Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in Hn and En. We show that the intersection C_∞ of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C_∞ is a smooth submanifold of _∂∞Hn. In the hyperbolic case, we show that for any k?(n−1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in En, and give asymptotic estimates as 1?k?n.     

On singularities of submanifolds of higher dimensional Euclidean spaces

Summary Generalizations of principle axes are found for surfaces in E⁴. The singularities generalize umbilics. The generic indicies are computed. For these computations the Thom Transversality Theorem as applied by Feldman to geometry is used. Hower we ? reduce the group ? rendering the calculations more tractible. Also we show that a torus or sphere cannot be immersed in E⁴ with everywhere nonzero curvature of the normal bundle. Entrata in Redazione il 19 novembre 1968.     

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

We study the topology of (properly) immersed complete minimal surfaces P ² in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ? ⁿ and in ? ⁿ (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.     

Higher dimensional isoperimetric functions in hyperbolic groups

We introduce the notion of an -combing and use it to show that hyperbolic groups satisfy linear isoperimetric inequalities for filling real cycles in each positive dimension. S. Gersten suggested the concept of metabolicity (over or ) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of -cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, -metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff for any normed vector space V and any . Received December 9, 1998     

Higher dimensional multifractal analysis of non-uniformly hyperbolic systems

A. Johansson, T.M. Jordan, A. Öberg, and M. Pollicott (2010) [7] have studied the multifractal analysis of a class of one-dimensional non-uniformly hyperbolic systems. By introducing some new techniques, we extend the results to the case of high dimension.     

On a new embedding theorem and the CLR-type inequality for Euclidean and hyperbolic spaces

The goal of this note is to provide a new embedding theorem and to derive from this embedding the CLR-type inequality for a potential belonging to a proper subspace of integrable functions.     

On the numbers of n-tangencies of polygons in odd dimensional Euclidean spaces

The purpose of this paper is to show relationships among the numbers of (2n+1)-tangencies of a polygon in the Euclidean (2n+1)-space, generalizing the results for smooth curves in three space in [3], [4] and [6].     

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of and the relation between these spaces and traces of classical Sobolev spaces.     

Finite dimensional polish spaces are extreme boundaries of convex bodies in Euclidean space

It is proved that everyn-dimensional Polish space is homeomorphic to the set of extreme points of a compact convex set inR ^18(n+1). The contribution of M. Levin to this paper is a part of his Ph.D. thesis prepared at the University of Haifa under the supervision of Y. Sternfeld.     

Metric tangent spaces to Euclidean spaces

We prove that the metric spaces pretangent to a finite-dimensional Euclidean or unitary space E are isometric to E. As a consequence of this result, we describe the metric pretangent spaces at the nonsingular points of smooth surfaces. It is also proved that there exist the spaces pretangent to the Hilbert space l ₂ , which are not isometric to it.     

Higher dimensional cable knots and their finite cyclic covering spaces

As an analogue of the classical cable knot, the p-cable n-knot about an n-knot K, where p is an integer and n?2, is defined, and some basic properties of higher dimensional cable knots are described. We show that for p>0 then p-fold branched cyclic covering space of an (n+2)-sphere branched over the p-cable knot about an n-knot K is an (n+2)-sphere or a homotopy (n+2)-sphere which is the result of Gluck-surgery on the composition of p copies of K according as if p is odd or even. At the same time, we prove that for any n?2 and p?2, the composition of p copies of any n-knot K is the fixed point set of a Z_p-action on an (n+2)-sphere. This is another counterexample to the higher dimensional Smith conjecture.     

Gromov hyperbolic spaces

A mini monograph on Gromov hyperbolic spaces, which need not be geodesic or proper.     

Euclidean and hyperbolic lengths of images of arcs

Let f be a function that is analytic in the unit disc. We givenew estimates, and new proofs of existing estimates, of theEuclidean length of the image under f of a radial segment inthe unit disc. Our methods are based on the hyperbolic geometryof plane domains, and we address some new questions that follownaturally from this approach.     

Bergman spaces of higher‐dimensional hyperbolic polyhedron‐type domains I

In this paper we develop explicit formulas for the Green's function and the monogenic reproducing Bergman kernel function of some hyperbolic polyhedron‐type domains that generalize the fundamental domain of the modular group SL(2,?) to higher dimensions. Copyright © 2005 John Wiley & Sons, Ltd.     

Higher dimensional knot spaces for manifolds with vector cross products

Vector cross product structures on manifolds include symplectic, volume, G₂- and Spin(7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces.For the complex case, the holomorphic volume form on a Calabi-Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi-Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.