## Graham Smith [1]

We define a k-surface in hyperbolic type to be a complete immersed surface of constant extrinsic curvature equal to k. We say that this surface is of finite type when, in addition, it has finite area. Individual finite-type k-surfaces have very precise structures which we can use to study the local geometry of their moduli space. Since this moduli space naturally identifies with the space of (pointed) ramified coverings of the sphere, this in turn yields a number of interesting structures over the latter. In this talk we will describe the main tool used to study these surfaces, namely an asymptotic decomposition of their ends when viewed as graphs over complete geodesic rays.