JournalscmhVol. 91, No. 2pp. 317–356

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

  • Subhojoy Gupta

    Indian Institute of Science, Bangalore, India
  • Michael Wolf

    Rice University, Houston, USA
Quadratic differentials, half-plane structures, and harmonic maps to trees cover

A subscription is required to access this article.

Abstract

Let (Σ,p)(\Sigma,p) be a pointed Riemann surface and k1k\geq 1 an integer. We parametrize the space of meromorphic quadratic differentials on Σ\Sigma with a pole of order k+2k+2 at pp, having a connected critical graph and an induced metric composed of kk Euclidean half-planes. The parameters form a finite-dimensional space LRk×S1\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 Tg,1×L\mathcal{T}_{g,1} \times \mathcal{L}, a unique metric spine of the surface that is a ribbon-graph with kk infinite-length edges to pp. The proofs study and relate the singular-flat geometry of the quadratic differential, and the infinite-energy harmonic map from Σp\Sigma \setminus p to a kk-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