Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $ \mathbb{E}_2^4 $ \mathbb{E}_2^4 and in neutral pseudo 4-sphere S ₂⁴ (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H ₂⁴ (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H ₂⁴ (−1). Conversely, every parallel Lorentz surface in H ₂⁴ (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.     

Submanifolds in de sitter space-time satisfying ΔH=λH

In [3] the author initiated the study of submanifolds whose mean curvature vectorH is an eigenvector of the Laplacian Δ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. The classification of surfaces with ΔH=λH in a Euclidean 3-space was done by the author in 1988. Moreover, in [4] the author classified such submanifolds in hyperbolic spaces. In this article we study this problem for space-like submanifolds of the Minkowski space-timeE ₁ ^m when the submanifolds lie in a de Sitter space-time. As a result, we characterize and classify such submanifolds in de Sitter space-times.     

New examples of imbedded spherical soap bubbles inS N (1)

A. D. Alexandrov proved that the only imbedded soap bubbles in euclidean or hyperbolicn-space are round spheres. This also holds for ann-dimensional hemisphere. In this paper we give many new examples of differentiable (n–1)-spheres imbedded as soap bubbles in sphericaln-space. Thus we show Alexandrov's theorem is false for sphericaln-space even under the added assumption that the soap bubbles are differentiable (n–1)-spheres.The author would like to thank the referee for many helpful comments and corrections     

Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces in the de Sitter space.     

THE GEOMETRY OF HYPERSURFACES IN A KAEHLER MANIFOLD

Let M be a complex analytic manifOld there is given a positive definite quadratic differentialform[1]dS2 = gjkdZJdZ* (1)where the Latin indices j, k take the values 1,2,' l n; 1, 2,' t n. Assume the Greek indices o, Ptake the value8 1,2,', n and.=. l cr if i= al = 1 cr if i= a, (2)t cr if i = cr.Assume now that the symmetric tensor gjk is selfadoint(see below Definitioll l), that is-- --gap = g95 = go0 = g9rr, (3)g.0 = gPcr = gap = g05. (4)and 8atisfiesgoP = gap = 0. (5)From the com…     

Exceptional sets on del Pezzo surfaces with one log-terminal singularity

New full exceptional sets of coherent sheaves on a certain family of log-terminal del Pezzo surfaces, which is treated as a smooth stack, are constructed. These surfaces are not toroidal and can be represented as hypersurfaces in weighted projective 3-space.     

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

We prove a sharp inequality for hypersurfaces in the n‐dimensional anti‐de Sitter‐Schwarzschild manifold for general n ≥ 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three‐dimensional euclidean space and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric inequality established by the first author in [3].© 2015 Wiley Periodicals, Inc.     

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.     

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.     

Biconservative surfaces in BCV‐spaces

Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.     

A generalization of translation surfaces with constant curvature in the isotropic space

In this paper we study the translation surfaces generated by a space curve and a planar curve in the isotropic 3-space \({\mathbb{I}^{3}}\). We completely classify such surfaces in \({\mathbb{I}^{3}}\) with constant curvature. Several examples are also given by figures.     

Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution, one of the Riemann minimal examples, or a generalized cone. Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain     

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space. Received: 25 January 2001 / Accepted: 23 February 2001 / Published online: 25 June 2001     

New bounds for lower envelopes in three dimensions,with applications to visibility in terrains

We consider the problem of bounding the complexity of the lower envelope ofn surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope ofn such surface patches is , for some constantc depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the “lower envelope” of the space of allrays in 3-space that lie above a given polyhedral terrainK withn edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is for some constantc; in particular, there are at most that many rays that pass above the terrain and touch it in four edges. This bound, combined with the analysis of de Berget al. [4], gives an upper bound (which is almost tight in the worst case) on the number of topologically different orthographic views of such a terrain. Work on this paper by the first author has been supported by a Rothschild Postdoctoral Fellowship. Work on this paper by the second author has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Verallgemeinerte Regelflächen im Grossen II

In Euclidean n-space we prove two vertex theorems for simply closed generalized ruled surfaces. The statements refer to the minimum of generators with special geometrical properties, and generalize theorems of SABAN[9] about ruled surfaces in Euclidean 3-space. Two examples are given.

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Herrn Prof. Dr. Karl Strubecker zum 80. Geburtstag     

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.     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

Total Curvature for <Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript>-Singular Surfaces with Limiting Tangent Bundle

The total curvature of a compact C^∞-immersed surface in Euclidean 3-space ³ can be interpreted as the average number of critical points for a linear ‘height’ function. The Morse inequalities provide an intrinsic topological lower bound for the total curvature and ‘tight’ surfaces, which realize equality, have been an active topic of research. The objective of this paper is to describe the natural notion of total curvature for C^∞-singular surfaces which fail to immerse on C^∞-embedded closed curves, but which have a C^∞-globally defined unit normal (e.g. caustics, or critical images for mappings of 3-manifolds into Euclidean 3-space). For such surfaces total curvature consists of a sum of two-dimensional and one-dimensional integrals, which have various lower bounds. Large sets of LT-surfaces which realize equality are then constructed. As an application, the orthogonal projection of an immersed tight hypersurface in Euclidean 4-space is shown to have LT-tight critical image, and several related inequalities are given. Mathematics Subject Classifications (2000): 57N65, 14P99, 53C21, 53B25, 53B20.