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

Abstract

Let $(\Sigma ,p)$ be a pointed Riemann surface and $k\ge 1$ 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.