The Behnke-Stein theorem for open Riemann surfaces

Abstract

Using the Riemann-Roch theorem and the set-topological part of Bishop's polyhedron lemma, we show that the usual Runge approximation theorem for compact subsets of the Riemann sphere is valid word-for-word on any compact Riemann surface X, with meromorphic functions on X playing the role of rational functions; this result is essentially equivalent to the Behnke-Stein approximation theorem. The Behnke-Stein generalization of the Runge approximation theorem [1], which is the basic tool for many existence questions on open Riemann surfaces, can be stated in several equivalent ways, for instance: Let X be a Riemann surface, and K a compact subset of X. Then, in order that every holomorphic function in a neighborhood of K be uniformly approximable on K by holomorphic functions on X, it is necessary and sufficient that X K have no connected component with compact closure in X. As is well known, the necessity part of the above theorem follows from the sufficiency part (using the theory of compact Riemann surfaces). In this note, we wish to point out how the famous special polyhedron lemma of Bishop [2], which is an elementary set-topological result, can be used (together with the Riemann-Roch theorem for compact Riemann surfaces) to give a simple proof of the following theorem. Theorem 1.1. Let X be a compact Riemann surface, and K c X a compact subset. Let Q be any subset of X K which contains (precisely) one point qi from each connected component W of X K. Then any holomorphic function on a neighborhood of K can be approximated uniformly on K by meromorphic functions on X whose poles lie in Q. Again it is well known and easy to see that Theorem 1.1 implies the sufficiency part of Behnke-Stein. The main fact needed for this implication is the following: Received by the editors March 24, 1988. 1980 Mathematics Subject Classification (1 985 Revision). Primary 30E 10, 30F10.