The Riemann mapping theorem from Riemanns viewpoint

Abstract

This article presents a rigorous proof of the Riemann mapping theorem via Riemann’s method, uncompromised by any appeals to topological intuition.

The Riemann mapping theorem is one of the most remarkable results of nineteenth-century mathematics. Even today, more than a hundred fifty years later, the fact that every proper simply connected open subset of the complex plane is biholomorphically equivalent to every other seems deep and profound. This is not a result that has become in any sense obvious with the passage of time and the general expansion of mathematics. And at the time, the theorem must have been truly startling. Even Gauss, never easily impressed, viewed the result favorably, though he had reservations about the summary nature of Riemann’s writings. At the time, Riemann’s method appeared hard to carry out in detail. And indeed there have been those since who believed it could not be carried out in detail at all.

Thus, when a different proof arose later on using Montel’s idea of normal families, this proof established itself as standard [2]. Indeed, it is rare to find any other proof than the normal families one in contemporary texts on complex analysis. Only if the student of complex analysis goes on to study uniformization of open Riemann surfaces is Riemann’s original idea likely to be encountered. At best, the original proof idea is relegated to exercises or brief summaries in texts on basic complex analysis (cf., e.g., Exercise 73, p. 251 in [4], or Section 5.2, p. 249–251 in [1]).

And yet, in the historical view, Riemann’s proposed method of proof was as interesting as and perhaps even more important than the result itself. It would have been almost impossible for anyone listening to Riemann’s presentation in 1851 to have imagined that what they were hearing was the first instance of a mathematical method that would become a massive part of geometric mathematics in the decades to come and that continues to be a vitally active subject today. But so it was, for Riemann’s proof method for his mapping theorem marked the introduction of the use of elliptic equations and the solution of elliptic variational problems to treat geometric questions. The analytic theory of Riemann surfaces via harmonic forms and Hodge’s generalization to algebraic varieties in higher dimensions; the circle of results known by the name the Bochner technique; the theory of minimal submanifolds and its applications to topology of manifolds; the use of elliptic methods in 4-manifold theory; and, most recently, the proof of the Poincaré Conjecture and the geometrization conjecture that extends it—all these and much more could not have been anticipated in any detail on that historic day at the time Riemann presented his mapping theorem. But in retrospect, when Riemann suggested constructing the biholomorphic map to the unit disc that his result called for by solving an elliptic variational problem, the whole development began. The fact that Riemann could not in fact actually prove what he called Dirichlet’s Principle is almost beside the point. He had found the way into the thicket. Chopping the path onward could be and would be done by others.

Thus, it seemed to the authors unfortunate that finding a precise and complete discussion of how actually to carry out Riemann’s argument is not easy. Osgood’s proof [6] (See also [16]) of the Riemann Mapping theorem—usually regarded as the first reasonably complete proof, correct except for certain topological details being brushed over—does indeed use Riemann’s general idea. But it is made more difficult than need be today because he was not in possession of the Perron method of solving the Dirichlet problem. Thus, he had to work with piecewise linear approximations from the interior and take limits of the piecewise linear (even piecewise real analytic) case of the Dirichlet problem that had been solved by Schwarz already at that time [12].

Our goal in this article is to present a clear proof