On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

Inverse Hyperbolic Problems with Time-Dependent Coefficients

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R ⁿ with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C ⁿ denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C ⁿ →C ⁿ and for holomorphic automorphisms of C ⁿ on discrete subsets of C ⁿ.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C ⁿ.For each closed complex submanifold (or subvariety) M ⊂ C ⁿ of complex dimension m < n we construct a domain Ω ⊂C ⁿ containing M and a biholomorphic map F: Ω → C ⁿ onto C ⁿ with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C ^n−m →C ⁿ at infinitely many points. If m = n − 1, we construct F as above such that C ⁿ ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C ^m →C ^m−1 such that the complement C ^m+1 ∖F(C ^m )is hyperbolic.     

A dyadic decomposition approach to a finitely degenerate hyperbolic problem

Abstract We use the Littlewood-Paley decomposition technique to obtain a C^∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration. Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C^∞-well-posedness, Approximate energy method     

Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle

We prove that stable ergodicity is C ^r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞]. The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C ^r open and dense, and essential accessibility implies ergodicity. Mathematics Subject Classification (2000) Primary: 37D30, Secondary: 37A25     

On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b (in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).     

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t→-∞in the energy norm, and to show it has a free profile as t→ ∞. Our approach is based on the work of [11]. Namely we use a weighted L∞norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.     

On the constructibility of a negatively curved complete surface of constant mean curvature in H<Superscript>3</Superscript> determined by a prescribed Hopf differential

The role that a prescribed holomorphic Hopf (Quadratic) differential A(z) dz dz plays in the construction of a negatively curved immersed simply connected complete surface ∑₀ of prescribed constant mean curvature c ∈ (−1, 1)in the hyperbolic 3-Space H ³ is investigated in this work. When a holomorphic function A(z), which is the coefficient function of the Hopf differential, is prescribed on a unit disk |z| < 1,it is shown that the unit disk |z| < 1can be immersed in the hyperbolic 3-Space H ³ as a negatively curved complete surface of constant mean curvature c ∈ (−1, 1),provided that |A(z)| satisfies a certain growth condition. Moreover, it is shown that the unit disk |z| < 1can be uniquely embedded in H ³ when the holomorphic function A(z) has a certain admissible structure.     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups. The first author was supported in part by NSF Grant DMS-0504251. The second author was supported in part by an NSF Mathematical Sciences Post-doctoral Research Fellowship. Both authors thank the NSF for their support. Most of this work was done while both authors were Taussky-Todd Fellows at Caltech.     

C k-rigidity for hyperbolic flows II

In this note we prove the following result: Any conjugating homeomorphism between two geodesic flows for compact negatively curved compactC ^∞ surfaces is necessarilyC ^∞. This extends a result of Feldman and Ornstein. We also discuss some related results for hyperbolic flows and diffeomorphisms.     

Resonances on Some Geometrically Finite Hyperbolic Manifolds

Abstract

We first prove the meromorphic extension to ? for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the quotients Γ\ ⁿ⁺¹ with rational nonmaximal rank cusps previously studied by Froese-Hislop-Perry.     

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degreen. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the ‘one-point compactifications of the strata indexed by a large class of number partitions, including λ=(k ^m , 1 ^r ), form≥2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.     

Densest Packing of Equal Spheres in Hyperbolic Space

Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Triangle subgroups of Kleinian groups

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF ²=∏/Γ _∏ is always embedded in the hyperbolic 3-orbifoldM ³=ℍ³/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup. We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups. for J. A. Kalman on the occasion of his 65th birthday Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.     

Densest Packing of Equal Spheres in Hyperbolic Space

   Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.     

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍ ⁿ⁺¹, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of ℍ ⁿ⁺¹ in terms of the Hausdorff dimension of the limit set of Γ.