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.     

Gaussian curvature on singular surfaces

We consider prescribing Gaussian curvature on surfaces with conical singularities in both critical and supercritical cases. First we prove a variant of Kazdan-Warner type necessary conditions. Then we obtain sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai singular metric. We only require that the values of the function are not too large at singular points of the metric with the smallest angle, say, less or equal to 0, or less than its average value. To prove the results, we apply some new ideas and techniques. One of them is to estimate the total curvature along a certain minimizing sequence by using the “Distribution of Mass Principle” and the behavior of the critical points at infinity.     

Decay of the Fourier transform of surfaces with vanishing curvature

We prove L ^p -bounds on the Fourier transform of measures μ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that \({\hat{\mu}\in L^{4+\beta}}\) , β > 0, and we give a logarithmically divergent bound on the L ⁴-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, \({e(p)= \sum_1^3 [1-\cos p_j]}\) , of the discrete Laplace operator on the cubic lattice. We briefly explain our motivation for this bound originating in the theory of random Schrödinger operators.     

On square integrability of mean curvature for surfaces with positive Gaussian curvature

We show that {ie319-1} H ²dµ = for any complete surface M R ³ which has positive curvature outside a compact subset of R ³. This proves a conjecture of Friedrich.     

On Peterson surfaces of constant Gaussian curvature

The surfaces of constant Gaussian curvature bearing conjugate networks of conic lines are found.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 3–5.     

Ruled surfaces with small class

Some progress in the study of smooth complex projective ruled surfaces S?P^Nwith low class m is obtained by means of adjunction theory. In particular surfaces of degree d and class m ≤ 2d + 2 are completely classified and the result is extended to higher dimensions.     

On diameters of convex surfaces with Gaussian curvature bounded from below

Translated from Sibirskii Matematicheskii, Vol. 36, No. 1, pp. 93–101, January–February, 1995.     

The semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly

A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.     

On stable hypersurfaces with vanishing scalar curvature

We will prove that there are no stable complete hypersurfaces of $\mathbb {R}^4$ with zero scalar curvature, polynomial volume growth and such that $\frac{(-K)}{H^3}\ge c>0$ everywhere, for some constant $c>0$ , where K denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. Our second result is the Bernstein type one there is no entire graphs of $\mathbb {R}^4$ with zero scalar curvature such that $\frac{(-K)}{H^3}\ge c>0$ everywhere. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and $\frac{(-K)}{H^3}\ge c>0$ everywhere, that is, with volume growth larger than polynomial growth of order four, then its tubular neighborhood is not embedded for suitable radius.     

Sasakian manifolds with vanishing C-Bochner curvature tensor

Supported by TGRC-KOSEF.     

Ruled Laguerre minimal surfaces

A Laguerre minimal surface is an immersed surface in ${\mathbb{R}^3}$ being an extremal of the functional ${\int (H^2/K-1)dA}$ . In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces ${\mathbf{R}(\varphi,\lambda) = ( A\varphi,\, B\varphi,\, C\varphi + D\cos 2\varphi\, ) + \lambda\left(\sin \varphi,\, \cos \varphi,\, 0\,\right)}$ , where ${A,B,C,D\in \mathbb{R}}$ are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.     

Gaussian graphical models with toric vanishing ideals

Annals of the Institute of Statistical Mathematics - Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout...     

On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak’s classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak’s theorem to the case of finite characteristic.     

Extension of a Riemannian metric with vanishing curvature

Let $\text{Ω}$ be a connected and simply-connected open subset of $\text{R}{}^{\mathrm{n}}$ such that the geodesic distance in $\text{Ω}$ is equivalent to the Euclidean distance. Let there be given a Riemannian metric (g_ij) of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$, such that the functions g_ij and their partial derivatives of order $\text{?}\text{2}$ have continuous extensions to $\text{Ω}$. Then there exists a connected open subset $\text{Ω}$ of $\text{R}{}^{\mathrm{n}}$ containing $\text{Ω}$ and a Riemannian metric $\text{(}\text{g}\text{?}{}_{\mathrm{ij}}\text{)}$ of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$ that extends the metric (g_ij). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).