Stable and unstable manifolds for hyperbolic bi-semigroups

We show the existence of local Lipschitzian stable and unstable manifolds for the ill-posed problem of perturbations of hyperbolic bi-semigroups. We do not assume backward nor forward uniqueness of solutions. We do not use cut-off functions because we do not assume global smallness conditions on the nonlinearities. We introduce what we call dichotomous flows which recovers the symmetry between the past and the future. Thus, we need to prove only a stable manifold theorem. We modify the Conley–McGehee–Moeckel approach to avoid appealing to Wazewski principle and Brouwer degree theory. Hence we allow both the stable and unstable directions to be infinite dimensional. We apply our theorem to the elliptic system $u_{\xi \xi }+\mathrm{\Delta }u=g(u,u_{\xi })$ in an infinite cylinder $\mathbb{R}\times \Omega$.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.     

Transversality of stable and unstable manifolds for parabolic problems arising in composite materials

In this paper we study one dimensional parabolic problems that arise from composite materials. We show that the eigenvalues and eigenfunctions of the associated linear unbounded operator have the Sturm-Liouville property and the nonincrease of the lap number along the solutions. These results are used to show that the stable and unstable manifolds of equilibrium points are transversal.     

Coxeter groups and hyperbolic manifolds

The theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, and in one case of smallest possible volume.The author is grateful to Patrick Dorey for a number of helpful conversations.Revised version: 22 December 2003     

Finite groups and hyperbolic manifolds

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n≥2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n=2 and n=3 have been proven by Greenberg (1974) and Kojima (1988), respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Cusp closing of hyperbolic manifolds

A general method to produce infinitely many pairwise homotopically nonequivalent closed manifolds using the cusp closing construction is presented. An infinite sequence of closed homotopically nonequivalent real analytic Riemannian 5-manifolds with uniformly bounded volumes and uniformly bounded nonpositive sectional curvatures, which are allowed to vanish along codimension two submanifolds only, is constructed using this method.Both authors were supported in part by the Grant R24000 from the International Science Foundation.     

Quasiconformal homogeneity of hyperbolic manifolds

We exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold. In particular, if n3, a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. Moreover, if n3, we show that there is a constant K_n>1 such that if M is a hyperbolic n-manifold, other than which is K–quasiconformally homogeneous, then KK_n.Mathematics Subject Classification (2000): 30C60Research supported in part by NSF grant 070335 and 0305704.Research supported in part by NSF grant 0203698.Research supported in part by the NZ Marsden Fund and the Royal Society (NZ).Research supported in part by NSF grant 0305704.     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

Bending and stamping deformations of hyperbolic manifolds

The paper deals with some geometric approaches (from the viewpoint of (n + 1)- dimentional Lobachevsky geometry) to the deformation theory for uniformized conformal (i.e., flat conformal) structures on a hyperbolic n-manifold M with finite volume. Namely, two kinds of deformations are studied: bendings and stampings along totally geodesic submanifolds of M. The construction of the last deformation disproves a conjecture of C. Kourouniotis.     

L 2-torsion of hyperbolic manifolds

We show that the L ²-torsion of odd dimensional hyperbolic manifolds, which is proportional to the volume, is non-zero. This proves a conjecture of Lott. Received: 27 March 1998     

Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

Counting commensurability classes of hyperbolic manifolds

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v ^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi-isometry classes of lattices in SO(n, 1). Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.     

On the structure of hyperbolic manifolds

Forn≥2, we quantify the Margulis constant ε(n) giving rise to a thick and thin decomposition of hyperbolicn-manifolds of finite volume. As a consequence, we obtain new universal lower bounds for the volume and Gromov's invariant as well as a geometrical inequality between injectivity radius and diameter for compact manifolds. Finally, we concretise the upper bound for the counting function of hyperbolic manifolds of dimension >4 as described by Burger, Gelander, Lubotzky and Mozes. Partially supported by Schweizerischer Nationalfonds No. 20-61379.00 and 20-67619.02.     

Asymptotic flexibility of globally hyperbolic manifolds

In this short Note, a question of patching together globally hyperbolic manifolds is addressed which appeared in the context of the construction of Hadamard states.