Ruled surfaces of Weingarten type in Minkowski 3-space, II

In our previous paper of the same title, we did not study the ruled surfaces of Weingarten type M : x(s, t)=α(s)+t β (s) in Minkowski 3-space with vector fields β and β′ along the base curve β such that β is nowhere null but β′ is null everywhere. We here fulfill our project by investigating this remaining case.     

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Introducing Weingarten cyclic surfaces in R3

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.     

Linear Weingarten Surfaces in ℝ<Superscript>3</Superscript>

 In this paper we study properties of linear Weingarten immersions and graphs related to non-existence problems and behaviour of its curvatures. The main results are obtained giving a harmonic representation of linear Weingarten surfaces and by proving optimal estimates of the height and curvatures that the immersion must satisfy, characterizing the spherical caps as the only ones achieving these bounds. Received January 25, 2001; in revised form April 4, 2002     

Lorentzian Clairaut submersions

We investigate a class of semi-Riemannian submersions satisfying a Lorentzian analogue of the classical Clairaut's relation for surfaces of revolution. We show that a Lorentzian submersion with one-dimensional fibers is Clairaut if and only if the fibers are totally umbilic with a gradient field as the normal curvature vector field. We also investigate the behavior of timelike and null geodesics in Lorentzian Clairaut submersions. In particular, every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic in the base space with respect to a conformally related metric on the base space if and only if the integrability tensor of the submersion vanishes.     

Minding isometries of ruled surfaces in pseudo-Galilean space

In this paper we study the Minding isometries of skew ruled surfaces in pseudo-Galilean space. The Minding isometries are isometries of ruled surfaces which are also generator-preserving. The obtained results can be easily transfered to the ruled surfaces in Galilean space.     

Ruled surfaces of finite type in 3- dimensional minkowski space

We show that a ruled surface M in E₁ ³ is of finite type if and only if M is minimal, or M is a part of a circular cylinder, or M is a part of a hyperbolic cylinder, or M is an isoparametric surface with null rules. We also give a complete classification of isoparametric ruled surfaces with null rules.     

On the Gauss map of B-scrolls in 3-dimensional Lorentzian space forms

In this note we show that B-scrolls over null curves in a 3-dimensional Lorentzian space form are characterized as the only ruled surfaces with null rulings whose Gauss maps G satisfy the condition being a parallel endomorphism of .     

Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles 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 such a surface must be a surface of revolution, one of the Riemann minimal examples, or a generalized cone. Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain     

Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations

On any space-like Weingarten surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution to the Lund-Regge reduction problem for space-like Weingarten surfaces in Minkowski space. We apply this theory to linear fractional space-like Weingarten surfaces and obtain the natural non-linear partial differential equations describing them. We obtain a characterization of space-like surfaces, whose curvatures satisfy a linear relation, by means of their natural partial differential equations. We obtain the ten natural PDE’s describing all linear fractional space-like Weingarten surfaces.     

Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s

The authors derive best possible maximum principles for some combinations of solutions of a class of fully nonlinear elliptic PDEs and their gradients. These maximum principles are then applied to establish various inequalities of interest in the theory of Weingarten surfaces.     

Lie geometry of linear Weingarten surfaces

We show how linear Weingarten surfaces appear as special Ω-surfaces and give a characterization of those linear Weingarten surfaces that allow a Weierstrass type representation.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Transforms for minimal surfaces in the 5-sphere

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction considered in a previous paper, and illustrate this explicitly in the case of bipolar minimal surfaces.     

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere. Received: 23 January 1998 / Revised version: 23 October 1998     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

A characterization of Weingarten surfaces in hyperbolic 3-space

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 ?³ 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 ?³ 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 ?³.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.