Ulrich Dierkes

Representation Formulas and Examples of Minimal Surfaces

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.

Bernstein Theorems and Related Results

This chapter provides a fairly comprehensive—although still not complete—presentation of results which lead to Bernstein theorems for solutions of the n-dimensional minimal surface equation as well as for stationary points of singular integrals of the type (E_{alpha}(u):=int_{Omega}u^{alpha}sqrt{1+|Du|^{2}},dx), called α-minimal hypersurfaces. The basic results are integral curvature estimates and pointwise curvature estimates for minimal hypersurfaces. Essential tools are formulae for the first and second variation, Simons’s identity for the second fundamental form and “Jacobi’s field equation”, a discussion about the nonexistence of stable cones, monotonicity formulae, and “Sobolev-type” estimates by Michael and Simon. In particular, a proof of the celebrated curvature estimate of Schoen–Simon–Yau is presented which generalizes Heinz’s estimate from n=2 to n≤5.

The Plateau Problem and the Partially Free Boundary Problem

This chapter is the central part of Volume 1, as it treats the existence theory for Plateau’s problem, which can be viewed as the simplest of the boundary value problems to be studied in this treatise. A version of this problem is: Given a closed rectifiable Jordan curve Γ , find a minimal surface spanning Γ.

Stable Minimal- and H-Surfaces

The principal topic in this chapter is the study of stable minimal surfaces and of stable surfaces of prescribed mean curvature (H-surfaces). In this context it is important to investigate the Gauss map of such a map and its relation to stability. One is led to Bernstein-type results, to Nitsche’s uniqueness theorem, and to the finiteness of the number of minimizers proved by F. Tomi. Furthermore, the above results will enable us in Chapter 7 to solve the nonparametric equation of prescribed mean curvature via the solution of Plateau’s problem for parametric surfaces of prescribed mean curvature. Using and extending the ideas presented in Chapter 4, this more general Plateau problem for H-surfaces will be solved in Vol. 2, Chapter 4.

Euler Characteristic and Morse Theory for Minimal Surfaces

The main goal of this chapter is to define an Euler characteristic for minimal surfaces solving Plateau’s problem in ℝ n , n≥3, and to derive its main properties. The most interesting case n=3 is much more difficult than the case n≥4 since, according to the index theorem, for n=3 the nondegenerate minimal surfaces are not generic. Nevertheless it will be useful to have a theory for n≥4 as one can connect ℝ3-contours via homotopies in ℝ n with ℝ n -contours, n≥4, bounding only nondegenerate minimal surfaces. To carry out this programm, first some properties of Fredholm vector fields W:ℳ→Tℳ on a Hilbert manifold ℳ are recalled, and then a special class of Palais–Smale vector fields is defined. For such a vector field V one can define an Euler characteristic χ(V) if its zeros are nondegenerate, and for two vector fields V,W of this kind one has χ(V)=χ(W) if V and W are properly homotopic.

The General Problem of Plateau: Another Approach

In this chapter we shall present another solution of the general problem of Plateau, i.e. of the Douglas problem. This is the question whether a configuration 〈Γ 1,Γ 2,…,Γ k 〉 of several closed curves Γ j may bound multiply connected minimal surfaces that could be of higher genus and even nonorientable; for a special case this problem was studied in Vol. 1. As background, an approach to Teichmuller theory of compact oriented Riemann surfaces is outlined which then is extended to oriented Riemann surfaces with boundary. These results are basic for the variational treatment of the general Plateau problem, as one has to deal with variations of the complex structure of two-manifolds. The second basic tool is a compactness theorem of Mumford which is combined with Courant’s condition of cohesion to obtain nondegenerate solutions of the variational problem connected with the general Plateau problem. Then the crucial ideas for solving this problem are outlined and some of the principal existence results due to Douglas, Courant, and Shiffman are stated. Finally, the general Plateau problem for orientable surfaces is handled in combination with an obstacle problem in order to obtain a sufficient condition for the solvability of the general Douglas–Plateau problem that can be formulated in geometric-topological terms.

Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

In this chapter we give a brief introduction to the differential geometry of surfaces in three-dimensional Euclidean space. The main purpose of this introduction is to provide the reader with the basic notions of differential geometry and with the essential formulas that will be needed later on.

Minimal Surfaces with Supporting Half-Planes

In Vol. 3, the regularity of stationary minimal surfaces with a partially free boundary was discussed. It was shown that, for a uniformly smooth surface S with a smooth boundary ∂ S, the stationary surfaces X belong to the class C 1,1/2(B∪I,ℝ3). One of the consequences of results proved in the present chapter will be that this regularity result is optimal. The nonoriented tangent of the free trace Σ X changes continuously which, in particular, means that the free trace cannot have corners at points where it attaches to the border of the supporting surface S. On the other hand, since isolated branch points of odd order cannot be excluded, there might exist cusps on the free trace. In fact, experimental evidence suggests that cusps do appear for certain shapes of the boundary configuration 〈Γ,S〉.

The Index Theorems for Minimal Surfaces of Zero and Higher Genus

In 1981 R. Bohme and A. Tromba proved an index theorem for branched minimal surfaces of disk type in Euclidean space ℝ n . One of the consequences of their result is the finiteness of the number of branched minimal disks spanning a contour in general position. Later Tomi and Tromba proved an index theorem for minimal surfaces of higher topological type spanning one boundary contour. In the present chapter these two index theorems are derived. To present the details one needs here as well as in Chapters 4 and 6 a number of notations and theorems from Teichmuller theory and Global analysis; either detailed references are given or results are proved on the spot if they cannot easily be found in the literature. Clearly these chapters are profoundly influenced by ideas of S. Smale and his approach to global nonlinear analysis. Therefore these results can be viewed as a truly nonlinear application of Global analysis.

Unstable Minimal Surfaces

Here it is shown that the existence of two minimal surfaces in a closed rectifiable contour Γ which are local minimizers of Dirichlet’s integral D guarantees the existence of a third minimal surface bounded by Γ which is unstable, i.e. of non-minimum character. Results of this kind were first proved by M. Shiffman and simultaneously by M. Morse and C. Tompkins. The method used in this chapter is based on ideas due to Courant and proceeds by reduction of the problem to a finite-dimensional one if the boundary contour is a polygon. Then, following an approach by Shiffman, the results are carried over from polygons to general boundaries. In this context an isoperimetric property of harmonic mappings and a convergence theorem for such mappings, based on the Douglas functional, play an essential role.