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Using the mathematical MAPLE package, we construct models of surfaces of revolution of constant Gaussian curvature and geodesic lines on such surfaces.

We classify all surfaces with constant Gaussian curvature K in Euclidean 3-space that can be expressed by an implicit equation of type $$f(x)+g(y)+h(z)=0$$ , where f, g and h are real functions of one variable. If $$K=0$$ , we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If $$K\not =0$$ , we prove that the surface is a surface of revolution.

In this paper, we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution. We employ symbolic manipulations to derive the contact constraint equations from an ordered process, and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations (DAEs), leaving nine coupled differential equations, six contact equations, two holonomic constraint equations and four nonholonomic constraint equations. We then present a complete description of hands-free circular motions, in which the time-dependent variables are eliminated through a rotation transformation. We find that the circular motions, similar to those of the bicycle moving on a horizontal surface, nominally fall into four solution families, characterized by four curves varying with the angular speed of the front wheel. Then, we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface. Furthermore, we directly linearize the nonlinear DAEs, from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates. The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position. We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families. Finally, based on the numerical simulation of the original nonlinear DAEs system, we show that the stable circular motion is not asymptotically stable.

A Ricci flow (M, g(t)) on an n-dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds \((S(t), g_E)\) of a fixed Euclidean space \(E = \mathbb {R}^{n+k}\) is called an extrinsic representation in \(\mathbb {R}^{n+k}\) of (M, g(t)) if there exists a smooth one-parameter family of isometries \((S(t), g_E) \rightarrow (M, g(t))\). When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in \(\mathbb {R}^3\). This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in \(\mathbb {R}^3\). In particular, we exhibit the first explicit extrinsic representations in \(\mathbb {R}^4\) of the Ricci flows initialized by toroidal surfaces of revolution immersed in \(\mathbb {R}^3\).

The cupola (dome) of Santa Maria del Fiore in Florence was ingeniously constructed by Brunelleschi using a conical bricklaying, radial-oriented toward a focus point on the central axis. Therefore, the dome is built as a surface of revolution but with parts cut away to leave the octagonal cluster vault form. This circular arrangement is compared with an octagonal horizontal corbelling in models where the dome is schematized as an assembly of rigid-blocks in frictional contact, analyzed with a Non-Smooth-Contact-Dynamics approach. The high indeterminacy of the contact reactions implies considerable difficulties in their determination, which are faced via a regularization procedure by adding a compliance at the contact points in representation of the deformability of the mortar joints. Numerical experiments, performed with a custom software, highlight the uniform flow of forces in the Brunelleschi arrangement, but evidence the disturbances induced by the herringbone spirals, mainly used for construction purposes, which are overloaded along the meridians and very weak in the direction of the parallels. This is due to the vertical narrow disposal of the blocks, which increases the stiffness in meridional direction, but diminishes the capacity of the friction-induced forces to equilibrate the hoop stress.

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or $$\mathbb {R}^N$$ .

This article proves the analytical inflexibility of regular developable surfaces and doubly connected surfaces of revolution, which are fixed along a curve on the surface simultaneously with respect to a point and a plane.

Calculus of variations and partial differential equations 60(1), 32 (2021). doi:10.1007/s00526-020-01875-6, Published by Springer, Heidelberg

We introduce some new approaches for the graphical assessment of distribution of the data that supplement the existing graphical methods. Analogous to Q–Q plots and P–P plots, we introduce plots based on arc length and area of surface of revolution of the density function. Thus, our method indirectly makes use not only of density assumed but also of the derivatives thereof. We illustrate, by using several examples, that these plots help us identify the correct distribution and also rule out the incorrect possibilities. We further consider the problem of assessing the behavior of the data toward the tail and develop graphical tools to identify the closest potential probability distribution for the tail. Examples based on real data are provided.

A special feature of the Filament Winding (FW) process is known as pattern: diamond-shaped mosaic that results from the sequence of movements of the mandrel and tow delivery eye. One of the main factors to generate different patterns is the return path of the tow and, for a non-geodesic trajectory, the path depends on the friction between tow and mandrel. Aiming at a practical description of the FW process, a novel geometric approach on pattern construction is presented. Pattern generation, skip configurations and definitions of geodesic and non-geodesic trajectories in regular winding and return regions are described based on developed surfaces, residue classes and modular arithmetic. The influence of mandrel’s length, mandrel’s rotation angle and variation of the winding angle in the return region are presented, for they are important parameters of the process. Examples of winding angle, mandrel rotation and non-geodesic path in cylindrical and non-cylindrical surfaces of revolution are shown and discussed.

