Friedrich Sauvigny

Introduction of isothermal parameters into a Riemannian metric by the continuity method

We shall introduce conformal parameters into a nonanalytic Riemannian metric on the closed disc by the continuity method. Here we solve first order equations by methods from I.N.Vekua and consider second order systems by methods from E.Heinz for the proof of this uniformization theorem. AMS Subject Classification: 35F05, 35J60.

Curvature estimates for immersions of minimal surface type via uniformization and theorems of Bernstein type

We give an adequate parametric description of surfaces of minimal surface type, satisfying the weighted relation ϱ1κ1+ϱ2κ2 with the positive factors ϱj for their principal curvatures κj, by the introduction of weighted conformal parameters. We then establish apriori estimates of the principal curvatures for certain classes of surfaces. These estimates imply new theorems of Bernstein type.

On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's problem

Assuming stability and integral conditions we show that a sequence of immersed surfaces of constant mean curvatureH converges to an immersedH-surface. The latter theorem depends on an oscillation estimate forH-surfaces based on an isoperimetric inequality. These compactness results are utilized to prove that certain Jordan curvesΓ only bound finitely many stable and unstable, immersed, smallH-surfaces.

ON THE TOTAL NUMBER OF BRANCH POINTS OF QUASI-MINIMAL SURFACES BOUNDED BY A POLYGON

The minimal surfaces spanning a polygon can be embedded into a family of quasi-minimal surfaces, which are harmonic but not necessari ly conformally parametrized and sa t i s f y certain generalized boundary condit ions. For these surfaces we can express the total number of branch points by the i r boundary-angles and a certain curvature integra l . AMS-Class i f icat ion: 49F10, 53A10. 1. The main resu l t As in the introduction of [3] l e t B: = {w = u + iv e C||w| < 1} be the unit disc and Γ<=Κ^ (ρ ä 2) a polygon with the consecutive vert ices

On the Morse index of minimal surfaces in ℝp with polygonal boundaries

The minimal surfaces spanning a polygon in ℝp (p≧2) correspond to the critical points of an analytic function Θ in finitely many variables, namely Shiffmans function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of Θ at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of Θ have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.

An energy estimate for the difference of solutions for the n-dimensional equation with prescribed mean curvature and removable singularities

Abstract We derive an energy bound, estimating a weighted Dirichlet integral of two solutions for the nonparametric equation with prescribed mean curvature in n dimensions in terms of the L1-norm for the difference of their values on the boundary. Furthermore, a similar estimate is established for solutions of the equation divFp(·,∇u)=n H(·,u), where F(x,p) denotes an elliptic Lagrangian with linear growth in p. These results are used to remove singularities of solutions to these equations.

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.

A new proof of the gradient estimate for graphs of prescribed mean curvature

The aim of this paper is to give a new proof of the gradient estimate for graphs of prescribed mean curvatureH=H(x,y,z). Similarly as in [2] where the caseH=H(x,y) is studied, we introduce conformal parameters for the surface. Then we employ the differential equation for the unit normal of the surface derived in [3] Satz 1. By this method, which is contained in [4] Satz 4, we prove the following

Nonlinear Partial Differential Equations

In this chapter we consider geometric partial differential equations, which appear for two-dimensional surfaces in their state of equilibrium. Here we give the differential-geometric foundations in Section 1 and determine in Section 2 the Euler equations of 2-dimensional, parametric functionals. In Section 3 we present the theory of characteristics for quasilinear hyperbolic differential equations, and Section 4 is devoted to the solution of Cauchy’s initial value problem with the aid of successive approximation. In Section 5 we treat the Riemannian integration method for linear hyperbolic differential equations. Finally, we prove S. Bernstein’s analyticity theorem in Section 6 using ideas of H. Lewy.

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.