Helicoidal surfaces under the cubic screw motion in Minkowski 3-space

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H²=K and prove that these surfaces have equal constant principal curvatures.     

Notes sur la fonction ζ de Riemann, 4

Opération fondamentale de l'arithmétique, familière depuis des millénaires, la division euclidienne n'a pas livré tous ses secrets. Ainsi, notons pour k et a entiers positifs, le reste de la division euclidienne de k par a, et imaginons un instant que, par un choix convenable d'un entier n et de réels c₂,…,c_n, nous sachions rendre arbitrairement petite la quantité

     

Equations différentielles invariantes sur les groupes et algèbres de Lie réductifs II

Dans un article précédent, nous avons démontré que si D est un opérateur différentiel bi-invariant sur un groupe réductif G vérifiant la condition de Benabdallah-Rouvière, alors on peut résoudre l’équation différentielle Du=v dans l'espace des distributions G-invariantes (par automorphismes intérieurs) d'ordre fini; nous allons montrer ici que, sous la même hypothèse, on peut résoudre cette équation dans l'espace de toutes les distributions G-invariantes. D'autre part, nous donnons un exemple dans qui montre que les équations différentielles invariantes dans les algèbres de Lie réductives ne sont pas toujours résolubles dans l'espace des fonctions indéfiniment différentiables invariantes.     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result of this paper is Theorem 4 stating that the round sphere of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric?n−1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to .     

The space of scalarly integrable functions for a Fréchet-space-valued measure

The space of all scalarly integrable functions with respect to a Fréchet-space-valued vector measure ν is shown to be a complete Fréchet lattice with the σ-Fatou property which contains the (traditional) space L¹(ν), of all ν-integrable functions. Indeed, L¹(ν) is the σ-order continuous part of . Every Fréchet lattice with the σ-Fatou property and containing a weak unit in its σ-order continuous part is Fréchet lattice isomorphic to a space of the kind .     

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem

     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.     

Exact number of solutions of a prescribed mean curvature equation

We study the exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equation

     

A kind of helicoidal surfaces in 3-dimensional Minkowski space

In this paper we constructed a helicoidal surface with a light-like axis with prescribed mean curvature or Gauss curvature given by smooth function in 3-dimensional Minkowski space and solved an open problem left by Beneki, Kaimakamis, and Papantoniou in [J. Math. Anal. Appl. 275 (2002) 586-614].     

Variational method based on finite dimensional approximation in a generalized prescribed mean curvature problem

An elementary existence proof based on variational and finite dimensional approximation methods is proposed for nontrivial solutions of the generalized prescribed mean curvature boundary value problem