Zur Theorie verallgemeinerter torsaler Strahlflächen

In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . Some results which are well-known for developable surfaces are proved for generalized ruled surfaces: Generalized developable surfaces are locally either cyclinders, cones or tangent surfaces. Each regular surface on a generalized ruled surface is locally Euclidean if and only if is developable. Each locally Euclidean hypersurface is a generalized developable hypersurface. Furthermore, the hypersurfaces with vanishing Gaussian curvature and the locally Euclidean hypersurfaces on generalized rule hypersurfaces will be characterized.     

Minding-Isometrien bei (k+1)-Regelflächen

In this paper we investigate (k+1)-dimensional generalized ruled surfaces generated by a oneparameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE. Minding-isometries of ruled surfaces are special isometries, which let the generating space invariant. Some new results will be given, concerned with Mindingisometries of generalized ruled surfaces of arbitrary codimension. In this investigations quadratic hypersurfaces in the normal spaces are of great importance.     

Spherical 2-type hypersurfaces

A hypersurface (not necessarily compact) of a hypersphereS ⁿ⁺¹ of a Euclidean spaceE ⁿ⁺² is of 2-type if and only if it has constant nonzero mean curvature inS ⁿ⁺¹ and constant scalar curvature, unless it is a portion of a small hypersphere inS ⁿ⁺¹. This shows that the 2-type compact hypersurfaces of a hypersphere are mass-symmetric.     

A pre-hilbert space consisting of classes of convex sets

An equivalence relation is defined in the set of all bounded closed convex sets in Euclidean spaceE ⁿ. The equivalence classes are shown to be elements of a pre-Hilbert spaceA ⁿ, and geometrical relationships betweenA ⁿ andE ⁿ are investigated.     

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS ₁ ⁿ⁺¹ (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result. Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua is supported by NNSFC No. 19571005     

An inequality for a simplex and its applications

In this paper, we improve some inequalities for ann-dimensional simplex in Euclidean spaceE ⁿ and give some applications.     

Zur axiomatischen charakterisierung des steinerpunktes konvexer körper

It is shown that the Steiner point is the only point that can be associated additively, homothety-equivariantly and continuously with any convex body in Euclidean spaceE ⁿ.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^P-norm Curvature in R^n＋1

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.     

Three geometric inequalities for a simplex

In this paper, we obtain three new geometric inequalities for ann-dimensional simplex in then-dimensional Euclidean spaceE ⁿ . As special cases we find two known inequalities from L. Fejes Tóth and M. S. Klamkin, respectively.     

An extension of Takahashi's theorem

A classical result of T. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ^m . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E ^m are eigenfunctions of their Laplacian.Partially supported by a CAICYT Grant PR84-1242-C02-02 Spain.     

Stability index jump for constant mean curvature hypersurfaces of spheres

It is known that the totally umbilical hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S ⁿ⁺¹, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface ${M \subset S^{{n+1}}}$ with cmc cannot take the values 1, 2, 3 . . . , n.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Immersions of Finite Geometric Type in Euclidean Spaces

In this paper, we introduce the class of hypersurfaces of finitegeometric type. They are defined as the ones that share the basicdifferential topological properties of minimal surfaces of finite totalcurvature. We extend to surfaces in this class the classical theorem ofOsserman on the number of omitted points of the Gauss mapping ofcomplete minimal surfaces of finite total curvature. We give aclassification of the even-dimensional catenoids as the only even-dimensional minimal hypersurfaces of R ⁿ of finite geometric type.     

Reconstructing Finite Sets of Points in Rup to Groups of Isometries

We prove reconstruction results for finite sets of points in the Euclidean spaceRⁿthat are given up to the action of groups of isometries that contain all translations and for which the origin has a finite stabilizer.     

An upper bound for the minimum diameter of integral point sets

For alln> d there existn points in the Euclidean spaceE ^d such that not all points are in a hyperplane and all mutual distances are integral. It is proved that the minimum diameter of such integral point sets has an upper bound of 2 ^{c logn log logn} .     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \mathbb Sⁿ⁺¹{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.     

Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces M ⁿ with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N ⁿ⁺¹. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M ⁿ , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with , respectively, has only one end.     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001