Bergman interpolation on finite Riemann surfaces. Part II: Poincar\'e-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, Berntsson, Ortega Cerd\`a, and a number of other authors. We then prove necessary and sufficient conditions for interpolation, and also some sufficient conditions under even weaker hypotheses. The results extend work of Ortega Cerd\`a, 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 punctures, thereby linking in a crucial way with the previous article in this two-part series.