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In this paper, we prove a Ros–Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in ℝ+3 with planar convex boundary is a topological disk under mild suitable assumptions. [ABSTRACT FROM AUTHOR]

Architectural surface panelling often requires fabricating molds for panels, a process that can be cost-inefficient and material-wasteful when using traditional methods such as CNC milling. In this paper, we introduce a novel solution to generating molds for efficiently fabricating architectural panels. At the core of our method is a machine that utilizes a deflatable membrane as a flexible mold. By adjusting the deflation level and boundary element positions, the membrane can be reconfigured into various shapes, allowing for mass customization with significantly lower overhead costs. We devise an efficient algorithm that works in sync with our flexible mold machine that optimizes the placement of customizable boundary element positions, ensuring the fabricated panel matches the geometry of a given input shape: (1) Using a quadratic Weingarten surface arising from a natural assumption on the membrane's stress, we can approximate the initial placement of the boundary element from the input shape's geometry; (2) we solve the inverse problem with a simulator-in-the-loop optimizer by searching for the optimal placement of boundary curves with sensitivity analysis. We validate our approach by fabricating baseline panels and a facade with a wide range of curvature profiles, providing a detailed numerical analysis on simulation and fabrication, demonstrating significant advantages in cost and flexibility. [ABSTRACT FROM AUTHOR]

In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space R³. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.

We classify Weingarten conoids in the real special linear group SL (2 , R) . In particular, there is no linear Weingarten nontrivial conoids in SL (2 , R) . We also prove that the only conoids in SL (2 , R) with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup N is a Weingarten surface. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we have a tendency to investigate a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space E 3 satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature, and therefore the second Gaussian curvature. On the premise of those equations, some canal surfaces are introduced.

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

In this paper, we construct and obtain the necessary condition of Weingarten and linear Weingarten translation surfaces in E3. Special cases of these types are investigated and plotted.

In this paper we study a large class of Weingarten surfaces M with prescribed hyperbolic Gauss map in the Hyperbolic 3-space, which are the analogous to the Laguerre minimal surfaces in Euclidean space, these surfaces will be called Generalized Weingarten surfaces of harmonic type (HGW-surfaces), this class includes the surfaces of mean curvature one and the linear Weingarten surfaces of Bryant type (BLW-surfaces). We obtain a Weierstrass type representation for this surfaces which depend of three holomorphic functions. As applications we classify the HGW-surfaces of rotation and we obtain a Weierstrass type representation for surfaces of mean curvature one with prescribed hyperbolic Gauss map which depend of two holomorphic functions. Moreover, we classify a class of complete mean curvature one surfaces parametrized by lines of curvature whose coordinates curves has the same geodesic curvature up to sign. [ABSTRACT FROM AUTHOR]

We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in R 3 that satisfies an arbitrary elliptic Weingarten equation W (κ 1 , κ 2) = 0 , and study the singularities of such examples. As global applications of this classification, we prove a sharp halfspace theorem for general elliptic Weingarten equations of finite order, and a classification of peaked elliptic Weingarten ovaloids with at most 2 singularities. In the case that W is not elliptic, we give a negative answer to a question by Yau regarding the uniqueness of rotational ellipsoids. [ABSTRACT FROM AUTHOR]

In this paper, Cyclic surfaces are introduced using the foliation of circles of curvature of a space curve. The conditions on a space curve such that these cyclic surfaces are of type Weingarten surfaces or HK-quadric surfaces are obtained. Finally, some examples are given and plotted. [ABSTRACT FROM AUTHOR]

In this paper, we study a special kind of tube surfaces, so-called tubelike surface in 3-dimensional Euclidean space E3 is generated by sweeping a space curve along another central space curve. This study investigates a tubelike surface satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature. Furthermore, some important theorems are obtained. Finally, an example of tubelike surface is used to demonstrate our theoretical results and graphed. [ABSTRACT FROM AUTHOR]

Canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. In Minkowski 3-space, many authors studied canal surfaces. However, when one investigates the papers, it is obvious that the parametrizations of the canal surfaces were found with respect to only pseudo sphere $${S_{1}^{2}(r)}$$ . In this paper, we reconsider the canal surfaces for all Lorentz spheres which are pseudo sphere $${ S_{1}^{2}(r)}$$ , pseudo-hyperbolic sphere H( r) or lightlike cone C and we find the parametrizations of the surfaces. Moreover, we found the parametrization of the tubular surfaces with respect to all Lorentz spheres. Also, we study Weingarten and linear Weingarten type spacelike tubular surface obtained from pseudo-hyperbolic sphere $${H_{0}^{2}(r)}$$ and the singular points of the spacelike tubular surface obtained from pseudo-hyperbolic sphere $${H_{0}^{2}(r)}$$ . [ABSTRACT FROM AUTHOR]

