On the Existence of Jenkins-Strebel Differentials Using Harmonic Maps from Surfaces to Graphs

Abstract

We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a Jenkins-Strebel differential $\Phi$ on a Riemann surface $\SR$ with prescribed heights of cylinders by considering the harmonic map from $\SR$ 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 Morrey (\cite{Mo}).

8 pages, 2 figures available upon request

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• Conformal surface embeddings and extremal length
• Elastic Graphs
• Variation of extremal length functions on Teichmuller space
• The Schwarzian derivative and measured laminations on Riemann surfaces
• Harmonic Splittings of Surfaces
• Elementary construction of some Jenkins-Strebel differentials
• Harmonic Maps and Teichmueller Theory