From Schwarz to Pick to Ahlfors and Beyond, Volume 46, Number 8

868 NOTICES OF THE AMS VOLUME 46, NUMBER 8 I n his pivotal 1916 paper [P], Georg Pick begins somewhat provocatively with the phrase, “The so-called Schwarz Lemma says...”, followed by a reference to a 1912 paper of Carathéodory. Pursuing that lead, one finds a reference to the original source in the expanded notes from a lecture course at the Eidgenössische Polytechnische Schule in Zürich given by Hermann Amandus Schwarz during 1869–70 ([S], 108–132). The lecture notes start by stating the Riemann Mapping Theorem from Riemann’s doctoral dissertation of 1851 and noting that Riemann’s argument did not provide a fully rigorous proof. The goal of the lecture course is to provide the first complete proof for a general class of domains. The Riemann mapping theorem states that any simply connected plane domain other than the entire plane can be mapped one-to-one and conformally onto the interior of the unit circle. (The plane domains considered at the time appear to have been domains bounded by a simple closed curve.) The lecture notes prove the theorem for domains bounded by a closed convex curve. Schwarz’s proof is based on his earlier work on domains bounded by polygons and what is now known as the Schwarz-Christoffel formula. Schwarz’s paper ([S], 65–83) appeared in 1869. In it he notes that back in 1863–64, when he attended a course by Weierstrass on the theory of analytic functions, he did not know of a single special case of a plane figure given in advance for which one could establish a conformal mapping onto the unit disk. He decides to start with the simplest case of a square. (It is in that context that he proves his famous “reflection principle” for analytic functions.) He goes on to give a general formula, noting that Christoffel developed it independently. He credits Weierstrass for filling in the details of showing that the arbitrary constants involved in the integral expression can be chosen to give the desired mapping for a polygon of any number of sides, whereas Schwarz himself had succeeded for just four sides. Once in possession of a mapping for polygons, Schwarz proceeds to approximate an arbitrary convex domain by domains bounded by polygons and to show that the corresponding mappings converge to a limit mapping with the desired properties. This proof has long been forgotten, since the result was superseded by more general results, leading eventually to a proof of the full theorem. However, the first step in his argument for convex domains is precisely the statement and proof of an early version of what eventually became known as the “Schwarz Lemma”.