Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe

TL;DR

This paper establishes a geometric correspondence between polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe, leading to new minimal Lagrangian maps between hyperbolic ideal polygons.

Contribution

It introduces a novel geometric construction linking quadratic differentials to light-like polygons and applies this to generate minimal Lagrangian maps in hyperbolic geometry.

Findings

Homeomorphism between moduli space and light-like polygons

Construction of minimal Lagrangian maps between ideal polygons

New insights into the geometry of quadratic differentials

Abstract

We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane.