Bergman interpolation on finite Riemann surfaces. Part I: Asymptotically Flat Case

TL;DR

This paper investigates the conditions for interpolation in weighted Bergman spaces on open Riemann surfaces derived from compact surfaces by removing points, equipped with asymptotically flat metrics with zero curvature outside a compact set.

Contribution

It establishes necessary and sufficient conditions for Bergman space interpolation on asymptotically flat Riemann surfaces, advancing understanding of function theory on such surfaces.

Findings

Characterization of interpolation conditions in weighted Bergman spaces

Development of criteria for asymptotically flat Riemann surfaces

Extension of Bergman space theory to non-compact surfaces with flat metrics

Abstract

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.