Minimal Surfaces and The Weierstrass-Enneper Representation

The field of minimal surfaces is an intriguing study, not only because of the exotic structures that these surfaces admit, but also for the deep connections among various mathematical disciplines. Minimal surfaces have zero mean curvature, and their parametrizations are usually quite complicated and nontrivial. It was shown however, that these exotic surfaces can easily be constructed from a careful choice of complex-valued functions, using what is called the Weierstrass-Enneper Representation. In this paper, we develop the necessary tools to study minimal surfaces. We will prove some classical theorems and solve an interesting problem that involves ruled surfaces. We will then derive the Weierstrass-Enneper Representation and use it to construct a well-known minimal surface, along with three, new and exciting minimal surfaces that have not been parametrized before