Stefan Hildebrandt

Minimal Surfaces II

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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.

On the boundary behavior of minimal surfaces with a free boundary which are not minima of the area

The well known boundary regularity results of H. Lewy and W. Jäger for area minimizing minimal surfaces with a free boundary are shown to be true also for minimal surfaces which are only stationary points of the Dirichlet integral with respect to a given boundary configuration.

Optimal boundary regularity for minimal surfaces with a free boundary

Let x(w), w=u+iv ∈ B, be a minimal surface in ℝ3 which is bounded by a configuration 〈Γ, S〉 consisting of an arc Γ and of a surface S with boundary. Suppose also that x(w) is area minimizing with respect to 〈Γ, S〉. Under appropriate regularity assumptions on Γ and S, we can prove that the first derivatives of x(u, v) are Hölder continuous with the exponent α=1/2 up to the “free part” of ∂B which is mapped by x(w) into S. An example shows that this regularity result is optimal.

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.

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.

Conformal mapping of multiply connected Riemann domains by a variational approach

Abstract We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in ℝ2 can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebes mapping theorem.

The Partially Free Boundary Problem for Parametric Double Integrals

We prove the existence of conformally paramaterized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class \( H_{loc}^{2,2} \cap C^{1,\alpha } ,0 < \alpha < 1 \), up to the free boundary under the assumption that there exists a perfect dominance function in the sense of Morrey.

Dominance Functions for Parametric Lagrangians

We discuss the concept of dominance functions for parametric Lagrangians together with important examples and various applications to the existence and regularity theory for minimizers of parametric functionals, and for the construction of unstable stationary surfaces. The focus lies on the construction of a perfect dominance function based on ideas of Morrey.

Mathematical aspects of ?t Hooft?s eigenvalue problem in two-dimensional quantum chromodynamics: Part I. A variational approach, and nodal properties of the eigenfunctions

We consider the eigenvalue problem of t′ Hooft for the meson spectrum in 2-dimensional QCD. Various alternative formulations are discussed, and their equivalence is proved. Then, a variational characterization of the eigenfunctions and the eigenvalues is derived yielding that the spectrum is discrete and consists of denumerably many positive eigenvalues tending to infinity. The corresponding eigenfunctions are real analytic, and form a complete system in L2 Finally, the number of nodes of each eigenfunctions is estimated.