Four

Abstract

Courant’s dissertation was entitled “On the application of Dirichlet’s principle to the problems of conformal mapping.” In it he was able to modify and simplify Hilbert’s approach to the principle and to apply it to certain fundamental problems of geometric function theory concerning, among other things, the conformal mapping of Riemann surfaces of higher genus. He was able to prove in a new and different way the uniformization theorem which Koebe had proved in 1908. He was also able to prove another important theorem, known as the slit theorem, which had been stated very generally in 1909 by Hilbert. He developed as well an estimate which later led him to the statement of a lemma that was to become one of his most powerful and most frequently used tools.