On Realizing Measured Foliations via Quadratic Differentials of Harmonic Maps to R-trees

We give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) 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. The theorem of Thurston [Th] that the space of classes of projective measured foliations is a 6g − 7 dimensional sphere follows immediately by Riemann-Roch. Our argument involves relating each representative of a class of measured foliations to an equivariant map from e R to an R-tree, and then finding an energy minimizing such map by the direct method in the calculus of variations. The normalized Hopf differential of this harmonic map is then the desired differential.