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On the existence of Jenkins-Strebel differentials using harmonic maps from surfaces to graphs. (English) Zbl 0846.30019
Summary: We give a new proof of the existence [J. Hubbard and H. Masur, Acta Math. 142, 221-224 (1979; Zbl 0415.30038), H. Renelt, Math. Nachr. 73, 125-142 (1976; Zbl 0347.30017)] of a Jenkins-Strebel differential $$\Phi$$ on a Riemann surface $$\mathcal R$$ with prescribed heights of cylinders by considering the harmonic map from $$\mathcal R$$ to the leaf space of the vertical foliation of $$\Phi$$, thought of as a Riemannian graph. The novelty of the argument is that it is essentially Riemannian as well as elementary; moreover, the harmonic maps existence theory on which it relies is classical, due mostly to [C. B. Morrey, Ann. Math. 49, 807-851 (1958; Zbl 0090.38401)].

##### MSC:
 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 58E20 Harmonic maps, etc.
##### Keywords:
Jenkins-Strebel differential
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