Quadratic differentials, measured foliations and metric graphs on punctured surfaces

TL;DR

This paper establishes a correspondence between measured foliations, quadratic differentials, and metric graphs on punctured Riemann surfaces, extending classical theorems to the meromorphic case with poles.

Contribution

It introduces the concept of asymptotic directions at poles and proves existence and uniqueness results for meromorphic quadratic differentials with prescribed foliations and pole data.

Findings

Unique determination of complex structures from measured foliations and asymptotic data.

Existence of meromorphic quadratic differentials with prescribed horizontal foliations.

Extension of classical theorems to meromorphic differentials with poles.

Abstract

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic direction at each pole, and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner-Masur, for meromorphic quadratic differentials. We also prove an analogue of the Hubbard-Masur theorem, namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular-flat geometry at the poles.