On genus one minimal surfaces

Abstract

A surface S in R is said to be minimal if the mean curvature H is everywhere zero. In this dissertation we discuss the basic theory of minimal surfaces, particulary the Enneper-Weierstrass representation and how this can be used to construct new minimal surfaces. Special properties of finite total curvature minimal surfaces are then discussed. These properties concern the Gauss map and the Enneper-Weierstrass representation. There is also a relation between the gems, the number of ends and the degree of the Gauss map for complete minimal surfaces of finite total curvature. Two genus one finite total curvature examples are Chen-Gackstatter's surface and Costa's surface which are non-embedded and embedded minimal surfaces, respectively.