到欧氏空间的等距极小浸入

本文研究了欧氏空间极小子流形的测地球的体积增长,给出了一个黎曼流形可等距极小浸入到欧氏空间的一个必要条件,并给出了具非正截曲率的欧氏空间极小超曲面的一个分类．     

Some Bernstein-type rigidity theorems

We prove some Bernstein-type rigidity theorems for complete submanifolds in a Euclidean space and space-like submanifolds of a Lorentzian space. In particular, we obtain a Bernstein rigidity theorem for complete minimal submanifolds of arbitrary codimension in Euclidean space.     

Isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space

In this article we consider the problem of isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space. For each of the possible topological types the corresponding minimal dimension of the extending Euclidean space is indicated. See [3].Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 427–429, March, 1973.     

Embedding problems: Geometric and topological aspects

A survey of results obtained after 1976 on questions of the isometric immersions of Riemannian spaces in a Euclidean space, the immersions and embeddings of differential manifolds, and the immersions with minimal absolute curvature is presented.     

Euclidean semi-matchings of random samples

A linear programming relaxation of the minimal matching problem is studied for graphs with edge weights determined by the distances between points in a Euclidean space. The relaxed problem has a simple geometric interpretation that suggests the name minimal semi-matching. The main result is the determination of the asymptotic behavior of the length of the minimal semi-matching. It is analogous to the theorem of Beardwood, Halton and Hammersley (1959) on the asymptotic behavior of the traveling salesman problem. Associated results on the length of non-random Euclidean semi-matchings and large deviation inequalities for random semi-matchings are also given.Research supported in part by NSF Grant #DMS-8812868, ARO contract DAAL03-89-G-0092.P001, AFOSR-89-08301.A and NSA-MDA-904-89-2034.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H^n

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space Hn is studied.     

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of . We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.     

Simply Connected Minimal Surfaces in R3 Transverse to Horizontal Planes

We consider simply connected minimal surfaces in Euclidean space and we give a characterisation of the helicoid.     

Calculating a minimal sphere containing a polytope defined by a system of linear inequalities

We will propose an algorithm for calculating a minimal sphere containing a polytope defined by a system of linear inequalities in low dimensional Euclidean space. This algorithm is a straightforward application of the algorithm for maximizing a convex quadratic function over a polytope. It will be shown that this algorithm successfully generates a minimal sphere when the dimensions of the underlying space is up to five.International Digital Communication Inc.     

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa (Kyushu J Math 52(1):167?C185, 1998) and Jiang (Chin Ann Math Ser. 8A:376?C383, 1987) states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper, Caddeo et?al. (Isr J Math 130:109?C123, 2002) showed that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic.     

Invariant singular minimal surfaces

We classify all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations.     

Laguerre minimal surfaces,isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.     

偶数维复Clifford代数中的Dirac旋量空间

本文引入了偶数维欧氏空间的复结构及Witt基,在此基础上讨论了偶数维复Clifford代数中的Dirac旋量空间.由Fock空间的结果我们得到了Dirac旋量空间视为复Clifford代数中极小左理想,最后我们研究了Dirac旋量空间的对偶空间.     

关于空间型中的正螺面

将正螺面引入到一般的空间型中,在Beltrami-Klein坐标系下统一加以表示,并一致地详尽考察其度量、法向、第二基本形式及主曲率,推出正螺面是极小曲面.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

Sensitivity analysis in vector optimization

For a vector optimization problem that depends on a parameter vector, the sensitivity analysis of perturbation, proper perturbation, and weak perturbation maps is dealth with. Each of the perturbation maps is defined as a set-valued map which associates to each parameter value the set of all minimal, properly minimal, and weakly minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering cone. Using contingent cones in a finite-dimensional Euclidean space, we investigate the relationship between the contingent derivatives of the three types of perturbation maps and three types of minimal point sets for the contingent derivative of the feasible-set map in the objective space. These results provide quantitative informations on the behavior of the perturbation maps.The authors would like to thank the referees for their valuable comments and suggestions.     

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H n

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space H^n is studied.