On Discrete Constant Mean Curvature Surfaces

Recently, a curvature theory for polyhedral surfaces has been established that associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gaussian image face. Therefore, a study of constant mean curvature (cmc) surfaces requires studying pairs of polygons with some constant nonvanishing value of the discrete mean curvature for all faces. We focus on meshes where all faces are planar quadrilaterals or planar hexagons. We show an incidence geometric characterization of a pair of parallel quadrilaterals having a discrete mean curvature value of ?1. This characterization yields an integrability condition for a mesh being a Gaussian image mesh of a discrete cmc surface. Thus, we can use these geometric results for the construction of discrete cmc surfaces. In the special case where all faces have a circumcircle, we establish a discrete Weierstrass-type representation for discrete cmc surfaces.     

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.     

Translationsflächen in der äquiaffinen Differentialgeometrie

We discuss with equiaffine methods the surfaces of translation with plane generating curves in the three-dimensional affine space. Using (pseudo-) isothermic parameters we determine in this class all the affine minimal surfaces (which include the affine spheres, the quadrics and the ruled surfaces), all the surfaces with vanishing affine Gauss curvature (which include the surfaces with constant non vanishing affine mean curvature), and all the surfaces with only one family of affine lines of curvature.     

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.     

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

Minimal surfaces,Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S ^4m .     

Semi-Discrete Isothermic Surfaces

We study mappings of the form ${x : \mathbb{Z}\times\mathbb{R}\to\mathbb{R}^3}$ which can be seen as a limit case of purely discrete surfaces, or as a semi-discretization of smooth surfaces. In particular we discuss circular surfaces, isothermic surfaces, conformal mappings, and dualizability in the sense of Christoffel. We arrive at a semi-discrete version of Koenigs nets and show that in the setting of circular surfaces, isothermicity is the same as dualizability. We show that minimal surfaces constructed as a dual of a sphere have vanishing mean curvature in a certain well-defined sense, and we also give an incidence-geometric characterization of isothermic surfaces.     

Laguerre geometry of surfaces with plane lines of curvature

We study surfaces with plane lines of curvature in the framework of Laguerre geometry and provide explicit representation formulae for these surfaces in terms of a potential function. As an application, we explicitly integrate allL- minimal surfaces with plane curvature lines. Partially supported by MURST 40.     

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.     

Enclosure theorems for extremals of elliptic parametric functionals

In the following paper we study parametric functionals. First we introduce a generalized mean curvature (so called F-mean curvature). This enables us to describe extremals of parametric funcionals as surfaces of prescribed F-mean curvature. Furthermore we give a differential equation for arbitrary immersions generalizing and apply this equation to surfaces of vanishing and prescribed F-mean curvature. Especially we prove non-existence results for such surfaces generalizing Theorems by Hildebrandt and Dierkes [3], [6]. Received: 11 May 2001 / Accepted: 11 July 2001 / Published online: 12 October 2001     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

Minimal Surfaces in the Heisenberg Group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot–Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.     

Generalized surfaces with constant <Emphasis Type="Italic">H</Emphasis>/<Emphasis Type="Italic">K</Emphasis> in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Björling formula to construct such surfaces with prescribed point or curve singularities.     

Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in ?³ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.     

Penrose-type inequalities with a Euclidean background

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.     

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.

     

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.