Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

Torsions of 3-Dimensional Small Covers

In this paper,it is shown that for a 3-dimensional small cover M over a polytope P,there are only 2-torsions in H1(M;Z).Moreover,the mod 2 Betti number growth of finite covers of M is studied.     

Hyperbolicity and one-dimensional waves in compressible two-phase flow models

Hyperbolic models for compressible two-phase flows including a conservative symmetric hyperbolic model are reviewed. The basis for a theory of shock waves is developed within the framework of the latter. The analysis of small amplitude discontinuities allows us to conclude that in general there are two types of shocks corresponding to two sound waves. The problem of transition between a pure phase and a mixture (the phase vacuum problem) is analysed. It is proved that for some models the smooth centred wave solution can not provide such a transition. Within the framework of our conservative model there is the possibility of constructing discontinuous solutions which can resolve the phase vacuum problem.PACS: 47.55Kf, 47.40.-xE. Romenski: On leave from Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk 630090, Russia     

Hyperbolic metric and normal family

In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

On the hyperbolic order

We define the hyperbolic order of any locally injective holomorphic function between arbitrary hyperbolic domains of the complex plane and study the relation between the hyperbolic order and the Schwarzian derivative for locally injective holomorphic functions from the unit disk into itself.     

LIMITATIONS ON PRESCRIBING THE HOPF DIFFERENTIAL FOR MINIMAL SURFACES IN H3

ABSTRACT

The Hopf map from minimal surfaces in H ³ to holomorphic quadratic differentials on H ² is considered. Whether or not a given quadratic differential is in the image of this map is determined by the global existence/non-existence of solutions to a certain quasi-linear PDE. It is shown that this map is not surjective.     

Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.