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

### Subhojoy Gupta

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

Rice University, Houston, USA

## Abstract

Let $(Σ,p)$ be a pointed Riemann surface and $k≥1$ an integer. We parametrize the space of meromorphic quadratic differentials on $Σ$ 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 $L≅R_{k}×S_{1}$ 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 $T_{g,1}×L$, 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 $Σ∖p$ 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