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# Commentarii Mathematici Helvetici

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**Volume 91, Issue 2, 2016, pp. 317–356**

**DOI: 10.4171/CMH/388**

Published online: 2016-04-26

Quadratic differentials, half-plane structures, and harmonic maps to trees

Subhojoy Gupta^{[1]}and Michael Wolf

^{[2]}(1) Indian Institute of Science, Bangalore, India

(2) Rice University, Houston, USA

Let $(\Sigma,p)$ be a pointed Riemann surface and $k\geq 1$ an integer. We parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $k+2$ at $p$, having a connected critical graph and an induced metric composed of $k$ Euclidean half-planes. The parameters form a finite-dimensional space $\mathcal{L} \cong \mathbb{R}^{k} \times S^1$ that describe a model singular-flat metric around the puncture with respect to a choice of coordinate chart. This generalizes an important theorem of Strebel, and associates, to each point in $\mathcal{T}_{g,1} \times \mathcal{L}$, a unique metric spine of the surface that is a ribbon-graph with $k$ infinite-length edges to $p$. The proofs study and relate the singular-flat geometry of the quadratic differential, and the infinite-energy harmonic map from $\Sigma \setminus p$ to a $k$-pronged tree, having the same Hopf differential.

*Keywords: *Meromorphic differentials, singular-flat metrics, Riemann surfaces, harmonic maps

Gupta Subhojoy, Wolf Michael: Quadratic differentials, half-plane structures, and harmonic maps to trees. *Comment. Math. Helv.* 91 (2016), 317-356. doi: 10.4171/CMH/388