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Let be a pointed Riemann surface and an integer. We parametrize the space of meromorphic quadratic differentials on with a pole of order at , having a connected critical graph and an induced metric composed of Euclidean half-planes. The parameters form a finite-dimensional space 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 , a unique metric spine of the surface that is a ribbon-graph with infinite-length edges to . The proofs study and relate the singular-flat geometry of the quadratic differential, and the infinite-energy harmonic map from to a -pronged tree, having the same Hopf differential.
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Subhojoy Gupta, Michael Wolf, Quadratic differentials, half-plane structures, and harmonic maps to trees. Comment. Math. Helv. 91 (2016), no. 2, pp. 317–356