Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Symbolic coding for the geodesic flow associated to a word hyperbolic group

 Let Γ be a word hyperbolic group M. Gromov has constructed a compact space equipped with a flow which is defined up to orbit-equivalence and which is called the geodesic flow of Γ. In the special case where Γ is the fundamental group of a Riemannian manifold of negative sectional curvature, is the unit tangent bundle of the manifold equipped with the usual geodesic flow. In this paper, we construct, for every hyperbolic group Γ, a subshift of finite type and a continuous map from the suspension of this subshift onto , which is uniformly bounded-to-one and which sends each orbit of the suspension flow onto an orbit of the geodesic flow. Received: 25 January 2002 / Revised version: 20 August 2002 Mathematics Subject Classification (2000): 20F67, 20F65, 20F69, 53C23, 53C21, 37D40, 37B10, 54H20     

Dynamical borel-cantelli lemma for hyperbolic spaces

We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr _t, provided the integral ∫ ₀ ^∞ r _t ⁿ −1 dt diverges (resp. converges).     

Indice local pour des ensembles symétriques par arcs

Let M be a complete Riemannian manifold with sectional curvature and dimension . Given a unit vector and a point we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension . Received April 13, 1998; in final form July 23, 1999 / Published online October 11, 2000     

Cusp Cross-Sections of Hyperbolic Orbifolds by Heisenberg Nilmanifolds I

Long and Reid [Algebr. Geom. Topol. 2: 285–296, 2002] have shown that the diffeomorphism class of every Riemannian flat manifold of dimension n≥ 3 arises as a cusp cross-section of a complete finite volume real hyperbolic (n+1)-orbifold. For the complex hyperbolic case, McReynolds [Algebr. Geom. Topol. 4: 721–755, 2004] proved that every 3-dimensional infranilmanifold is diffeomorphic to a cusp cross-section of a complete finite volume complex hyperbolic 2-orbifold. Moreover, he gave a necessary and sufficient condition for a Heisenberg infranilmanifold to be realized as a cusp cross-section of finite volume (arithmetically) complex hyperbolic orbifold. We study these realization problems by using Seifert fibrations.     

Stability of hyperbolic manifolds with cusps under Ricci flow

We show that every finite volume hyperbolic manifold of dimension greater than or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we do not need to make any decay assumptions on this perturbation.     

A weak characterisation of the Delaunay triangulation

We consider a new construction, the weak Delaunay triangulation of a finite point set in a metric space, which contains as a subcomplex the traditional (strong) Delaunay triangulation. The two simplicial complexes turn out to be equal for point sets in Euclidean space, as well as in the (hemi)sphere, hyperbolic space, and certain other geometries. There are weighted and approximate versions of the weak and strong complexes in all these geometries, and we prove equality theorems in those cases also. On the other hand, for discrete metric spaces the weak and strong complexes are decidedly different. We give a short empirical demonstration that weak Delaunay complexes can lead to dramatically clean results in the problem of estimating the homology groups of a manifold represented by a finite point sample.      

Asymptotic Geometry and Conformal Types of Carnot-Carathéodory Spaces

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot—Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further applications are discussed, such as the relation between local and global invertibility properties of quasiconformal immersions (the global homeomorphism theorem). Submitted: November 1997, revised: November 1998.     

Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds

Summary We show that a closed embedded totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. In particular, we show that ifM is a hyperbolic 3-manifold containingn rank two cusps andk disjoint totally geodesic embedded closed surfaces, then the volume ofM is bigger than . We also derive a (hyperbolic) quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last result, we construct examples to illustrate the qualitative sharpness of our tubular neighborhood function in dimension three. As an application of our results we give an eigenvalue estimate.Oblatum IX-1992 & 18-VIII-1993Research supported in part by NSF Grant DMS-9207019     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.     

The Gauss–Bonnet Formula for Hyperbolic Manifolds of Finite Volume

Let M denote an even-dimensional noncompact hyperbolic manifold of finite volume. We show that such manifolds are candidates for minimal volume. Generalizing H. Hopf's ideas around the Curvatura integra for compact Clifford–Klein space forms, we present an elementary combinatorial-metrical proof of the Gauss–Bonnet formula for M. In contrast to former results of G. Harder and M. Gromov, our approach doesn't make use of the arithmetical and differential geometrical machinery.     

Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

The Gromov norm and foliations

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology. We show that this norm is non-trivial — i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold.?Using a homotopy-theoretic refinement, we show that a taut foliation whose leaf space branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut foliation whose leaf space branches in both directions. Our technology also lets us produce examples of taut foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover.?Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties. Submitted: October 1999, Revision: December 1999, Revision: July 2000, Final version: September 2000.     

Sur l'homologie de cycles géodésiques dans des variétés hyperboliques compactes

The notion of an l-geodesic cycle in a compact hyperbolic n-manifold M generalises, in dimension l, the one of a closed geodesic. In this Note we show that when l ≥n/2, such a cycle lifts to a finite cover of M as an embedded totally geodesic submanifold non zero homologous. It enables us to prove that the compact hyperbolic manifolds constructed by Gromov and Piateski-Shapiro (see [4]) have infinite virtual Betti numbers and to give a new proof of the same fact for the compact arithmetic hyperbolic manifolds constructed by Borel in [3].     

Construction and recognition of hyperbolic 3-manifolds with geodesic boundary

We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition.

Our theory has been exploited to classify all the orientable finite-volume hyperbolic -manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.

     

Le bas du spectre d'une variété hyperbolique est un point selle

Besson, Courtois and Gallot proved that on a compact manifold, a hyperbolic metric has minimal entropy among the metrics of same volume. Here we show that a hyperbolic metric is a saddle point for the bottom of spectrum of the universal cover. More precisely, the bottom of its spectrum is maximal among the conformal metrics of same volume, and minimal in its Ebin class.     

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C^∞ Anosov splitting and preserves a C^∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C^∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\text{T}{}^4$, or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Quadratic divergence of geodesics in CAT(0) spaces

A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.Partially supported by NSF grant DMS-9200433