On realizing measured foliations via quadratic differentials of harmonic maps to \(\mathbf R\)-trees (Q1922215)

The author gives an elementary and analytic proof of a theorem of Hubbard and Masur that every class of measured foliations on a compact Riemann surface \(R\) of genus \(g\) can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential on \(R\). His proof involves the direct method in the calculus of variations, Weyl's lemma, the definition of an equivalence class of a measured foliation, and the equivariant map from the universal cover of \(R\) to a real tree. A direct corollary is the theorem of W. P. Thurston that the space of classes of projective measured foliations is a \(6g-7\)-dimensional sphere.