Representation Formulas and Examples of Minimal Surfaces

Abstract

In this chapter the classical theory of minimal surfaces is presented. The central point of this theory is the representation formula of Enneper and Weierstrass which expresses a given minimal surface in terms of integrals involving a holomorphic function μ and a meromorphic function ν. Conversely, any pair of such functions μ, ν can be used to define minimal surfaces provided that μν2 is holomorphic. In the older literature this representation was mostly used for a local discussion of minimal surfaces. Following Osserman, the representation formula has become very important for the treatment of global questions for minimal surfaces. As an example of this development the results concerning the omissions of the Gauss map of a complete regular minimal surface are described. These results are the appropriate generalization of Picard’s theorem in function theory to differential geometry and culminate in the remarkable theorem of Fujimoto that the Gauss map of a nonplanar complete and regular minimal surface cannot miss more than four points on the Riemann sphere. Furthermore the solution of Bjorling’s problem by H.A. Schwarz is described. This is just the Cauchy problem for minimal surfaces with an arbitrarily prescribed real analytic initial strip, and it is known to possess a unique solution due to the theorem of Cauchy–Kovalevskaya. Schwarz found a beautiful integral representation of this solution which can be used to construct interesting minimal surfaces, such as surfaces containing given curves as geodesics or as lines of curvature. As an interesting application of Schwarz’s solution his reflection principles for minimal surfaces is treated. Finally a few of the classical minimal surfaces are discussed, and a brief survey of recent results on complete minimal surfaces is given.