Unfolding of Surfaces

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of ) with a triangulation. We prove the following result: let T_n be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of T_n tend to the normals of S, then the shape of the unfolding of T_n tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges.     

On the convergence of metric and geometric properties of polyhedral surfaces

We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace–Beltrami operators. Additionally, we derive convergence of minimizing geodesics, mean curvature vectors, and solutions to the Dirichlet problem. This work was supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

On the approximation of a smooth surface with a triangulated mesh

We approximate the normals and the area of a smooth surface with the normals and the area of a triangulated mesh whose vertices belong to the smooth surface. Both approximations only depend on the triangulated mesh (which is supposed to be known), on an upper bound on the smooth surface's curvature, on an upper bound on its reach (which is linked to the local feature size) and on an upper bound on the Hausdorff distance between both surfaces.

We show in particular that the upper bound on the error of the normals is better when triangles are right-angled (even if there are small angles). We do not need every angle to be quite large. We just need each triangle of the triangulated mesh to contain at least one angle whose sinus is large enough.     

Die windschiefen Flächen konstanten Dralls in der Normalenkongruenz einer Fläche

In the congruence of surface normals of a given surface, the ruled surfaces for which the parameter of distribution has the constant value =0 or = are known to be developables and their base curves on the surface are the lines of curvature. In this paper a characterization is given of the ruled surfaces in the congruence of normals for which the parameter of distribution has a constant value (o,). If the given surface is developable the base curves may be characterized by a simple integral representation. If the given surface is not developable a characterization of the base curves is possible by means of the strips circumscribed along the base curves. Moreover the striction lines and torsal generators of these ruled surfaces are studied.

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Herrn Professor Dr. W. Wunderlich zum 60. Geburtstag gewidmet     

Harmonicity of gradient mappings of level surfaces in a real affine space

In our previous paper [4] we have investigated level surfaces of a non-degenerate function in a real affine space A ⁿ⁺¹ by using the gradient vector field . We gave characterizations of by means of the shape operatorS, the transversal connection , and studied the difference between and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and =0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].     

Gradient-like and integrable vector fields on ℝ2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ² null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.Partially supported by a grant from the Weldon Springs endowment of the University of Missouri-Columbia     

Nontrivial minimal surfaces in a hyperbolic Randers space

The contribution of this paper is two‐fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing vector field, by considering the Busemann–Hausdorff measure and Holmes–Thompson measure simultaneously. The second one is to obtain the explicit local expressions of two types of nontrivial rotational BH‐minimal surfaces in a Randers domain of constant flag curvature , which are the first examples of BH‐minimal surfaces in the hyperbolic Randers space.     

Intrinsic third derivatives for Plateau's problem and the Morse inequalities for disc minimal surfaces in ℝ3

Summary A simply branched minimal surface in ³ cannot be a non-degenerate critical point of Dirichlet's energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Böhme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is ±2 ^p , wherep is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in ³, thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in ⁴ arbitrarily close to the given contour must span at least 2 ^p disc minimal surfaces.     

On the Projective Normality of Smooth Surfaces of Degree Nine

The projective normality of smooth, linearly normal surfaces of degree 9 in ^N is studied. All nonprojectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given.     

Uniqueness of Noncompact Spacelike Hypersurfaces of Constant Mean Curvature in Generalized Robertson–Walker Spacetimes

On any spacelike hypersurface of constant mean curvature of a Generalized Robertson–Walker spacetime, the hyperbolic angle between the future-pointing unit normal vector field and the universal time axis is considered. It is assumed that has a local maximum. A physical consequence of this fact is that relative speeds between normal and comoving observers do not approach the speed of light near the maximum point. By using a development inspired from Bochner's well-known technique, a uniqueness result for spacelike hypersurfaces of constant mean curvature under this assumption on , and also assuming certain matter energy conditions hold just at this point, is proved.     

Edge Flips in Surface Meshes

Little theoretical work has been done on edge flips in surface meshes despite their popular usage in graphics and solid modeling to improve mesh equality. We propose the class of \((\varepsilon ,\alpha )\)-meshes of a surface that satisfy several properties: the vertex set is an \(\varepsilon \)-sample of the surface, the triangle angles are no smaller than a constant \(\alpha \), some triangle has a good normal, and the mesh is homeomorphic to the surface. We believe that many surface meshes encountered in practice are \((\varepsilon ,\alpha )\)-meshes or close to being one. We prove that flipping the appropriate edges can smooth a dense \((\varepsilon ,\alpha )\)-mesh by making the triangle normals better approximations of the surface normals and the dihedral angles closer to \(\pi \). Moreover, the edge flips can be performed in time linear in the number of vertices. This helps to explain the effectiveness of edge flips as observed in practice and in our experiments. A corollary of our techniques is that, in \(\mathbb {R}^2\), every triangulation with a constant lower bound on the angles can be flipped in linear time to the Delaunay triangulation.     

Surface Reconstruction by Voronoi Filtering

We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs. Received August 25, 1998, and in revised form March 31, 1999.     

Classification of marginally trapped Lorentzian flat surfaces in and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in with light-like mean curvature vector.     

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric g _h on given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (). By following the ideas and techniques by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group , arXiv:math.DG/0512547) we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot–Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C ² compact, connected, embedded surfaces in () with empty singular set and constant mean curvature H such that is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (). A. Hurtado has been partially supported by MCyT-Feder research project MTM2004-06015-C02-01. C. Rosales has been supported by MCyT-Feder research project MTM2004-01387.     

Quadrature formulas with minimum error for certain classes of functions

Problems connected with optimum quadrature formulas for the function classes H and H ⁿ are investigated.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 577–586, May, 1968.     

Automatic continuity of linear maps on spaces of continuous functions

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map AC(T)C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).     

On the semidiscrete differential geometry of A-surfaces and K-surfaces

In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface??s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.     

Surfaces in $$\mathbb {R}^7$$ obtained from harmonic maps in $$S^6$$S6.

We will investigate the local geometry of the surfaces in the 7-dimensional Euclidean space associated to harmonic maps from a Riemann surface \(\varSigma \) into \(S^6\). By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions \(\varphi :\varSigma \rightarrow S^6\) whose associated immersions \(F:\varSigma \rightarrow \mathbb {R}^7\) belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.     

The Bernstein problem for complete Lagrangian stationary surfaces

In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and mean curvature has finite total absolute curvature.