Ruled Laguerre minimal surfaces

A Laguerre minimal surface is an immersed surface in ${\mathbb{R}^3}$ being an extremal of the functional ${\int (H^2/K-1)dA}$ . In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces ${\mathbf{R}(\varphi,\lambda) = ( A\varphi,\, B\varphi,\, C\varphi + D\cos 2\varphi\, ) + \lambda\left(\sin \varphi,\, \cos \varphi,\, 0\,\right)}$ , where ${A,B,C,D\in \mathbb{R}}$ are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.     

Arithmetic of diagonal quartic surfaces, II

The author investigates the solubility in rationals of equationsof the form where a₀a₁a₂a₃is a square, building on the ideas which Colliot-Thène,Skorobogatov and he have developed; see Invent. Math. 134 (1998)579–650. He obtains sufficient conditions for solubility,which appear to be related to the absence of a Brauer–Maninobstruction. This represents the first large family of K3 surfaceswhich almost satisfy the Hasse principle, in the sense thatthe auxiliary condition which ensures that local solubilityeverywhere implies global solubility is nearly always satisfied.1991 Mathematics Subject Classification: 10B10.     

Extending symmetric determinantal quartic surfaces

We give an explicit construction for the extension of a symmetric determinantal quartic K3 surface to a Fano sixfold. Remarkably, the moduli of the sixfold extension are in one-to-one correspondence with the moduli of the quartic surface. As a consequence, we determine a 16-parameter family of surfaces of general type with \(p_g=1, q=0\) and \(K^2=2\) as weighted complete intersections inside Fano sixfolds.     

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings \(X \rightarrow {\mathbb {P}}^3\) ramified in the degeneration locus \(D\) of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface \(D,\) one can associate an Enriques surface \(S\) which is the factor of the blowup of \(D\) by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface \(S\) is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of \(X\).     

Ruled 4-manifolds and isotopies of symplectic surfaces

We study symplectic surfaces in ruled symplectic 4-manifolds which are disjoint from a given symplectic section. As a consequence, in any symplectic 4-manifold, two homologous symplectic surfaces which are C ⁰ close must be Hamiltonian isotopic.     

On the classification of quartic half-arc-transitive metacirculants

A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1 and n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ^m has at least one fixed vertex. In a recent paper Maruši? and the author showed that each connected quartic half-arc-transitive metacirculant belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph relative to the semiregular automorphism ρ. One of these classes coincides with the class of the so-called tightly-attached graphs, which have already been completely classified. In this paper a complete classification of the second of these classes, that is the class of quartic half-arc-transitive metacirculants for which the quotient graph relative to the semiregular automorphism ρ is a cycle with a loop at each vertex, is given.     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

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.     

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 minimal surfaces in the Berger sphere

We show that any ruled minimal surface in the Berger sphere is a helicoid whose axis is a Hopf fiber by solving the ruled minimal surface equation in the parametric form.     

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.