Jørgensen’s inequality gives a necessary condition for the discreteness of a non-elementary group of isometries of hyperbolic 3-space. The main idea of the proof may be generalised widely but the statement is quite specialised. Here we give a scheme for restating Jørgensen’s inequality for Möbius transformations of a metric space. This unifies many previously published versions of Jørgensen’s i...

In a previous paper [6], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces Lp(Ω,Σ, μ) on which the hyperbolic topology induced by the norm ∥·∥p coincides with the norm topology. We show the following. (1) The hyperbolic topol...

Abstract The space of Euclidean cone metrics on centrically symmetric octahedra with fixed angles ?i < 2 ? , total surface area 1, has a natural hyperbolic metric, and is locally isometric to 3-space. metric completion the ideal tetrahedron whose dihedral are half cone-deficits ? .

Let H1, H2 be the universal covers of two compact Riemannian manifolds (of dimension 6= 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism. As a consequence, every self quasiconformal map of a Heisenberg group (equipped with the Carnot metric and viewed as the ideal boundary of complex hyperbolic space) of dimen...

In the half-space model of hyperbolic space, that is, R+ = {(x, y, z) ∈ R ; z > 0} with the hyperbolic metric, a translation surface is a surface that writes as z = f(x) + g(y) or y = f(x) + g(z), where f and g are smooth functions. We prove that the only minimal translation surfaces (zero mean curvature in all points) are totally geodesic planes. MSC: 53A10

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured su...

In this paper we present a new kind of B-splines, called hyperbolic B-splines generated over the space spanned by hyperbolic functions and we use it to interpolate an arbitrary function on a set of points. Numerical tests for illustrating hyperbolic B-spline are presented.

we prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of aliouche, et al. [2], rao et al. [14] and [15].

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality. A smooth Riemannian manifold (M, g) is said [1] to be Einstein if its Ricci curvature is a constant multiple of g. Any i...

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core. Let S be a surface of finite topological type, obtained by removing finitely many points from a compact surface without boun...