On realizing measured foliations via quadratic differentials of harmonic maps toR-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 surfaceR of genusg can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential onR. 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 to anR-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. Partially supported by NSF grant DMS9300001; Alfred P. Sloan Research Fellow.