Einstein metrics onS 3,R 3 andR 4 bundles

Starting from a 4n-dimensional quaternionic Kähler base space, we construct metrics of cohomogeneity one in (4n+3) dimensions whose level surfaces are theS ² bundle space of almost complex structures on the base manifold. We derive the conditions on the metric functions that follow from imposing the Einstein equation, and obtain solutions both for compact and non-compact (4n+3)-dimensional spaces. Included in the non-compact solutions are two Ricci-flat 7-dimensional metrics withG ₂ holonomy. We also discuss two other Ricci-flat solutions, one on theR ⁴ bundle overS ³ and the other on anR ⁴ bundle overS ⁴. These haveG ₂ and Spin(7) holonomy respectively.     

Generalized Robertson-Walker metrics and some of their properties. II

The Robertson-Walker (RW) metrics, of dimensionality four and signature –2, are generalized to metrics of dimensionality (n+1) and of arbitrary signature,n (> 1) being an arbitrary integer. In canonical coordinates (t, x ¹,x ², ...,x ⁿ ) these generalized Robertson-Walker (GRW) metrics are functions of the coordinatet. The following statements are proved to be equivalent: The GRW metrics are (a) expressible int-independent form, (b) of constant curvature, (c) Einstein spaces. Furthermore, there are six, and only six, classes of GRW metrics satisfying these three statements. The coordinate transformations which transform these metrics to theirt-independent form are given explicitly. Two of these classes of GRW metrics reduce, in theirt-independent form, to the same flat (generalized Minkowski) metrics, three reduce to the samet-independent metrics which are generalizations of the de Sitter space-time metric, and the last class tot-independent metrics which are generalizations of the anti-de Sitter space-time metric.     

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝ ⁿ (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity. Received: 8 November 1999 / Accepted: 27 March 2000     

Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mⁿ_i, of riemannian manifolds with special holonomy. We show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension 4. Additional results on the the detailed structure of the singular set support our main conjecture that if the Mⁿ_i are compact and a certain characteristic number, C(Mⁿ_i), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.The first author was partially supported by NSF Grant DMS 0104128 and the second by NSF Grant DMS 0302744.     

Generalization of the Gross-Perry Metrics

A class of SO(n+1) symmetric solutions of the (N+n+1)-dimensional Einstein equations is found. It contains 5-dimensional metrics of Gross and Perry and Millward.     

Finite-Time Blow-up of Solutions of an Aggregation Equation in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We consider the aggregation equation in R ⁿ , n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) = e ^−|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.     

Orientifolds and slumps in G2 and Spin(7) metrics

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by , which are complete on a complex line bundle over . The principal orbits are S⁷, described as a triaxially squashed S³ bundle over S⁴. The behaviour in the S³ directions is similar to that in the Atiyah–Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S⁴. We then consider new G₂ metrics which we denote by , which are complete on an bundle over T^1,1, with principal orbits that are S³×S³. We study the metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S² cycles, and both carry magnetic charge with respect to the R–R vector field. We also discuss some four-dimensional hyper-Kähler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(∞) Toda equation, which can provide a way of studying their interior structure.     

Higher Dimensional Extension of Charged and Neutral Fluid Spheres with Pressure

General exact higher-dimensional (n+2), n>2 solutions in general theory of relativity of Einstein-Maxwell field equations for spherically symmetric distribution of charged pressure perfect fluid are expressed in terms of pressure extending 4-dimensional solutions presented by Bijalwan (Astrophys. Space Sci. 2011, doi:). Subsequently, metrics (e ^λ and e ^υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω (=c ₁+c ₂ r ²), where c ₁ and c ₂ (≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category. Also, these solutions satisfy barotropic equation of state relating the radial pressure to the energy density. But we noticed that none of these solutions in terms of pressure for charged fluids has a well behaved neutral counter part for a spatial component of metric e ^λ i.e. choosing same spatial component for charged and neutral fluid. To illustrate the approach, we discovered a new solution for extended charged analogues of Schwarzschild interior solution in higher dimensions which is found to be well behaved only for n=2. The maximum mass found to be 1.512 M _Θ with linear dimension 14.964 km. Physical quantities pressure, energy density, red-shift, velocity of sound and p/c ² ρ are well behaved and monotonically decreasing towards the surface while adiabatic index and charge density are monotonically increasing. For brevity we don’t discuss the numerical results in detailed.     

Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M²{{\mathcal M}^2} which can be realized as isometric immersions into \mathbbR³{\mathbb{R}^3}. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in \mathbbR³{\mathbb{R}^3}. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in \mathbbR³{\mathbb{R}^3}. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C ^{1, 1} isometric immersion of the two-dimensional manifold in \mathbbR³{\mathbb{R}^3} satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the a priori estimates including the L ^∞ control and H ⁻¹–compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into \mathbbR³{\mathbb{R}^3} satisfying our initial conditions. The theory is applied to a specific example of the metric associated with the catenoid.     

