Representation Formulas and Examples of Minimal Surfaces

Abstract

In this chapter we present the elements of the classical theory of minimal surfaces developed during the last century. We begin by representing minimal surfaces as real parts of holomorphic curves in ℂ³ which are isotropic. This leads to useful and handy formulas for the line element, the Gauss map, the second fundamental form and the Gauss curvature of minimal surfaces. Moreover we obtain a complete description of all interior singular points of two-dimensional minimal surfaces as branch points of ℂ³-valued power series, and we derive a normal form of a minimal surface in the vicinity of a branch point. Close to a branch point of order m, a minimal surface behaves, roughly speaking, like an m-fold cover of a disk, a property which is also reflected in the form of lower bounds for its area. Other by-products of the representation of minimal surfaces as real parts of isotropic curves in ℂ³ are results on adjoint and associated minimal surfaces that were discovered by Bonnet.