On tube-log Riemann surfaces and primitives of rational functions

TL;DR

This paper provides an explicit geometric description of flat structures on the Riemann sphere induced by certain rational functions with simple poles, using a construction involving log-polygons and flat half-cylinders.

Contribution

It introduces a novel geometric construction for flat structures associated with a generic class of rational functions with simple poles.

Findings

Explicit description of flat structures on the Riemann sphere

Construction of flat structures via log-polygons and half-cylinders

Applicable to rational functions with simple poles

Abstract

For a generic class of rational functions, we give an explicit description of the flat structure on the Riemann sphere induced by a meromorphic 1-form R(z)dz, where R is a rational function. The rational functions in the generic class we consider have only simple poles. We show that the flat structure may be obtained by pasting isometrically flat half-cylinders to a 'log-polygon', which is a domain bounded by straight line segments in a simply connected finite sheeted branched cover of C.