Bergman interpolation on finite Riemann surfaces. Part II: Poincar\'e-Hyperbolic Case

TL;DR

This paper extends Bergman-type interpolation theory to finite open Riemann surfaces covered by the unit disk, providing necessary and sufficient conditions for interpolation that generalize previous results and adapt to the geometry near punctures.

Contribution

It formulates a generalized interpolation problem on finite Riemann surfaces and establishes comprehensive criteria for when interpolation is possible, extending prior work to more complex geometries.

Findings

Established necessary and sufficient conditions for Bergman-type interpolation on finite Riemann surfaces.

Extended previous results to include surfaces with punctures and more complex boundary structures.

Linked the interpolation problem to the geometric modifications near punctures, enriching the theoretical framework.

Abstract

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.