Quadratic differentials with prescribed singularities

TL;DR

This paper proves that most local invariants of meromorphic quadratic differentials on compact Riemann surfaces can be realized, classifying exceptions and analyzing realizability in different strata using flat metric techniques.

Contribution

It establishes a near-complete classification of realizable local invariants for quadratic differentials, including exceptions and realizability in nonconnected strata.

Findings

Most local invariants are realizable with few exceptions.

Exceptions occur only in genus zero and one.

Bounds on the number of disjoint cylinders are provided.

Abstract

The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface. The exceptions are completely classified and only occur in genera zero and one. Moreover, in the case of a nonconnected stratum, we show that, with three exceptions in genus one, every invariants can be realized in each connected component of the stratum. These results are obtained using the flat metric induced by the differentials. We give an application by bounding the number of disjoint cylinders on a primitive quadratic differential.