Meromorphic quadratic differentials and measured foliations on a Riemann surface

TL;DR

This paper characterizes measured foliations on Riemann surfaces induced by meromorphic quadratic differentials, extending classical results to include poles with prescribed principal parts, using harmonic map techniques.

Contribution

It generalizes Hubbard and Masur's theorem to meromorphic differentials with prescribed principal parts at poles, establishing uniqueness of the foliation realization.

Findings

Every measured foliation from a meromorphic quadratic differential is uniquely realized with prescribed principal parts.

The proof employs analysis of infinite-energy harmonic maps to trees with prescribed boundary behavior.

The results extend the classical correspondence between quadratic differentials and measured foliations to the meromorphic setting.

Abstract

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to $\mathbb{R}$-trees of infinite co-diameter, with prescribed behavior at the poles.