Thick-thin decomposition for quadratic differentials

TL;DR

This paper explores the relationship between flat and hyperbolic geometries on Riemann surfaces with quadratic differentials, establishing length comparisons for curves within thick subsurfaces.

Contribution

It introduces a comparison theorem linking flat and hyperbolic lengths of curves in thick regions of quadratic differential surfaces.

Findings

Hyperbolic length is comparable to flat length times flat size in thick subsurfaces.

Provides a quantitative relationship between flat and hyperbolic geometries.

Enhances understanding of geometric structures on Riemann surfaces.

Abstract

We compare the flat geometry associated to a quadratic differential with the hyperbolic geometry associated to the underlying Riemann surface. We show that if a curve is contained in a thick subsurface, then its hyperbolic length is comparable to its flat length times the flat size of the subsurface.