A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by ε >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.Mathematics Subject Classiffications (2000). primary: 30F45; secondary: 32G07     

Counting Hyperbolic Manifolds with Bounded Diameter

Let ρ _n (V) be the number of complete hyperbolic manifolds of dimension n with volume less than V . Burger et al [Geom. Funct. Anal. 12(6) (2002), 1161–1173.] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV log V ≤ log ρ _n (V) ≤ bV log V, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially with volume. Additionally, this bound holds in dimension 3.     

Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics

A hyperbolic 3-manifold is said to have the spd-property if all its closed geodesics are simple and pairwise disjoint. For a 3-manifold which supports a geometrically finite hyperbolic structure we show the following dichotomy: either the generic hyperbolic structure has the spd-property or no hyperbolic structure has the spd-property. Both cases are shown to occur. In particular, we prove that the generic hyperbolic structure on the interior of a handlebody (or a surface cross an interval) of negative Euler characteristic has the spd-property. Simplicity and disjointness are consequences of a variational result for hyperbolic surfaces. Namely, the intersection angle between closed geodesics varies nontrivially under deformation of a hyperbolic surface.     

Spectral Convergence for Degenerating Sequences of Three Dimensional Hyperbolic Manifolds

For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.

     

Asymptotic Geometry and Growth of Conjugacy Classes of Nonpositively Curved Manifolds

Let X be a Hadamard manifold and Γ⊂Isom(X) a discrete group of isometries which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the limit set of Γ in order to derive a new asymptotic estimate for the growth rate of closed geodesics in not necessarily compact or finite volume manifolds. Mathematics Subject Classifications (2000): 20E45, 53C22, 37F35     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

Upper and Lower Bounds on Resonances for Manifolds Hyperbolic Near Infinity

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r ⁿ⁺¹) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an r ⁿ⁺¹ lower bound on the counting function for scattering poles.     

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.     

弱双曲流形的逼近

弱双曲流形是一类包含中心流形作为其特例的不变流形.本文讨论了它的逼近问题,不仅给出了计算其逼近的方法,还估计了使得这一逼近有效的它的局部性半径. 同时给出了一个具体实例来展示这些计算和估计.     

Trace Equivalence in SU(2, 1)

In this paper it is shown that one can choose an arbitrarily large number of inconjugate elements of the group Z/2Z*Z/2Z*Z/2Z which have the property that, under all representations of the group in SU(2,1) as a discrete complex hyperbolic ideal triangle group, the elements are hyperbolic and correspond to closed geodesics of equal length on the associated complex hyperbolic surface. This is an analogue of the geometric fact that the multiplicity of the length spectrum of a Riemann surface is never bounded or the equivalent algebraic phenomenon that an arbitrarily large number of conjugacy classes in a free group can have the same trace under all representations in SL(2,R ).     

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

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.     

Geometrical Decomposition of the Free Loop Space on a Manifold with Finitely Many Closed Geodesics

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.     

Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space R ⁿ (corresponding to constrained optimization) and the R ⁿ space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than R ⁿ , the new algorithms are very efficient. Convergence rates are obtained. Applications are discussed. This paper is based on part of the Ph.D Thesis of the author under the supervision of Professor Tits, University of Maryland, College Park, Maryland. The author is in debt to him for invaluable suggestions on earlier versions of this paper. The author is grateful to the Associate Editor and anonymous reviewers, who pointed out a number of papers that have been included in the references; they made also detailed suggestions that lead to significant improvements of the paper. Finally, the author thanks Dr. S.T. Smith for making available his Ph.D Thesis.     

Behavior of Distant Maximal Geodesics in Finitely Connected Complete Two-Dimensional Riemannian Manifolds II

We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ² the Cohn–Vossen theorem states that the total curvature of M, say c(M), is 2. We already studied the case c(M)<2 in our previous paper. So we study the behavior of geodesics in M with total curvature 2 in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.     

Bounded Classes in the Cohomology of Manifolds

We discuss the image of a natural homomorphism from the bounded cohomology to the ordinary cohomology of a manifold. We give a necessary and sufficient condition for a Haken 3-manifold M to have the property that any class in the nth cohomology of M is bounded if n > 1.The first author was partially supported by JSPS.     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s=0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523-540], to a noncompact case.