Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz–Minkowski space

In this paper we introduce the notion of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz?CMinkowski space which is analogous to the notion of evolutes of curves on the hyperbolic plane. We investigate the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface.     

The equidistant involution of the hyperbolic plane and two models of the Euclidean plane geometry

In this paper we define an involution of the hyperbolic plane corresponding to an equidistant curve and a point of its base line that keeps a certain subset of the equidistant curves invariant. Based on this mapping we present two models of the Euclidean geometry in the hyperbolic plane.     

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In [10] one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane \mathbb E²{{\mathbb E}^2} were obtained. Moreover, the relations between the complex velocities of one-parameter motion in the complex plane were provided by [10]. One-parameter planar homothetic motion was defined in the complex plane, [9]. In this paper, analogous to homothetic motion in the complex plane given by [9], one-parameter planar homothetic motion is defined in the hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale h identically equal to 1, the results given in [15] are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.     

Hyperbolic Polygons and Real Schottky Groups

In a work due to Aigon and Silhol (also Buser and Silhol) a construction of 10 genus two closed Riemann surfaces is done from a given right angled hyperbolic pentagon. In this note we construct real Schottky groups uniformizations of the corresponding constructions. In particular, we are able to write down the algebraic curves obtained in the above work in terms of the parameters of the real Schottky group. We generalize such a construction for any right angled hyperbolic polygon and also consider an example for a nonright angled pentagon in the last section.     

Self-similar solutions to the hyperbolic mean curvature flow

This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.     

On solutions to the wave equation on a non-globally hyperbolic manifold

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of special form (the Minkowski plane with a handle) containing closed time-like curves. We prove that the classical solution of the Cauchy problem exists and is unique for initial data satisfying a specific set of additional requirements. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 273–287.     

The hyperbolic meaning of the Milnor–Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

Coordinates in the hyperbolic plane

Abstract This note shows that in the hyperbolic plane three kinds of coordinates are possible. Keywords: Hyperbolic plane, Quasi-regular quadrangle, Pseudo-parallelogram, Hypercycle, Hypocycle Mathematics Subject Classification (2000): 20N05     

Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow

This paper concerns the hyperbolic mean curvature flow (HMCF) for plane curves. A quasilinear wave equation is derived and studied for the motion of plane curves under the HMCF. Based on this, we investigate the formation of singularities in the motion of these curves. In particular, we prove that the motion under the HMCF of periodic plane curves with small variation on one period and small initial velocity in general blows up and singularities develop in finite time. Some blowup results have been obtained and the estimates on the life-span of the solutions are given.     

Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces

We show that closed Chen-Willmore rotational hypersurfaces of nonnegative curved real space forms are shaped on closed hyperelastic curves of the hyperbolic plane. Then, we study the variational problem associated to this class of curves, proving that there exist a rationally dependent family of closed solutions. They give rise to the first non-trivial examples of Chen-Willmore hypersurfaces in real space forms. Research partially supported by a MCYT-FEDER grant No. BFM2001-2871-C04 and by a UPV grant 9/UPV00127.310-13574/01.     

Breaking hyperbolicity for smooth symplectic toral diffeomorphisms

We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on π ₁-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.     

Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff

We present an axiom system for plane hyperbolic geometry in a language with lines as the only individual variables and the binary relation of line-perpendicularity as the only primitive notion. It was made possible by results obtained by K. List and H.L. Skala. A similar axiomatization is possible for n-dimensional hyperbolic geometry with n≥4. We also point out that plane hyperbolic geometry admits a AE-axiomatization in terms of line-perpendicularity alone, an axiomatization we could not find.     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Möbius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.     

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner??s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).     

Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.