Riemann Mapping Theorem - A Proof Sketch

A Proof Sketch

Given and, we want to construct a function which maps to the unit disk and to . For this sketch, we will assume that is bounded and its boundary is smooth, much like Riemann did. Write

where is some (to be determined) holomorphic function with real part and imaginary part . It is then clear that z₀ is the only zero of f. We require for on the boundary of, so we need

on the boundary. Since is the real part of a holomorphic function, we know that is necessarily a harmonic function; i.e., it satisfies Laplace's equation.

The question then becomes: does a real-valued harmonic function exist that is defined on all of and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of u has been established, the Cauchy-Riemann equations for the holomorphic function allow us to find (this argument depends on the assumption that be simply connected). Once and have been constructed, one has to check that the resulting function does indeed have all the required properties.