Standard model of fermions on the domain wall in five-dimensional space-time

The mechanism of generation of the Standard Model for fermions on the domain wall in five-dimensional space-time is presented. As a result of self-interaction of five-dimensional fermions and gravity induced by matter fields, in the strong coupling regime, in the model there arises a spontaneous translational symmetry breaking, which leads to localization of light particles on a 3 + 1-dimensional domain wall (“3-brane”) that is embedded into a five-dimensional anti-de Sitter space-time (AdS₅). Appropriate low-energy, effective action, classical kink-like vacuum configurations for the gravity and scalar fields are investigated. Mass spectra for light composite particles and their coupling constants interaction in ultra-low-energy, which localize on the brane, are explored. We establish estimates of characteristic scales and constants interactions of the model and also a relation between the bulk five-dimensional gravitational constant, curvature of AdS₅ space-time, and brane Newton’s constants. The induced cosmological constant on the brane exactly vanishes in all orders of the theory perturbation. We find out that scalar interaction is strongly suppressed at ultra-low-energy, and the brane fluctuations (branons) are suitable “sterile” canditates for explanation of the phenomenon of Dark Matter. Bibliography: 21 titles. Dedicated to the 100th birthday of M. P. Bronstein __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 5–29.     

On chen surfaces in a minkowski space time

A-submanifo1ds of a pseudo-Euclidean space E _s ^m+1 are considered. A characterization for them is given. A theorem onA-submanifolds contained in a de Sitter space-time S _s ⁿ or an anti-de Sitter space-time H _s–1 ⁿ is proved. A number of non-trivial examples ofA-surfaces in a Minkowski space-time E ₁ ⁴ are studied. Some classification theorems are proved forA-surfaces contained in S ₁ ³ or H³.     

Free equations for massive matter fields in (2 + 1)-dimensional anti-de sitter space from a deformed oscillator algebra

We reformulate the free equations of motion for massive spin-0 and spin-1/2 matter fields in (2+1)-dimensional anti-de Sitter space in the form of some constant curvature conditions. The infinite-dimensional representation of the anti-de Sitter algebra, which is basic to this formulation, admits a natural realization in terms of the algebra of deformed “oscillators” with a deformation parameter related to the mass parameter. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 372–384, March, 1997.     

On the Uniqueness of AdS Space-Time in Higher Dimensions

In this paper, based on an intrinsic definition of asymptotically AdS space-times, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension 8. Instead of using the positive energy theorem for asymptotically hyperbolic spaces with spin our approach appeals to the classic positive mass theorem for asymptotically flat spaces. Communicated by Piotr T. ChruscielSubmitted 17/10/03, accepted 07/11/03     

Constant Mean Curvature Foliations of Globally Hyperbolic Spacetimes Locally Modelled on <Emphasis Type="Italic">AdS</Emphasis><Subscript>3</Subscript>

We prove that every three-dimensional maximal globally hyperbolic spacetime, locally modelled on the anti-de Sitter space AdS ₃, with closed orientable Cauchy surfaces, admits a unique CMC time function.     

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space ${H^{n+1}_1(-1)}$ . we prove that if a complete space-like hypersurface with constant mean curvature ${x:\mathbf M\rightarrow H^{n+1}_1(-1) }$ has two distinct principal curvatures ??, ??, and inf|?? ? ??|?>?0, then x is the standard embedding ${ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}$ in anti-de Sitter space ${ H^{n+1}_1 (-1) }$ .     

On the geometry of conformally stationary Lorentz spaces

We study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending a result of J. Simons on the minimality of cones in Euclidean space to these spaces, and apply it to the construction of complete, noncompact minimal Lorentz submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong r-stability of spacelike hypersurfaces of constant r-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.     

Holographic estimation of multiplicity and the collision of membranes in modified AdS5 spaces

The quark-gluon plasma formed as a result of heavy-ion collisions is currently investigated actively both theoretically and experimentally. According to the holographic approach, forming a quark-gluon plasma in the four-dimensional world is associated with creating black holes in a five-dimensional anti-de Sitter (AdS) space. The multiplicity of particles produced in heavy-ion collisions is then determined by the entropy of the five-dimensional black hole, which is estimated by the area of the trapped surface. In this approach, we can model the dependence of the entropy on the energy of the colliding ions and thus the dependence of the multiplicity on the energy, and we can also compare the theoretical results with experimental data. To obtain a variety of model dependences on the energy, we consider the formation of black holes in modified AdS spaces, namely, in AdS spaces with different b factors. We find dynamics of the change of the trapped surface area depending on the energy for each investigated space.     

Formation of trapped surfaces in the collision of nonexpanding gravitational shock waves in an AdS<Subscript>4</Subscript> space-time

We study the formation of marginally trapped surfaces in the head-on collision of two shock waves in anti-de Sitter space-time. We compare the obtained results with the corresponding results for de Sitter space-time. To clarify this comparison, we use coordinates that allow studying AdS/dS cases in a universal way. We also analyze the dependence of the area of the trapped surface on the choice of the regularization of the shock wave metric.     

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999.     

Lorentz spacetimes of constant curvature

This paper is unpublished work of Geoffrey Mess written in 1990, which gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions.      

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the 3-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.     

Hawking radiation as tunneling with back-reaction for the Kerr-anti-de Sitter black hole

Medved and Vagenas demonstrated how the Angheben-Nadalini-Vanzo-Zerbini analysis can be adapted in describing the effects of the gravitational back-reaction for the generic spherically symmetric black hole. We extend the Medved-Vagenas quantum tunneling method to the case of the stationary axisymmetric Kerr black hole in anti-de Sitter space-times. We find a correction to the Hawking radiation by considering the effects of the gravitational back-reaction and also energy conservation and angular momentum conservation.     

Maximal surfaces and the universal Teichmüller space

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdS _n+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS₃, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.     

Spacelike submanifolds of codimension two in anti-de Sitter space

In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.     

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.     

Generalized relativistic kinematics

We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincaré group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity c) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants R and c. The limit transitions lead to the ordinary (R → ∞) or alternative (c → ∞) but nevertheless relativistic kinematics.     

Ultrarelativistic hadrons as cosmic membranes in TeV-gravity models

We regard an ultrarelativistic hadron moving on a brane embedded in a higher-dimensional space-time as a membrane propagating on a three-brane. For hadron sizes exceeding the compactification radius and characteristic scales of five-dimensional gravitational interaction, this membrane can be regarded as a cosmic membrane in the sense that it extends to the infinities. In the five-dimensional case, a cosmic membrane produces a topological defect similar to the defect produced by a cosmic string in the four-dimensional space-time. This representation allows calculating the effects of the influence of the hadron mass on the gravitational interaction of partons belonging to the colliding hadrons.     

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.