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

### Subhojoy Gupta

Indian Institute of Science, Bangalore, India### Michael Wolf

Rice University, Houston, USA

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## Abstract

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.

## Cite this article

Subhojoy Gupta, Michael Wolf, Quadratic differentials, half-plane structures, and harmonic maps to trees. Comment. Math. Helv. 91 (2016), no. 2, pp. 317–356

DOI 10.4171/CMH/388