LAUSR

We consider harmonic diffeomorphisms to a fixed hyperbolic target Y Y from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on Y Y . The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of the existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target of the same topological type.