The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Commentarii Mathematici Helvetici


Full-Text PDF (633 KB) | Metadata | Table of Contents | CMH summary
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