In this work, we study the $$L^p$$ estimates for a certain class of rough maximal functions with mixed homogeneity associated with the surfaces of revolution. Using these estimates with an extrapolation argument, we obtain some new results that represent substantially improvements and extensions of many previously known results on maximal operators.

Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.

In this paper, we investigate translating solitons for the inverse mean curvature flow (IMCF), which is a special solution deformed only for translation under the flow. The IMCF has been studied extensively not only as a type of a natural geometric flow, but also for obtaining various interesting geometric inequalities. We show that the translating solitons that are either ruled surfaces or translation surfaces are cycloid cylinders, and completely classify 2-dimensional helicoidal translating solitons and the higher dimensional rotationally symmetric translating solitons using the phase-plane analysis. The surface foliated by circles, which is called a cyclic surface, is regarded in terms of being the translating soliton for the IMCF, and then it is a surface of revolution whose revolution axis is parallel to the translating direction. In particular, we extend the result to a higher dimension, namely, the n-dimensional translating soliton foliated by spheres lying on parallel hyperplanes in $$\mathbb {R}^{n+1}$$ must be a rotationally symmetric hypersurface whose rotation axis is parallel to the translating direction.

We show that the construction of a digital sphere by circularly sweeping a digital semi-circle (generatrix) around its diameter results in the appearance of some holes (absentee-voxels) in its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration in the surface of revolution can ensure the required completeness. In this paper, we present a characterization of the absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semi-circular generatrix. Next, we design an algorithm to fill up the absentee-voxels so as to generate a spherical surface of revolution, which is complete and realistic from the viewpoint of visual perception. We also show how the proposed technique for absentee-filling can be used to generate a variety of digital surfaces of revolution by choosing an arbitrary curve as the generatrix. We further show that covering a solid sphere by a set of complete spheres also results to an asymptotically larger count of absentees, which is cubic in the radius of the sphere. A complete characterization of the absentee-voxels that aids the subsequent generation of a solid digital sphere is also presented. Test results have been furnished to substantiate our theoretical findings.

We will show a symmetry result concerning Willmore surfaces of revolution which satisfy symmetric Dirichlet data. The first step is to find a regular energy minimizing surface. We will do this by carefully modifying a minimizing sequence using an idea by Dall’Acqua, Deckelnick & Grunau. After this we will establish a priori estimates for non-symmetric solutions, in which an order-reduction argument by Langer & Singer will be essential.

We study the Gauss map G of surfaces of revolution in the 3-dimensional Euclidean space $${{\mathbb {E}}^3}$$ with respect to the so-called Cheng–Yau operator $$\square $$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map G satisfying $$\square G=AG$$ for some $$3\times 3$$ matrix A are the planes, right circular cones, circular cylinders, and spheres.

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Mobius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.

We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution. Розраховано енергію одиничного нормального векторного поля на рiмановiй поверхні M. Показано, що енергія одиничного нормального векторного поля не залежить від вибору ортогонального базиса в дотичному просторі. Визначено енергію поверхні. Більш того, розраховано енергію сфер, областей на прямому круговому циліндрі та торі і, більш загально, поверхонь обертання.

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.

This paper presents a method to assist in the tedious process of reconstructing ceramic vessels from excavated fragments. The method exploits vessel surface marking information coupled with a series of generic models constructed by the archaeologists to produce a virtual reconstruction of what the original vessel may have looked like. Generic models are generated based on the experts' historical knowledge of the period, provenance of the artifact, and site location. The generic models need not to be identical to the original vessel, but must be within a geometric transformation of it in most parts. By aligning the fragments against the generic models, the ceramic vessels are virtually reconstructed. The alignment is based on a novel set of weighted discrete moments computed from convex hulls of the markings on the surface of the fragments and the generic vessels. When the fragments have no markings on them, they are virtually mended to abutting fragments using intrinsic differential anchor points computed on the surface breaks and aligned using a set of absolute invariants. For axially symmetric objects, a global constraint induced by the surface of revolution is added to guarantee global mending consistency.

