Study on singularities of discrete Weingarten surfaces in Riemannian and Lorentzian spaceforms.
problem Analysis of singularities in discrete Weingarten surfaces.
method Defined and analyzed singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3D Riemannian and Lorentzian spaceforms.
result Discussed singularities of discrete surfaces with non-zero constant Gaussian curvature and parallel surfaces of discrete minimal and maximal surfaces.
Study classifies semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities.
problem Characterizing semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities.
method Established properties, classified, and analyzed the singularities of semi-discrete linear Weingarten surfaces in Riemannian and Lorentzian spaceforms.
result Defined and analyzed the singularities of semi-discrete linear Weingarten surfaces, including those with non-zero constant Gaussian curvature, parallel surfaces of minimal and maximal surfaces, and constant mean curvature 1 surfaces in de Sitter 3-space.
Discrete linear Weingarten surfaces in space forms are characterized as special discrete Ω-nets, a discrete analogue of Demoulin's Ω-surfaces. It is shown that the Lie-geometric deformation of Ω-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…
Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated…
We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …
Discretizes surface theory preserving integrable structure.
problem Discretizing surface theory in projective differential geometry.
method Introduces a canonical frame and derives a Backlund transformation for discrete Demoulin surfaces.
result Derives a two-component generalization of the integrable discrete Tzitzeica equation.
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using Ω-dual transform and lightlike Gauss maps in Laguerre geometry. result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
The paper classifies surfaces in isotropic space satisfying Weingarten conditions.
problem Classifying surfaces in isotropic space with specific curvature conditions.
method Analyzing rotational surfaces in isotropic 3-space with Weingarten conditions.
result Classification of surfaces of linear Weingarten type in isotropic space.
We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
Study of surfaces in space forms using Lie sphere geometry.
problem Investigate channel linear Weingarten surfaces in various space forms.
method Lie sphere geometric approach to uniform treatment of different ambient geometries.
result Any channel linear Weingarten surface is isothermic and a surface of revolution.
Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.
problem Characterizing uniformly elliptic Weingarten spheres in S2xR.
method Proving bounded second fundamental form and applying Hopf's result.
result Rotational uniformly elliptic Weingarten surfaces in S2xR are congruent to the canonical example.
New approach classifies rotational Weingarten surfaces in Lorentz-Minkowski space.
problem Classifying rotational Weingarten surfaces in Lorentz-Minkowski space.
method Using geometric linear momentum of generatrix curves with respect to axes of revolution.
result Unified framework for three causal types of rotation axes.
We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential eq…
In this study, we investigated the (K,H), (K,K_{II}), (H,K_{II})-Weingarten and (K,H),(K,K_{II}),(H,K_{II}) and (K,H,K_{II})-linear Weingarten canal surfaces in IR^3.
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property. First, we consider Weingarten surfaces in $\r^3$ that are foliated by circles, …
We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss curvature. We provide explicit parametrizations and deduce existence of complete hyperbolic linear Weingarten surfaces.
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
In this paper we will classify those translation surfaces in E3 involving polynomials which are Weingarten surfaces. We analyze Weingarten translation surfaces satisfying 2aH + bK = 0. We study also other types of translation surfaces, involving power functions, for which the second Gaussian curvature vanishes.
Estimates heights of special surfaces in warped products.
problem Estimating heights of special surfaces in warped products.
method Provided a vertical height estimate.
result Vertical height estimates for special Weingarten surfaces of elliptic type in warped products.
Study properties of Weingarten surfaces using curvature nets.
problem Characterizing Weingarten surfaces in 3D space.
method Analysis of curvature nets on central surfaces.
result Properties of Weingarten surfaces identified.
Study of 17 surface behaviors and singularities for elliptic Weingarten equations.
problem Characterizing and understanding elliptic Weingarten surfaces and their singularities.
method Phase space analysis and classification of surface behaviors.
result Classification of 17 possible qualitative behaviors for rotational surfaces.
The paper describes and classifies surfaces in isotropic 3-space.
problem Characterizing surfaces in isotropic 3-space.
method Analyzes Weingarten and linear Weingarten surfaces in \mathbb{I}^{3} using the position vector and Laplace operator.
result Classifies surfaces in \mathbb{I}^{3} satisfying certain equations.
A linear Weingarten surface in Euclidean space R3 is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH+bK=c, where a,b,c∈R. Such a surface is said to be hyperbolic when a2+4bc<0. In this paper we classify all rotational linear Weingarten surfaces of…
In this paper, we study meridian surfaces of Weingarten type in Euclidean 4-space E^4. We give the neccessary and sufficient conditions for a meridian surface in E^4 to become Weingarten type.
