Conformal Equivalence and Hyperbolic Geometry

In this chapter, we study conformal maps between domains in the extended complex plane \(\widehat{\mathbb{C}}\); these are one-to-one meromorphic functions. Our goal here is to characterize all simply connected domains in the extended complex plane. The first two sections of this chapter study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely, the group PSL(2, ℂ) and the projective special linear group. This group is also known as the Mobius group. In the third section we characterize simply connected proper domains in the complex plane by establishing the Riemann mapping theorem (RMT). This extraordi- nary theorem tells us that there are conformal maps between any two such domains.