Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

A global solution of the Einstein-Yang-Mills equation on the conformal space

The method used to construct the SU(2) Yang-Mills field on a compactified Minkowski space-M(which is equivalent to the conformal space) is generalized to construct an SU(N)(N > 2) Yang-Mills fieldFjκ on M. It is proved that both Fjκ and the invariant metric tensor gjκ of M satisfy the Einstein-Yang-Mills equation. The case of N →∞ is also discussed.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

On AdS/CFT correspondence of EYM fields

In the compactized Minkowski space, which is equivalent to the conformal spaceM ₄, we introduced a Lorentz metric d σ² and a Yang-Mills field θ. Later, we proved that dσ² and θ together satisfy the EYM (Einstein-Yang-Mills) equation. In this paper, it is proved that θ onM ₄ (which is the boundary of the anti-de-Sitter space AdS₅) can be extended to be a Yang-Mills field [^(q)]\hat \theta on AdS₅ such that Hua’s metric ds² on AdS₅, together with [^(q)]\hat \theta satisfies the EYM equation on AdS₅.     

The relationship between the Einstein and Yang-Mills equations

In this paper we prove that special requirements to Yang-Mills equations on a 4-dimensional conformally connected manifold allow one to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. We propose an algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang-Mills conformally connected manifold. As an application of the algorithm, we prove that the metric of any Einstein space and the Robertson-Walker metric are embeddable into the specified manifold.     

The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn-Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.     

The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds

In this paper, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that gives rise to critical points of the Yang-Mills functional. Moreover, we show that there is a relationship between this particular family of critical points of the Yang-Mills functional and Laplace's equation on multiplication-type, skew-symmetric elements of quantum Heisenberg manifolds; recall that Laplacian is the leading term for the coupled set of equations making up the Yang-Mills equation.     

Gauge fields and space-time geometry

Mappings between the SU (2) Yang-Mills gauge theory and the gauge gravity theory are discussed. The global aspects are considered in the framework of the fiber-bundle technique, and three different local constructions (depending on the existence of a fixed background metric) are described in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 249–262, November, 1998.     

A Lipschitz metric for the Hunter–Saxton equation

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.     

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

Summary. We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Lévy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Lévy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space. Received: 8 August 1996 / In revised form: 8 January 1997     

On a class of conformal metrics arising in the work of Seiberg and Witten

We examine a class of conformal metrics arising in the “N = 2 supersymmetric Yang-Mills theory” of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere does not admit a Seiberg-Witten metric, but for all there is a conformal metric on of regularity which is Seiberg-Witten off of a finite set. Received August 18, 1998     

Yang-Mills-Higgs soliton dynamics in 2+1 dimensions

Dimensional reduction of the self-dual Yang-Mills equation in 2+2 dimensions produces an integrable Yang-Mills-Higgs-Bogomolnyi equation in 2+1 dimensions. For theSU(1,1) gauge group, a t'Hooft-like ansatz is used to construct a monopole-like solution and an N-soliton-type solution, which describes both the static deformed monopoles and the exotic monopole dynamics including a transmutation. How the monopole solution results from the twistor formalism is shown. Multimonopole solutions are commented on. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 339–350, December, 1998.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

SU(3)规范场的恰当形式(欧空间)

应用数学机械化方法讨论SU(3)规范场的规范化问题.首先提出一种具有明确几何意义的Yang-Mills方程,称其为恰当的Yang-Mills方程.然后构造了一类线性微分变换,称之为SU(3)规范场的示性变换,它具体给出联络和截面之间的微分关系.经由示性变换,将非线性的恰当的YM-方程变为一组线性Laplace方程,即实现了规范场YM-方程的线性化.从而证明了SU(3)规范场包括8个独立的Yang-Mills规范场.     

Yang-Mills规范场的存在性、可解性(一)

应用数学机械化方法研究欧氏空间中$SU(2)$ Yang-Mills规范场的存在性问题.首先对YM--方程的结构进行了讨论,说明YM--方程由它的奇部份和偶部份联立组成.对于YM--方程构造了一类线性微分变换,称之为$SU(2)$规范场的示性变换.经示性变换,将非线性的YM--方程的奇部份变为一组Laplace方程,实现了$SU(2)$规范场方程的线性化.从而证明了$SU(2)$规范场存在3个独立的Yang-Mills规范场.     

Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

This paper is concerned with the 2D relativistic membrane equation evolving in curved space‐time, whose metric is prescribed as a small perturbation to the flat Friedmann‐Lemaître‐Robertson‐Walker (FLRW) metric with time‐dependent coefficients. Thanks to the partial “null structure” of the membrane equation and the properties of the background metric, we could prove the global stability of a class of time‐dependent solutions by weighted energy estimate.     

Surfaces of revolution associated with the kink-type solutions of the SIdV equation

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.     

Metric regularity of a positive order for generalized equations

This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Fréchet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.     

Positivity of relative canonical bundles and applications

Given a family of canonically polarized manifolds, the unique K?hler–Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle . We use a global elliptic equation to show that this metric is strictly positive on , unless the family is infinitesimally trivial.