These are meromorphic quadratic dierentials 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 dierentials with double-order poles, and associates metric spines with the Riemann surface.