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

Let (Sigma, p) be a pointed Riemann surface and k >= 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 L similar or equal to R-k x 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,T- 1 x 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 \textbackslash p to a k-pronged tree, having the same Hopf differential