The canal surfaces with time-like center curves in 3D Minkowski space were defined and the Weingarten canal surfaces were classified. Similar to the studying method for surfaces in Euclidean space, at first, the parametric equation of canal surfaces under pseudo orthogonal frame was built according to the Frenet frame of time-like curves and the geometric definition of canal surfaces, then the basic theories were obtained which include two fundamental quantities, the Gaussian curvature and mean curvature and so on. Using basic theories, the relationship between the Gaussian curvature and the mean curvature were found and the Weingarten canal surfaces were studied explicitly. The conclusion was achieved that a canal surface is a Weingarten surface if and only if it is a tube or a revolution surface. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that parallel surfaces of a non-developable ruled surface are not ruled surfaces by using fundamental forms. It has been shown that the parallel surfaces of a developable ruled surface is the developable ruled surfaces. It is obtained that parallel surfaces of ruled Weingarten surface are Weingarten surface. [ABSTRACT FROM AUTHOR]

In this paper, we construct and obtain the necessary condition of Weingarten and linear Weingarten translation surfaces in E 3 . Special cases of these types are investigated and plotted. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show that an embedded Weingarten surface in of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for embedded minimal tori. [Copyright &y& Elsevier]

In this paper, we study tubular surfaces in Minkowski 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Minkowski 3-space. [ABSTRACT FROM AUTHOR]

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. In the Galilean and pseudo Galilean spaces, some special surfaces such as factorable surfaces, ruled Weingarten surfaces, translation surfaces and tubular surfaces have been studied in [2-4, 18, 19]. A regular surface M2 in Galilean space is called a area minimizing surface (minimal surface) if and only if its mean curvature is zero at each point. In this paper, we describe, THA-surfaces (Translation and homothetical affine surfaces) in the Galilean space G3 having constant Gaussian and mean curvatures and completely classify minimal or at THA-surfaces. Also, we classify linear Weingarten THA-surfaces in the Galilean space G3. [ABSTRACT FROM AUTHOR]

Abstract: We consider 2-surfaces arising from the Korteweg–de Vries (KdV) hierarchy and the KdV equation. The surfaces corresponding to the KdV equation are in a three-dimensional Minkowski ( ) space. They contain a family of quadratic Weingarten and Willmore-like surfaces. We show that some KdV surfaces can be obtained from a variational principle where the Lagrange function is a polynomial function of the Gaussian and mean curvatures. We also give a method for constructing the surfaces explicitly, i.e., finding their parameterizations or finding their position vectors. [Copyright &y& Elsevier]

In this paper, we have defined canal surfaces in Galilean and Pseudo-Galilean 3-spaces. Then, we have studied Tubular surface in Galilean and Pseudo-Galilean 3-space satisfying some equations in terms of the Gaussian curvature and the mean curvature. We have discussed Weingarten, linear Weingarten conditions and HK-quadric type for this surface with respect to their curvatures. [ABSTRACT FROM AUTHOR]

We study spacelike surfaces in Minkowski spaceE13 that satisfy a linear Weingarten condition of typek1=mk2+n where m and n are constant andk1 and k2denote the principal curvatures at each point of the surface. We prove that if the surface is foliated by a uniparametric family of circles in parallel planes, then it is rotational or it is part of the family of Riemann examples of maximal surfaces. If the surface is rotational and n=0 we obtain a rst integration if the axis is timelike and spacelike and a complete description if the axis is lightlike. [ABSTRACT FROM AUTHOR]

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^{3})$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ℍ orthogonal to the geodesics of Σ. We prove that the induced metric on a Lagrangian surface in ${\mathbb{L}}({\mathbb{H}}^{3})$ has zero Gauss curvature iff the orthogonal surfaces in ℍ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in ${\mathbb{L}}({\mathbb{H}}^{3})$ and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ℍ. [ABSTRACT FROM AUTHOR]

In this paper, we studied the tubular W-surfaces that satisfy a Weingarten condition of type ak1 + bk2 = c, where a, b and c are constants and k1 and k2 denote the principal curvatures of M and Mi in Euclidean 3-space E3 and Minkowski 3-space E3 1, respectively. [ABSTRACT FROM AUTHOR]

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler's theorem about minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's theorem about constant Gauss curvature ones. [ABSTRACT FROM AUTHOR]

In this paper, we study a spacelike (timelike) ruled W-surface in Minkowski 3-space which satisfies nontrivial relation between elements of the set {K, KII, H, HII}, where (K,H) and (KII,HII) are the Gaussian and mean curvatures of the first and second fundamental forms, respectively. Finally, some examples are constructed and plotted. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a wide family of spacelike surfaces in which includes, for instance, constant mean curvature 1 surfaces and flat surfaces: those whose mean and Gauss–Kronecker curvatures verify the lineal relationship for . We show that these surfaces can be parametrized with holomorphic data and we use it to classify the complete surfaces from this family with non-negative Gaussian curvature. [Copyright &y& Elsevier]