##### Venue: Department of Mathematics, Lecture Hall I

Given a domain D in the complex plane and a compact subset K, Runge’s theorem provides conditions on K which guarantee that a given function that is holomorphic in some neighbourhood of K can be approximated on K by a holomorphic function on D. We look at an analogous theorem on non-compact Riemann surfaces, i.e., Runge’s approximation theorem, stated and proved by Malgrange. We revisit Malgrange’s proof of the theorem, invoking a very basic result in distribution theory: Weyl’s lemma. We look at two main applications of Runge’s theorem. Firstly, every open Riemann surface is Stein and secondly the triviality of holomorphic vector bundles on non-compact Riemann surfaces. Next, we look at the Gunning-Narasimhan theorem which states that every open (connected) Riemann surface can be immersed into $\\mathbb{C}$. We discuss the proof of this theorem as well, which depends on Runge’s theorem too. Finally we contrast the compact case to the non-compact case, by showing that every compact Riemann surface can be embedded into a large enough complex projective space.

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Last updated: 06 Mar 2020