Meromorphic quadratic differentials with complex residues and spiralling foliations

TL;DR

This paper characterizes measured foliations induced by meromorphic quadratic differentials with second order poles on compact Riemann surfaces, establishing existence and uniqueness results based on complex residues and extending classical theorems.

Contribution

It introduces the space of such foliations and proves their realization and uniqueness by quadratic differentials with prescribed residues, generalizing Hubbard-Masur and Strebel theorems.

Findings

Any such foliation is realized by a quadratic differential with prescribed residues.

The differential is uniquely determined by the residues if compatible with transverse measures.

The proof uses harmonic maps to real trees and exhaustion techniques.

Abstract

A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that for a fixed Riemann surface, any such foliation is realized by a quadratic differential with second order poles at marked points. Furthermore, such a differential is uniquely determined if one prescribes complex residues at the poles that are compatible with the transverse measures around them. This generalizes a theorem of Hubbard and Masur concerning holomorphic quadratic differentials on closed surfaces, as well as a theorem of Strebel for the case when the foliation has only closed leaves. The proof involves taking a compact exhaustion of the surface, and considering a sequence of equivariant harmonic maps to real trees that do not have a uniform bound on total energy.