Integral curvature of a curve in n-dimensional Euclidean space

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 3–22, January–February, 1988.     

The measure of solid angles in n-dimensional Euclidean space

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

Stationary rotating surfaces in Euclidean space

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)² + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.     

Maximal packing density of n-dimensional Euclidean space with equal balls

An estimate _n 2^{-n(0.5237+o(1))} is obtained for the maximal packing density of n-dimensional Euclidean space with equal balls for n . This result is based on an improvement in a corresponding upper estimate for the maximal packing density of the unit (n-1)-dimensional sphere with spherical caps of fixed angular radius.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 301–311, August, 1975.     

On maximal surfaces in the n-dimensional Lorentz-Minkowski space

Complete maximal surfaces in the n-dimensional Lorentz-Minkowski space are studied from the behaviour of their normal vectors. Moreover, several examples of maximal surfaces are constructed.Research partially supported by DGICYT Grant PS87-0115-C03-02.     

Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace–Beltrami operator on closed surfaces of unit area.     

Ends of negatively curved surfaces in Euclidean space

We examine the geometry of a complete, negatively curved surface isometrically embedded in ${\mathbb{R}^3}$ . We are especially interested in the behavior of the ends of the surface and its limit set at infinity. Various constructions are developed, and a classification theorem is obtained, showing that every possible end type for a topologically finite surface with at least one bowl end arises, as well as all infinite type surfaces with a single nonannular end. Some other examples are given with oddly behaved bowl ends.     

Tight surfaces in three-dimensional compact Euclidean space forms

In this paper we define and discuss tight surfaces -- smooth or polyhedral -- in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.

     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Cyclic and ruled Lagrangian surfaces in Euclidean four space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces.     

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.     

The differential properties of one class of surfaces in Euclidean space

We consider n-dimensional smooth surfaces of class ℰ¹ in the Euclidean space of dimension n + m satisfying the following condition. Given two distinct points of the surface, the surface normals at these points either are disjoint or meet at the distance from both of these points bounded below by some fixed positive constant. We establish that every surface of this type carries in a neighborhood of each point a parametrization with bounded second order generalized derivatives in the sense of Sobolev. The proof is based on using geometric properties of the surfaces of this form and on the proposition that establishes sufficient conditions for the existence of bounded second order generalized derivatives of an arbitrary real function. In the Appendix we prove an analog of this lemma in the case of derivatives of arbitrary order.