Meromorphic quadratic differentials with complex residues and spiralling foliations

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.