Anthony Tromba

Minimal Surfaces with Free Boundaries

This chapter is centered on the proof of existence theorems for minimal surfaces with completely free boundaries. The problem is approached by applying the direct methods of the calculus of variations, thus establishing the existence of minimizers with a boundary on a given supporting surface S. However, this method does not yield the existence of stationary minimal surfaces which are not area minimizing. The remaining part of the chapter deals with additional properties of minimal surfaces with free boundaries. For instance, such a surface has to intersect the free boundary surface perpendicularly and in a balanced way. This fact implies nonexistence in certain cases. Finally an extensive report on the existence of stationary minimal surfaces with free or partially free boundaries is given.

Regularity of minimal surfaces

Boundary Behaviour of Minimal Surfaces.- Minimal Surfaces with Free Boundaries.- The Boundary Behaviour of Minimal Surfaces.- Singular Boundary Points of Minimal Surfaces.- Geometric Properties of Minimal Surfaces.- Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities.- The Thread Problem.- Branch Points.

Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in ℝn

The algebraic number of disc minimal surfaces spanning a wire in ℝ3 is defined and shown to be equal to one.

Global analysis of minimal surfaces

Introduction.- Part I. Free Boundaries and Bernstein Theorems.- 1.Minimal Surfaces with Supporting Half-Planes.- 2.Embedded Minimal Surfaces with Partially Free Boundaries.- 3.Bernstein Theorems and Related Results.- Part II. Global Analysis of Minimal Surfaces.- 4.The General Problem of Plateau: Another Approach.- 5.The Index Theorems for Minimal Surfaces of Zero and Higher Genus.- 6.Euler Characteristic and Morse Theory for Minimal Surfaces.- Bibliography.- Index.

The Boundary Behaviour of Minimal Surfaces

This and the next chapter deal with the main topic of Vol. 2, the boundary behaviour of minimal surfaces, with particular emphasis on the behaviour of stationary surfaces at their free boundaries. This and the following chapter will be the most technical and least geometric parts of our lectures. They can be viewed as a section of the regularity theory for nonlinear elliptic systems of partial differential equations. Yet these results are crucial for a rigorous treatment of many geometrical questions, and thus they will again illustrate what role the study of partial differential equations plays in differential geometry. It will be proved that a minimal (or H-) surface spanning a given boundary configuration behaves as smoothly at its boundary as the regularity class of its fixed or free boundary contour indicates. Results of this kind are basic for many investigations. In addition asymptotic expansions for minimal surfaces at boundary branch points and a general version of the Gauss–Bonnet formula is derived.

New Brief Proofs of the Gulliver–Osserman–Royden Theorem

We would like to present very much simplified proofs of versions of the Gulliver–Osserman–Royden (GOR) theorem (1973), in the case Γ is C 2,α smooth. In the first proof instead of employing a topological theory of ramified coverings used in (GOR), we introduce a new analytical method of root curves. The surprising aspect of this proof is that it connects the issue of the existence of analytical false interior branch points with boundary branch points. We should note that this fact was also observed by F. Tomi (to appear) who has found his own very brief proof of (GOR) in the case Γ∈C 2,α which we also include.

The First Main Theorem; Non-exceptional Branch Points; The Non-vanishing of the L th Derivative of Dirichlet’s Energy

Let us state our main goal: Assuming that Open image in new window is a nonplanar minimal surface in normal form having w=0 as a branch point of order n and index m, we want to show that \(\hat{X}\) cannot be a weak relative minimizer of Dirichlet’s integral D in the class Open image in new window . Unfortunately this goal cannot be achieved for all branch points but only for non-exceptional ones and special kinds of exceptional ones. In this chapter we investigate the non-exceptional branch points, while in Chaps. 5 and 6 we deal with the exceptional ones. The main result of the present section – our First Main Theorem – is the following

Very Special Case; The Theorem for n +1 Even and m +1 Odd

In this chapter we want to show that a (nonplanar) weak relative minimizer\(\hat{X}\)of Dirichlet’s integralDthat is given in the normal form cannot havew=0 as a branch point if its ordernis odd and its indexmis even. Note that such a branch point is not exceptional since n+1 cannot be a divisor of m+1. We shall give the proof only under the assumptions n≥3 since n=1 is easily dealt with by a method presented in the next section. (Moreover it would suffice to treat the case m≥6 since 2m−2<3n is already treated by the Wienholtz theorem. So 2m≥3n+2≥11, i.e. m≥6 since m is even.)

Higher Order Derivatives of Dirichlet’s Energy

In this chapter we take the point of view of Jesse Douglas and consider minimal surfaces as critical points of Dirichlet’s integral within the class of harmonic surfaces X:B→ℝ3 that are continuous on the closure of the unit disk B and map ∂B=S 1 homeomorphically onto a closed Jordan curve Γ of ℝ3. It will be assumed that Γis smooth of classC ∞and nonplanar. Then any minimal surface bounded by Γ will be a nonplanar surface of class \(C^{\infty}(\overline {B},\mathbb{R}^{3})\), and so we shall be allowed to take directional derivatives (i.e. “variations”) of any order of the Dirichlet integral along an arbitrary C ∞-smooth path through the minimal surface.

Boundary Branch Points

In this chapter we first show that Dirichlet’s integral possesses intrinsic second and third derivatives at a minimal surface \(\hat{X}\) on the tangent space T X M of M:=H 2(∂B,ℝ n ) of \(X =\hat{X}|_{\partial B}\) on the space \(J(\hat{X})\) of forced Jacobi fields for \(\hat{X}\). In particular it will be seen that \(J(\hat{X})\) is a subspace of the kernel of the Hessian D 2 E(X) of Dirichlet’s integral E(X) defined in (8.1) below, and an interesting formula (see (8.16)) for the second variation of Dirichlet’s integral is derived.