3-Branes on Eguchi–Hanson 6D instantons

We use the approach used by Eguchi–Hanson in constructing four-dimensional instanton metrics and construct a class of regular six-dimensional instantons which are nothing but S ² × S ² resolved conifolds. We then also obtain D3-brane solutions on these EH-resolved conifolds.     

Formulas for An- and Bn-solutions of WDVV equations

The simplest non-trivial solutions of WDVV equations are A_n-and B_n-potentials, which describe metrics of Saito on spaces of versal deformation of A_n-and B_n-singularities. These are some polynomials, which were known for n≤4. In this paper, we find the potentials for all A_n-and B_n-singularities. We give a recurrence formula for coefficients of KP and n-KdV hierarchy.     

On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations

We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations in ℝ ⁿ (n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends Lions' existence theorem [15] to the case 1< γ <γ _n for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ _n = 3/2 for n= 2 and γ _n = 9/5 for n= 3. Dedicated to Professor Rolf Leis on the occasion of his 70th birthday Received: 17 January 2000 / Accepted: 3 July 2000     

On the reconstruction of a unitary matrix from its moduli

We study the problem of reconstructing a unitary matrix from the knowledge of the moduli of its matrix elements, first in the case of a symmetric matrix, which could be theS matrix forn coupled channels, second in the case of a non-symmetric matrix like the Cabibbo-Kobayashi-Maskawa matrix forn generations of quarks and leptons. In the symmetric case we find conditions under which the problem has solutions differing in a non-trivial way, but also situations where one has continuous ambiguities.In the non-symmetric case we show that forn>3 there may be continuous ambiguities, of which we give an exhaustic list forn=4. We give indications that there is also a set of moduli for which one has discrete solutions, but no rigorous proof.Unité associée au CNRS n^o 040768     

On the problem of one-dimensional disordered chain

Conclusion Some properties of a one-dimensional disordered homogeneous chain were studied in this paper. Using standard techniques of probability theory, expressions for the frequency distribution function (2) and the localization length (6) were derived. Having considered only pair correlations between atoms, both these expressions contained only one unknown function — the joint probability distribution of the massm _n and the ratiot _n ^± = –ku _n±1/u _n which could be found as a solution of the integral equation (5). Our approach to the problem was applied on the ideal lattice and the lattice with low concentration of impurities. In these cases the solutions of the integral equation (5) reduced to the functional form (7) were found analytically. Using these solutions, old well-known results for ( ²) and the local vibration of impurities were derived by this method.Derivation of all equations in this paper is straightforward from the equations of motion. The quantities we deal with have a clear physical meaning, which facilitated, for instance to find the solutions of functional equation (7) in the special case of the ideal crystal. This is what we consider to be the advantages of our approach.     

LETTER: On a Class of Solutions for Null Dust

In stationary metrics depending on two spacelike coordinates the field equations permit one to choose a radial coordinate r such that D ² := –g ₃₃ g ₄₄ + g ₃₄ ² = r ², leading to solutions with axial symmetry. However, solutions exist also for the case in which D ² = 1. These solutions are examined in this paper, and, if physically realistic, are found to refer to null dust or vacuum. This is at variance with the interpretation of Hoenselaers and Vishveshwara [4] who examined one of the solutions and concluded that it described non-null dust or vacuum.     

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ^N × S ⁿ universe calculated by Sahdev with the help of numerical methods.     

AdS solutions in gauge supergravities and the global anomaly for the product of complex two-cycles

Cohomological methods are applied for the special set of solutions corresponding to rotating branes in arbitrary dimensions, AdS black holes (which can be embedded in ten or eleven dimensions), and gauge supergravities. A new class of solutions is proposed, the Hilbert modular varieties, which consist of the 2n-fold product of the two-spaces H ⁿ /Γ (where H ⁿ denotes the product of n upper half-planes, H ², equipped with the co-compact action of Γ⊂SL(2,ℝ) ⁿ ) and (H ⁿ )^∗/Γ (where (H ²)^∗=H ²∪{cusp of Γ} and Γ is a congruence subgroup of SL(2,ℝ) ⁿ ). The cohomology groups of the Hilbert variety, which inherit a Hodge structure (in the sense of Deligne), are analyzed, as well as bifiltered sequences, weight and Hodge filtrations, and it is argued that the torsion part of the cuspidal cohomology is involved in the global anomaly condition. Indeed, in the presence of the cuspidal part, all cohomology classes can be mapped to the boundary of the space and the cuspidal contribution can be involved in the global anomaly condition.     

Algebraic classification of ℂP n instanton solutions

The finite condition for two-dimensional ℂP ⁿ models is discussed noting that one can impose boundary conditions such that the domain of the field is a compact Riemann surface S _g. Holomorphic maps φ: S _g →ℂP ⁿ give finite energy solutions of the classical field equations, which are classified according to standard methods of algebraic geometry. The moduli problem is discussed in detail for S _g =ℂP ¹=S ².