Pointed k-surfaces

Let S be a Riemann surface. Let H-3 be the 3-dimensional hyperbolic space and let partial derivative H-infinity(3) be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping phi : S -> partial derivative H-infinity(3) = C. If i : S -> H-3 is a convex immersion, and if N is its exterior normal vector field, we define the Gauss lifting, i, of i by i = N. Let (n) over right arrow : UH3 -> partial derivative(infinity) H-3 be the Gauss-Minkowski mapping. A solution to the Plateau problem (S,phi) is a convex immersion i of constant Gaussian curvature equal to k is an element of (0, 1) such that the Gauss lifting (S, i) is complete and (n) over right arrow o i = phi. In this paper, we show that, if S is a compact Riemann surface, if P is a discrete subset of S and if phi : S -> (C) over cap is a ramified covering, then, for all p0 is an element of P, the solution (S\P, i) to the Plateau problem (S \ P, phi) converges asymptotically as one tends to p0 to a cylinder wrapping a finite number, k, of times about a geodesic terminating at phi(p0). Moreover, k is equal to the order of ramification of phi at p0. We also obtain a converse of this result, thus completely describing complete, constant Gaussian curvature, immersed hypersurfaces in H-3 with cylindrical ends.