Leon Simon

Spacelike hypersurfaces with prescribed boundary values and mean curvature

We consider the boundary-value problem for the mean curvature operator in Minkowski space, and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions. Our main results hold for non-constant mean curvature, and make no assumptions about the smoothness of the boundary or boundary data.

Minimal surfaces with isolated singularities

For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone and is stable in case the cone satisfies a strict stability inequality.

Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals

Introduction In this paper we study the structure of n dimensional rectifiable currents in R n+l which minimize the integrals of parametric elliptic integrands. The existence of such minimizing surfaces is well known [7, 5.1.6] as is their regularity almost everywhere [7, 5.3.19]. In Par t I of the present paper we give a new geometric construction from which regularity estimates can be obtained for minimizing hypersurfaces. In this construction we replace the parametric problem for n dimensional surfaces in R ~§ by a nonparametric problem for which the minimizing hypersurfaee is a graph in R n§ with horizontal slices closely approximating in a certain sense the hypersuffaee(s) minimizing the original problem. Analysis of the associated Euler-Lagrange partial differential equation carried out in w 2 of Part I yields an upper bound for the integral of the square of the second fundamental form over the approximating graphs, hence over the regular parts of the original surface. Since a neighbourhood of a singular point must contribute substantially to this integral (see Theorem 1.3 and the remark following it), we can thus conclude by an argument similar to that given by Miranda [13] tha t the Hausdorff ( n 2)-dimensional measure of the interior singular set is locally finite (Theorem 3.1). In Par t I I of this work we show that the singular sets in question must have Hausdorff

A Hölder estimate for quasiconformal maps between surfaces in Euclidean space

In [2] C. B. Morrey proved a H61der estimate for quasiconformal mappings in the plane. Such a HSlder estimate was a fundamental development in the theory of quasiconformal mappings, and had very important applications to partial differential equations. L. Nirenberg in [3] made significant simplifications and improvements to Morreys work (in particular, the restriction that the mappings involved be 1 1 was removed), and he was consequently able to develop a rather complete theory for second order elliptic equation with 2 independent variables. In Theorem (2.2) of the present paper we obtain a H61der estimate which is analogous to that obtained by Nirenberg in [3] but which is applicable to quasiconformal mappings between surface~ in Euclidean space. The methods used in the proof are quite analogous to those of [3], although there are of course some technical difficulties to be overcome because of the more general setting adopted here. In w 3 and w 4 we discuss applications to graphs with quasiconformal Gauss map. In this case Theorem (2.2) gives a H61der estimate for the unit normal of the graph. One rather striking consequence is given in Theorem (4.1), which establishes the linearity of any C2(R *) function having a graph with quasiconformal Gauss map. This result includes as a special case the classical theorem of Bernstein concerning C2(R 2) solutions of the minimal surface equation, and the analogous theorem of Jenkins [1] for a special class of variational equations. There are also in w 3 and w 4 a number of other results for graphs with quasiconformal Gauss map, including some gradient estimates and a global estimate of H61der continuity. w 4 concludes with an application to the minimal surface system. One of the main reasons for studying graphs satisfying the condition that the Gauss map is quasieonformal (or (A1, A2)-quasiconformal in the sense of (1.8) below) is tha t such

Boundary regularity and embedded solutions for the oriented Plateau problem

BOUNDARY REGULARITY AND EMBEDDED SOLUTIONS FOR THE ORIENTED PLATEAU PROBLEM BY ROBERT HARDT AND LEON SIMON Any fixed C Jordan curve F in R is known to span an orientable minimal surface in several different senses. In the work of Douglas, Rado and Courant (see e.g. [3, IV, §4]) the minimal surface occurs as an area-minimizing mapping from a fixed orientable surface of finite genus and may possibly have self-intersections. In the work of Fédérer and Fleming (see e.g. [4, §5]) the minimal surface, which occurs as the support of an area-minimizing rectifiable current, is necessarily embedded (away from T) but was not previously known even to have finite genus. Our work in [7], which establishes complete boundary regularity for the latter surface, thus implies that there exists an orientable embedded minimal surface with boundary T. In fact:

Rectifiability of the singular set of energy minimizing maps

The regularity theory of Schoen and Uhlenbeck [SU] for energy minimizing maps u from a domain ~2 C R n (equipped with any smooth Riemannian metric) into a compact smooth Riemannian target manifold N , established that the singular set s ingu always has Hausdorff dimension _ 0, where ~ s denotes s-dimensional Hausdorff measure. Similar results were obtained independently by Giaquinta and Giusti [GG] in the case when the image is contained in a coordinate chart. In case the target manifold N is real analytic, the main theorems of this paper (Theorems 1-3 below) establish rectifiability properties for such singular sets in the dimension n 3, and in other dimensions m < n 3 in case N happens to be such that all energy minimizing maps u into N have dim sing u < m. Recall that a subset A C R n is said to be m-rectifiable if ,~flm(A) < oo, and if A has an approximate tangent space a.e. in the sense that for ~ m a . e . z E A there is an m-dimensional subspace Lz such that

Theorems on the Regularity and Singularity of Minimal Surfaces and Harmonic Maps

These lectures are meant as an introduction to the analytic aspects of the study of regularity properties and singularities of minimal surfaces and harmonic maps.

Regularity Theory for Harmonic Maps

Suppose that Ω is an open subset of ℝ n , n ≥ 2, and that N is a smooth compact Riemannian manifold of dimension m ≥ 2 which is isometrically embedded in some Euclidean space ℝ p . We look at maps u of Ω into N; such a map will always be thought of as a map u = (u1,…, u p ): Ω → ℝ p with the additional property that u(Ω) ⊂ N.

Approximation Properties of the Singular Set

In this chapter u continues to denote an energy minimizing map of Ω into N, with Ω an open subset of ℝn.