The Theory of Minimal Surfaces in R^3 with a look at the Area Minimizing Property

Abstract

In this thesis we look at minimal surfaces in R^3. We begin by looking at the theory of minimal surfaces. We look at the definition as well as examples of minimal surfaces. A minimal surface is defined as a surface that has zero mean curvature. We also look at the connection of minimal surfaces to harmonic functions, as well as the Weierstrass Enneper formulas. Then we look at the area minimizing property of minimal surfaces. A minimal surface does not always have minimal surface area. Thus we would like to find out when a minimal surface has minimal surface area. In order to look at this, we study the connection between minimal surfaces and the area functional. We also look at this problem by studying the connection between minimal surfaces and soap bubbles, which is given by Plateau's problem. This allows us to find some conditions that tell us when a minimal surface has minimal surface area. We demonstrate these concepts by looking at specific examples of minimal surfaces and finding out when they have minimal surface area. The examples we consider are the catenoid, Enneper's surface and higher order Enneper surfaces. We find out when these minimal surfaces have minimal surface area. We mention that this is a local property. By that we mean that minimal surfaces locally minimize their surface area.