We study the Bergman space interpolation problem of open Riemann surfaces
obtained from a compact Riemann surface by removing a finite number of points.
We equip such a surface with what we call an asymptotically flat conformal
metric, i.e., a complete metric with zero curvature outside a compact subset.
We then establish necessary and sufficient conditions for interpolation in
weighted Bergman spaces over asymptotically flat Riemann surfaces.Comment: The main result has been corrected: Sequences of density <1 are still
interpolating, but the density of an interpolation sequence is only shown to
be at most 1. The corrected result is sharp, by work of Borichev-Lyubarskii.
Also added a motivating section on Shapiro-Shields interpolation. Otherwise
typos and minor errors corrected. To appear in Journal d'Analys