A Characterization for Geodsic Spheres in Space Forms

We use the maximum principle for second-order elliptic operators to establish a sufficient condition for a compact hypersurface in a space form to be a geodesic sphere in terms of a pinching for the s-mean curvature.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Umbilical routes along geodesics and hypercycles in the hyperbolic space

Given a geodesic line γ in the hyperbolic space ${\mathbb{H}}^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to γ. Then we extend the result to γ being a hypercycle i.e. a geodesic on a hypersurface equidistant from the totally geodesic one.     

Trajectories for Sasakian magnetic fields on real hypersurfaces of type (B) in a complex hyperbolic space

On a real hypersurface in a Kähler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kähler structure. We treat trajectories under magnetic fields which are constant multiples of this 2-form. We consider a condition for them to be also curves of order 2 on tubes around totally geodesic real hyperbolic spaces in a complex hyperbolic space.     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

Embeddings of Gromov hyperbolic spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. Submitted: October 1998, Revised version: January 1999.     

Gromov hyperbolic spaces and the sharp isoperimetric constant

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.     

Extrinsic radius pinching in space forms of nonnegative sectional curvature

We first give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to these estimates. We show that under a suitable pinching condition, M is diffeomorphic and almost isometric to an n-dimensional sphere.      

Divergence of Geodesics in Teichmüller Space and the Mapping Class Group

We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence. The same statements are true for geodesic rays in the mapping class group. We explicitly describe efficient paths “near infinity” in both spaces.     

On Sampling Formulas on Symmetric Spaces

We give a sampling formula using the Radon transform along a maximal geodesic subspace of the Riemannian symmetric space. For the real hyperbolic space we can get a total sampling formula. To get this formula, we prepare a sampling formula for the sphere.     

Characterization of Totally Geodesic Totally Real 3-dimensional Submanifolds in the 6-sphere

In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.     

The grafting map of Teichmüller space

Grafting is a method of obtaining new projective structures from a hyperbolic structure, basically by gluing a flat cylinder into a surface along a closed geodesic in the hyperbolic structure, or by limits of that procedure. This induces a map of Teichmüller space to itself. We prove that this map is a homeomorphism by analyzing harmonic maps between pairs of grafted surfaces. As a corollary we obtain bending coordinates for the Bers embedding of Teichmüller space.

     

Unique ergodicity of horospheric foliations revisited

With each rational function on the Riemann sphere, Lyubich–Minsky construction (1997) associates an abstract topological space called the quotient hyperbolic lamination. The latter space carries the so-called vertical geodesic flow with Anosov property. Its unstable foliation is what we call the quotient horospheric lamination. We consider the case of hyperbolic rational function, and more generally, functions postcritically finite on the Julia set without parabolics, that do not belong to the following list of exceptions: powers, Chebyshev polynomials and Latt‘es examples. In this case the quotient horospheric lamination is known to be minimal, while restricted to the union of nonisolated hyperbolic leaves (Glutsyuk, 2007). In the present paper we prove its unique ergodicity. To this end, we introduce the so-called transversely contracting flows and homeomorphisms (on abstract compact metrizable topological spaces), which include the vertical geodesic flows under consideration and the usual Anosov flows and diffeomorphisms. We prove a version of our unique ergodicity result for the transversely contracting flows and homeomorphisms. Particular cases for Anosov flows and diffeomorphisms are given by classical results by Bowen, Marcus, Furstenberg, Margulis, et al. We give a new and purely geometric proof, which seems to be simpler than the classical ones (which use either Markov partitions, K-property, or harmonic analysis).     

Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein's model).     

The Radon transform on hyperbolic space

The Radon transform that integrates a function in ⁿ , the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.     

General curvature estimates for stable H-surfaces immersed into a space form

In this paper, we give general curvature estimates for constant mean curvature surfaces immersed into a simply-connected 3-dimensional space form. We obtain bounds on the norm of the traceless second fundamental form and on the Gaussian curvature at the center of a relatively compact stable geodesic ball (and, more generally, of a relatively compact geodesic ball with stability operator bounded from below). As a by-product, we show that the notions of weak and strong Morse indices coincide for complete non-compact constant mean curvature surfaces. We also derive a geometric proof of the fact that a complete stable surface with constant mean curvature 1 in the usual hyperbolic space must be a horosphere.     

Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001     

Collars in Complex and Quaternionic Hyperbolic Manifolds

We use the complex and quaternionic hyperbolic versions of Jørgensen's inequality to construct embedded collars about short, simple, closed geodesics in complex and quaternionic hyperbolic manifolds. In general, the width of these collars depend both on the length of the geodesic and on the rotational part of the group element uniformising it. For complex hyperbolic space we are able to use a lemma of Zagier to give an estimate based only on the length. We show that these canonical collars are disjoint from each other and from canonical cusps. We also calculate the volumes of these collars.