Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

We formulate the Bergman-type interpolation problem on finite open Riemann surfaces covered by the unit disk. Our version of the interpolation problem generalizes Bergman-type interpolation problems previously studied by Seip, Berndtsson, Ortega Cerdà, and a number of other authors. We then prove sufficient conditions for a sequence to be interpolating. When the curvature of the weight in question is bounded in an appropriate sense, we show that the sufficient conditions are almost necessary, but not quite. The results extend work of Ortega Cerdà, who resolved the case in which the boundary of the surface is pure 1-dimensional. Our version of the interpolation problem effectively changes the geometry of the underlying space near the 0-dimensional boundary components, or punctures, thereby linking in a crucial way with the previous article (Varolin, J d’Anal Math, 2015, to appear) in this two-part series.