Meromorphic quadratic differentials and measured foliations on a Riemann surface

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.