In this work, we propose an original resolution of Young–Laplace equation for capillary doublets from an inverse problem. We establish a simple explicit criterion based on the observation of the contact point, the wetting angle and the gorge radius, to classify in an exhaustive way the nature of the surface of revolution. The true shape of the admissible static bridges surface is described by parametric equations; this way of expressing the profile is practical and well efficient for calculating the binding forces, areas and volumes. Moreover, we prove that the inter-particle force may be evaluated on any section of the capillary bridge and constitutes a specific invariant.

We find the exact location of the weighted Fermat–Torricelli point of a geodesic triangle on flat surfaces of revolution (circular cylinder and circular cone) in the three dimensional Euclidean space by applying a cosine law of three circular helixes which form a geodesic triangle on a circular cylinder, an explicit solution of the corresponding weighted Fermat–Torricelli point in the dimensional Euclidean space by calculating some lengths of geodesic arcs and angles and by using some lengths of straight lines on a circular cone which connect the vertices of the geodesic triangle with the vertex of the circular cone.

We study the surfaces of revolution with the non-degenerate second fundamental form in Minkowski 3-space. In particular, we investigate the surfaces of revolution satisfying an equation in terms of the position vector field and the 2nd-Laplacian in Minkowski 3-space. As a result, we give some new examples of the surfaces of revolution with light-like axis in Minkowski 3-space.

While the Steiner problem has been extensively studied in the Euclidean plane, it remains an open problem to solve the Steiner problem on arbitrary non-planar (piecewise smooth) surfaces. We suggest an algorithm for solving the n-point Steiner problem on surfaces of revolution which have a non-decreasing generating function by constructing an isometric framework on a plane endowed with a weighted distance metric, thus propelling a new analytical avenue for studying the Steiner problem on surfaces with non-constant curvature.

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.

We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem (Hass et al., Electron. Res. Announc. Am. Math. Soc., 1(3):98–102, 1995; Hutchings et al., Ann. Math., 155(2):459–489, 2002; Reichardt, J. Geom. Anal., 18(1):172–191, 2008). As part of the proof, we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.

Often in the engineering practice, cutting tools bounded by primary peripheral surfaces of revolution are used because of their effectiveness. Among these, ring and tangential tools can be used for the generation of constant pitch cylindrical helical surfaces. In this paper, we present an algorithm for the profiling of these types of tools. The algorithm is based on the topological representation of the tool’s primary peripheral surface. The main goal is to devise a methodology for the profiling of tools whose surfaces are reciprocally enveloping with cylindrical helical surfaces. We present a numerical example for the numerical determination of the axial section form for this type of tools. The application method for this algorithm was developed in the CATIA graphical design environment within which the procedure is developed as a vertical application. In addition, we present a solution for the shape correction of the tool’s axial cross-section by considering the existence of singular points on the profile of the helical surface to be generated where multiple normals to the surface exist.

In this paper, we propose a method based on the surfaces enveloping theory developed as a vertical application in the CATIA graphical design environment. This method is created for profiling tools intended to generate cylindrical helical surfaces with constant pitch. The devised method is applicable for the profiling of disc tools which are bounded by a primary peripheral surface of revolution. A few examples for the determination of the tool’s 3D primary peripheral surface are presented such as helical flutes of cutting tools, worms with wide pitches, and gear involute flanks. The results are presented in both graphical and tabular form for completeness. Screen snapshots of the CATIA-based vertical application are also shown.

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.

We study uniqueness, nonuniqueness and support properties of nonnegative bounded solutions of initial value problems on surfaces of revolution with boundary, for a class of quasilinear parabolic equations with variable density. At the boundary, the density can either vanish or diverge or need not to have a limit. In dependence of the behavior of the density near the boundary, we provide simple conditions for uniqueness or nonuniqueness of solutions; moreover, supposing that the initial datum does not intersect the boundary, we give criteria so that the support of any solution intersects the boundary at some positive time or it remains always away from it.

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.

