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

الأصل · EN

Let (Σ,p) be a pointed Riemann surface of genus g≥ 1. For any integer k≥ 1, we parametrize the space of meromorphic quadratic differentials on Σ 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 Rᵏ × S¹ 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 decorated Teichmüller space Tg,₁ × 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 on the surface and the infinite-energy harmonic map from Σ p to a k-pronged graph, whose Hopf differential is that quadratic differential.