Explicit Examples of Strebel Differentials

TL;DR

This paper constructs explicit examples of Strebel differentials on various Riemann surfaces, including holomorphic and algebraic cases, highlighting their properties and differences from typical non-Strebel differentials.

Contribution

It provides explicit constructions of Strebel differentials on hyperelliptic curves and modifies existing examples to produce different transcendental period properties.

Findings

Explicit holomorphic Strebel differentials on hyperelliptic curves.

Modification of algebraic Strebel differentials to alter period transcendence.

Demonstration of the rarity and explicit construction of Strebel differentials.

Abstract

Let X be some Riemann surface, and let omega be a meromorphic quadratic differential form on X, that is, omega can be written in local coordinates as f(z) dz^2, for some meromorphic function f. We say that a curve gamma is part of a horizontal leaf of omega if for each t in I, we have that f(gamma(t)) (gamma'(t))^2 is real and positive. The differential omega is said to be a Strebel differential if its noncompact horizontal leaves form a set of measure 0. Now, most quadratic differential forms on any given Riemann surface are not Strebel (in fact, the space of Strebel differentials on the projective line as a subset of the space of all quadratic differentials on the projective line is a countable union of real lines in the complex plane. Because of this, this paper focuses on writing down explicit examples of Strebel differentials. Most papers that discuss Strebel differentials seem to focus on those that have only second order poles, since they define a cellular decomposition on X. However, we construct an explicit family of examples of holomorphic Strebel differentials on any hyperelliptic curve branched over the points 0, 1, infinity, and 1/2 + ir, for any real r. Later, we look at an example from a previous paper, where omega was an algebraic Strebel differential such that omega had integral residues, on a Riemann surface X defined over the algebraic closure of the rationals that had a transcendental period. We then modify this example to obtain another algebraic Strebel differential omega', defined on an algebraic Riemann surface X', all of whose periods are transcendental.