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.     

Remarks on Complete Spacelike Hyper-surfaces with Constant Mean Curvature in the de Sitter Space

Let M be an n(≥3)-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1 (n+1) (1) with constant mean curvature and non negative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥1. When 2(n-1)~(1.2)/n < H < 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.     

A CHARACTERIZATION OF l~1(Γ) TYPE BANACH SPACES

In this paper it turns out that a Banach lattice X is order isomorphic to l~1(Γ) for some nonempty set Γ iff it is a Schur space and all the infinitely dimensional, separable and closed ideals of X are order isomorphic.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Complete hypersurfaces in a 4-dimensional hyperbolic space

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4（-1）,whose scalar curvature is bounded from below.     

Interpolation on the complex Hilbert sphere

Let S∞ denote the unit sphere in some infinite dimensional complex Hilbert space (H,<·,·>)Let z1,z2,…,z1 be distinct points on S∞ This paper deals with interpolation of arbitrary data on the zj by a function in the linear span of the l functionswhen is a suitable function that operates on nonnegative definite matrices.Conditions for the strict positive definiteness of the kernel are obtained.     

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.     

Bers型空间和复合算子

For α∈（0,∞),let Hα^∞（or Hα^∞,0)denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|^2)^α=O(1)(or|f(z)|(1-|z|^2)^α=o(1) as |z|→1).Hα^∞,0)is called a Bers-type space (or a little Bers-type space).In this paper,we give some basic properties of Hα^∞，Cψ，the composition operator associated with a symbol function ψ which is an analytic self map of D,is difined by Cψf=f o ψ,We characterize the boundedness,and compactness of Cψ which sends one Bers-type space to another function space.     

Isometric immersions into warped product spaces

This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σ R, this is a large family of manifolds including the usual space forms R^m, S^m and H^m. We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space R^n×σ R, which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.     

THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3) Let B be a Banach space, B* be its dual, and denote the weak * topology of B* by σ(B*,B). Then the functional dimension of (B*,σ(B*,B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.     

An inner product formula on two-dimensional complex hyperbolic space

We study the Eisenstein series for a convex cocompact discrete subgroup on a two-dimensional complex hyperbolic space ℍ_ℂ². We find an inner product formula which gives the connection between Eisenstein series and automorphic Green functions on a two-dimensional complex hyperbolic space ℍ_ℂ². As an application of our inner product formula, we obtain the functional equations of Eisenstein series.     

A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝⁿ with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space ℍⁿ.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

Graphs with prescribed mean curvature on hyperbolic spaces

We study a quasilinear elliptic equation in the unit ball ofℝ ^m . Using this result we get the existence of graphs with prescribed curvature on hyperbolic spacesℍ ^m inℍ ^m ×ℝ.     

Intrinsic regular hypersurfaces in Heisenberg groups

We study the ℍ-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group ℍ ⁿ = ℂ ⁿ × ℝ = ℝ²ⁿ⁺¹ endowed with a leftinvariant metric d_∞ equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words “intrinsic” and “regular” we mean, respectively notions involving the group structure of ℍ ⁿ and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside ℍ ⁿ by studying the intrinsic regularity of the parameterizations and giving an areatype formula for their intrinsic surface measure.     

Weighted composition operators from F（p,q, s） spaces to Bers-type spaces in the unit ball

This paper deals with the boundedness and compactness of the weighted composition operators from the F（p, q, s） spaces, including Hardy space, Bergman space, Qp space, BMOA space, Besov space and α-Bloch space, to Bers-type spaces Hv^∞（ or little Bers-type spaces Hv,o∞ ）, where v is normal.     

On the Equicontinuity Region of Discrete Subgroups of <Emphasis Type="Italic">PU</Emphasis>(1,<Emphasis Type="Italic">n</Emphasis>)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ_ℂ ⁿ preserving the unit ball ℍ_ℂ ⁿ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ_ℂ ⁿ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ_ℂ ⁿ which are tangent to ∂ℍ_ℂ ⁿ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ_ℂ ⁿ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.     

The stable mapping class group and Q(ℂP∞+)

In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction ℤ×BΓ⁺ _∞ has an infinite loop space structure. This result and the tools developed in [BM] to analyse transfer maps, are used here to show the following splitting theorem. Let Σ^∞(ℂP ^∞ ₊)^∧ _p ≃E ₀∨...∨E _p-2 be the “Adams-splitting” of the p-completed suspension spectrum of ℂP ^∞ ₊. Then for some infinite loop space W _p ,?(ℤ×BΓ⁺ _∞)^∧ _p ≃Ω^∞(E ₀)×...×Ω^∞(E _p-3 )×W _p ?where Ω^∞ E _i denotes the infinite loop space associated to the spectrum E _i . The homology of Ω^∞ E _i is known, and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group. This splitting also detects all the Miller-Morita-Mumford classes. Our results suggest a homotopy theoretic refinement of the Mumford conjecture. The above p-adic splitting uses a certain infinite loop map?α_∞:ℤ×BΓ⁺ _∞?Ω^∞ℂP ^∞ _-1?that induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that α_∞ might be a homotopy equivalence. Oblatum 2-VIII-1999 & 28-III-2001?Published online: 18 June 2001