Exact string solutions in 2+1-dimensional de Sitter spacetime

Exact and explicit string solutions propagating in 2+1 dimensional de Sitter spacetime are presented and physically analized. (In this case the string equations reduce to a sinh-Gordon model.) Strings generically tend to inflate or either to collapse. The world-sheet time interpolates between the cosmic (±) and conformal (0) times. For 0, the typical string instability is found, while for ±, a new string behavior appears. In that regime, the string expands (or contracts) but not with the same rate as the universe does. The string constraints select periodic solutions of the sinh-Gordon equation associated to the lower boundary of the allowed zone, therefore excluding elliptic solutions.In memoriam M. C. PolivanovLaboratoire de Physique Théorique et Hautes Energies, Paris. Landau Institute for Theoretical Physics, Russian Academy of Sciences. Observatoire de Paris, Section de Meudon, Demirm. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 232–240, February, 1993.     

On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.     

de Sitter空间中第二基本形式模长平方为常数的类空超曲面

设Mn是de Sitter空间S1n+1+1(c)中具有第二基本形式模长平方‖h‖2是常数的类空超曲面,利用极大值原理得到了Mn是全脐超曲面的三个充分条件.     

Uniqueness of Kottler spacetime and the Besse conjecture

We establish a black hole uniqueness theorem for Schwarzschild–de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.     

Uniqueness of static decompositions

We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fulfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections onto the base. We show that the base itself is a static manifold and the warping function satisfies severe restrictions, leading us to our classification results. Moreover, we show that certain condition on the lightlike sectional curvature ensures the uniqueness of static decomposition for Lorentzian manifolds.     

Uniqueness for diffusions with piecewise constant coefficients

Summary LetL be a second-order partial differential operator inR ^d. LetR ^d be the finite union of disjoint polyhedra. Suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron, and that the drift coefficient is bounded and measurable. We show that the martingale problem associated withL is well-posed.The research of this author was partly supported by NSF Grant DMS 8500581     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

Approximate representation of the de Sitter group

The purpose of this paper is to develop a theory of approximate representations of the de Sitter group considered as a perturbation of the Poincaré group. This approach simplifies investigation of relativistic effects pertaining to the mechanics in the de Sitter universe. Utility of the approximate approach is manifest if one compares the transformations of the de Sitter group with their approximate representations.     

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414].     

On space-like hypersurfaces in the de Sitter space

This paper gives the intrinsic conditions for a compact space-like hypersurface in a de Sitter space to be isometric to a sphere.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.