This paper presents a parametric approach to updating workpiece surfaces in a virtual environment. The workpiece surfaces are represented by a series of discrete vectors, which may be orientated in different directions. The methodology is developed for multi-axis machining in which a tool can be arbitrarily oriented in space. The cutter is modeled as a surface of revolution, which is a canal surface formed by sweeping a sphere with varying radius along a spine curve. To define the tool swept envelope, the cutter surfaces are decomposed into a set of characteristic circles which are generated by a two-parameter family of spheres. Then, the grazing points, at which the discrete vectors can intersect the tool envelope, are obtained by considering the relationships between these circles and feed vector of the cutter. From this, the envelope-vector intersections are transformed into a single-variable function. Examples of this technique are generated for typical milling tools with both linear and circular spine curves. The vector/tool envelope intersection calculations for cutters with linear spine curves can be performed analytically. However, the intersection calculations for cutters having circular spine curves require solving a system of nonlinear equations. For this purpose, a root-finding analysis is developed for guaranteeing the root(s) in the given interval. Finally, to improve the efficiency when updating the workpiece, a vector localization scheme is developed based on the Axis-aligned Bounding Box method.

In the paper, we establish the L p (ℝn+1)-boundedness for a class of singular integral operators associated to surfaces of revolution “(y, γ (|y|), y ∈ ℝ n ” with rough kernels. We also give several applications of this inequality.

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 \; dA -\gamma \int\limits_{\Gamma} K \; dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and \({\gamma \in [0,1]}\) is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to \({\mathcal{W}_{\gamma}}\) and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.

We analyze the Ginzburg–Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in \({\mathbb R^3}\). We establish Γ-convergence to a reduced Ginzburg–Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold \({\mathcal{M}}\) is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field Hc1 is simply (4π/(area of \({\mathcal{M}}\))) ln κ for large values of the Ginzburg–Landau parameter κ. Merging this with the Γ-convergence result, we also obtain the same asymptotic value for Hc1 in 3d valid for large κ and sufficiently thin shells.

We study special infinitesimal geodesic deformations of the surfaces of revolution in the Euclidean space E 3.

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.

In this paper, the authors establish L p boundedness for several classes of multiple singular integrals along surfaces of revolution with kernels satisfying rather weak size condition. The results of the corresponding maximal truncated singular integrals are also obtained. The main results essentially improve and extend some known results.

In 1841, Delaunay classified surfaces of revolution with constant mean curvature in the Euclidean three space. As a byproduct of his result, one obtains: A surface of revolution has a periodic generating curve if and only if its mean curvature is non-zero. One hundred and forty years after Delaunay’s work, Hsiang and Yu extended this result to higher dimensions, by extending Delaunay’s idea of tracing the locus of a focus by rolling a given conic section along a line. In this paper, we give a new proof of their result using elementary ODE theory to obtain the periodicity of the solutions under consideration.

We consider some problems concerning the Lp,q-cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from Lpk (M) into Lk+1q (M). In the second part, we prove that the first Lp,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q.

The sequence of nodal count is considered for separable drums. A recently derived trace formula for this sequence stores geometrical information of the drum. This statement is demonstrated in detail for the Laplace-Beltrami operator on simple tori and surfaces of revolution. The trace formula expresses the cumulative sum of nodal counts This sequence is expressed as a sum of two parts: a smooth (Weyl like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical context of the nodal sequence is thus explicitly revealed.

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.

In this note we use the Hopf map … : S 3 ! S 2 to construct an in- teresting family of Riemannian metrics h f on the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve ∞ in S 2 is the complete lift … i1 (∞) of ∞, and we prove that any geodesic of this Hopf tube satisfles a Clairaut relation. A Hopf tube plays the role in S 3 of the surfaces of revolution in R 3 . Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S 3 . Finally, if we consider the sphere S 3 equipped with a family h f of Lorentzian metrics, then a new Clairaut relation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2μ) ∑ pi2 with hermitian form of Cartesian momentum pi (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2μ) ∑ (1/fi)pifipi where the functions fi are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this article, the explicit forms of the dummy functions fi for quantum motion on some 2D surfaces of revolution of spherical topology are given.

We consider the spectral problem for the Schrodinger operator describing a charged particle confined by a homogeneous magnetic field to a certain two-dimensional symmetric surface. Spectral asymptotic series are calculated for either strong or weak magnetic field.

In this paper the surfaces of revolution without parabolic points, in the 3-dimensional Lorentz-Minkowski space are classified under the condition $$ \Delta ^{{II}} \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} = A\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} $$ where Δ II is the Laplace operator with respect to the second fundamental form and A is a real 3 × 3 matrix. More precisely we prove that such surfaces are either minimal or Lorentz hyperbolic cylinders or pseudospheres of real or imaginary radius.

For the cylindrical norm onR 3, for which the isoperimetric shape is a cylinder rather than a round ball, there are analogs of the classical Delaunay surfaces of revolution of constant mean curvature.