Introduction

Abstract

This text on minimal surfaces is arranged in four parts. The first part serves as an introduction to differential geometry and to the classical theory of minimal surfaces and should more or less be readable for any graduate student. Its only prerequisites are the elements of vector analysis and some basic knowledge of complex analysis. After an exposition of the basic ideas of the theory of surfaces in three-dimensional Euclidean space given in Chapter 1, we begin Chapter 2 by introducing minimal surfaces as stationary points of the area functional. Then we show that any minimal surface can be represented both in an elementary and a geometrically significant way by conformal parameters. In general this representation will only be local. However, invoking the uniformization theorem, we are led to global conformal representations. This reasoning will suggest a new definition of minimal surfaces that includes the old one but is much more convenient: a minimal surface X(w) is defined as a nonconstant harmonic mapping from a parameter domain Ω in the complex plane into ℝ3 which satisfies the conformality relation <X w , X w > = 0. Other parts of Chapter 2 are concerned with basic features of nonparametric minimal surfaces such as Bernstein’s theorem, stating that entire solutions of the nonparametric minimal surface equation in ℝ2 have to be planes, and with foliations by one-parameter families of minimal surfaces and their significance in establishing the minimum property. Finally we derive the classical formula for the second variation of area.