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On realizing measured foliations via quadratic differentials of harmonic maps toR-trees

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Abstract

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 \( \tilde R \) 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.

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Partially supported by NSF grant DMS9300001; Alfred P. Sloan Research Fellow.

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Wolf, M. On realizing measured foliations via quadratic differentials of harmonic maps toR-trees. J. Anal. Math. 68, 107–120 (1996). https://doi.org/10.1007/BF02790206

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  • DOI: https://doi.org/10.1007/BF02790206

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