Extends theory of minimal surfaces to elliptic special Weingarten surfaces.
problem Classify surfaces of finite total curvature.
method Extends Jorge-Meeks formula and uses it to classify surfaces.
result Planes are the only elliptic special Weingarten surfaces with total curvature < 4π.
Study on properties and transformations of Weingarten surfaces in 3D space.
problem Characterize Weingarten surfaces and their transformations.
method Analyzes Weingarten relations from three perspectives: umbilic points, SL2(R) transformations, and variational formulations.
result Established bounds on the slope of Weingarten relations at umbilic points and showed transitivity of the action on semi-quadratic Weingarten surfaces.
The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.
problem Proving planes are the only complete uniformly elliptic Weingarten multigraphs.
method Proving planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.
result Proves planes are the only complete uniformly elliptic Weingarten multigraphs.
In this paper, we study the timelike tubular Weingarten surfaces in 3-dimensional Minkowski space IR13.We have obtained some conditions for being (KII,H), (KII,K), timelike tubular Weingarten surfaces where are the second Gaussian curvature the Gaussian curvature and the mean curvature, respectively.
The paper explores conditions for symmetries in Weingarten surfaces.
problem Conditions for symmetries in Weingarten surfaces.
method Analyzes surfaces satisfying a specific Weingarten equation and conditions on their boundary.
result Symmetries of the boundary curve are inherited by the surface under certain conditions.
The techniques used in this paper are based on the exterior calculus of Maurer-Cartan forms, and Weingarten surfaces are used to illustrate the methods that apply to quadratic exterior equations with constant coefficients. Isothermic {\it surfaces of constant astigmatism} (non-linear Weingarten surfaces whose differenc…
New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
Method constructs Bryant surfaces in hyperbolic space.
problem Constructing Bryant type surfaces in hyperbolic space.
method Bianchi-Calo type construction method
result Constructs Bryant type surfaces in hyperbolic space.
This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.
problem Extension of quadric surfaces of revolution to 3-sphere.
method Rigorous classification and characterization using spherical angular momentum.
result Spherical ellipsoids, hyperboloids, and paraboloids are Weingarten surfaces with a specific cubic relation between principal curvatures.
Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.
problem Characterize elliptic Weingarten surfaces in warped product spaces with minimal type curvature conditions.
method Analyze surfaces with mean curvature and extrinsic curvature satisfying a specific relationship under radial symmetry of the warping function.
result Existence and uniqueness of rotationally-invariant elliptic Weingarten surfaces of minimal type in RimeshR. A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form aκ1+bκ2=c or aH+bK=c, where $a,b,c\in \r$ and, as usual, κi are the principal curvatur…
Study on rotational surfaces in de Sitter space with specific curvature conditions.
problem Characterizing rotational surfaces in de Sitter space with Weingarten conditions.
method Analyzing spacelike and timelike rotational surfaces in 3D de Sitter space, determining profile curves, and classifying surfaces based on curvature relations.
result Classification of Weingarten rotational surfaces in de Sitter space with specific curvature relations.
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
The paper studies special surfaces in 4D space forms with specific geometric properties.
problem Investigating biconservative surfaces with flat normal bundles in 4D space forms.
method Analyzing compatibility conditions, prescribing flat connection, and determining specific surface properties.
result Existence and characterization of biconservative Weingarten surfaces with flat normal bundles.
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via ano…
New surface class defined using osculating circles.
problem Defining a new surface class in Euclidean space.
method Using osculating circles of curves and classification of specific types.
result Classification of canal and Weingarten surfaces.
Researchers classify cmc surfaces using Jacobi elliptic functions.
problem Classifying rotational cmc surfaces in non-Euclidean space forms.
method Lie sphere geometric description of rotational linear Weingarten surfaces.
result Explicit parametrizations of cmc surfaces in hyperbolic space.
Study surfaces in S²xR, proving two classes of Weingarten surfaces.
problem Characterize surfaces in radial conformally flat spaces.
method Characterize surfaces of rotation, prove isometry, and derive curvature relations.
result Two classes of Weingarten surfaces in the Radial Model.
The aim of this paper is to present a complete description of all rotational linear Weingarten surface into the Euclidean sphere S3. These surfaces are characterized by a linear relation aH+bK=c, where H and K stand for their mean and Gaussian curvatures, respectively, whereas a; b and c are real constants.
In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type aH+bK=c, where a,b and c are constant and H and K denote the mean curvature and the Gauss curvature respectively. We prove that a such surface must be a surface of revolution, a…
Study surfaces with specific curvature properties in 3D space.
problem Classify surfaces with prescribed linear Weingarten curvature.
method Analyze immersed surfaces in R3 with specific curvature relations. result Classify rotational surfaces under certain curvature conditions.