A characterization of Möbius transformations by use of hyperbolic regular polygons

In this paper we present a new characterization of Möbius transformations by use of hyperbolic regular polygons.     

A Remark About Hyperbolic Equations with Log-Lipschitz Coefficients

We prove the well-posedness of the Cauchy problem for strictly hyperbolic equations and systems with Log-Lipschitz coefficients in the time variable.     

正多边形对称群的子群

利用Lagrange定理和正多边形对称群的性质,首先对正多边形对称群的子群的性质进行了研究,其次讨论了正多边形对称群的子群的结构,由此完全确定了正多边形对称群的子群,最后应用所得结论求出了正六边形对称群的所有子群.     

Hyperbolic surfaces in

We show a class of perturbations of the Fermat hypersurface such that any holomorphic curve from into is degenerate. Applying this result, we give explicit examples of hyperbolic surfaces in of arbitrary degree , and of curves of arbitrary degree in with hyperbolic complements.

     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

Uniform Stability Estimates for Constant-Coefficient Symmetric Hyperbolic Boundary Value Problems

Answering a question left open in Métivier and Zumbrun (2005 Métivier , G. , Zumbrun , K. ( 2005 ). Variable multiplicities, hyperbolic boundary value problems for symmetric systems with variable multiplicities . J. Diff. Eq. 211 ( 1 ): 61 – 134 . [Google Scholar]), we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong L ² well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L ² well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier–Stokes or viscous MHD equations linearized about a constant state or even a viscous shock.     

Kazhdan Constants of Hyperbolic Groups

Let H be an infinite hyperbolic group with Kazhdan property (T) and let (H,X) denote the Kazhdan constant of H with respect to a generating set X. We prove that inf_X(H,X)=0, where the infimum is taken over all finite generating sets of H. In particular, this gives an answer to a Lubotzky question.     

Least Area Planes in Gromov Hyperbolic 3-Spaces with Co-compact Metric

Let M be a closed irreducible Riemannian 3-manifold such that π₁(M) is word hyperbolic, and p: X→ M the universal covering. Suppose that X has the Riemannian metric induced form that on M via p. In this paper, we will show that any Jordan curve in the boundary ∂X of X spans a properly embedded least area plane in X.Mathematics Subject Classiffications (2000). Primary 57M50; secondary 53A10     

Initial Value Problem for General Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

The authors consider the global existence and the blow-up phenomenon of classical solutions with small amplitude to the Cauchy problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity and given some applications.     

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.     

A Nonlocal Problem with Integral Conditions for the Quasilinear Hyperbolic Equation

In this paper, we consider a nonlocal problem with integral conditions for the quasilinear hyperbolic equation in a rectangular domain. The existence and uniqueness of the generalized solution are established.     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

Central configurations consist of two layer twisted regular polygons

In this paper, we study the existence of a special twisted regular polygonal central configuration in R₃.     

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.     

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. Mathematics Subject Classification (2000):57N10, 57M25.The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case

This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.     

Hyperbolic equations, function spaces with exponential weights and Nemytskij operators

Different problems in the theory of hyperbolic equations bases on function spaces of Gevrey type are studied. Beside the original Gevrey classes, spaces defined by the behaviour of the Fourier transform were also used to prove basic results about the well-posedness of Cauchy problems for non-linear hyperbolic systems. In these approaches only the algebra property of the function spaces was used to include analytic non-linearities. Here we will generalize this dependence. First we investigate superposition operators in spaces with exponential weights. Then we show in concrete situations how a priori estimates of strictly hyperbolic type lead to the well-posedness of certain semi-linear hyperbolic Cauchy problems in suitable function spaces with exponential weights of Gevrey type. Mathematics Subject Classification (2000) 46E35, 35L80, 35L15, 47H30     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.