The Goursat problem for the Einstein–Yang–Mills–Higgs system in weighted Sobolev spaces

We establish original Moser estimates to clarify and complete previous works of Christodoulou and Müller zum Hagen concerning local existence and uniqueness results for the Goursat problem associated to second order quasilinear hyperbolic systems. As an application we locally solve, in some weighted Sobolev spaces, the Goursat problem for the Einstein–Yang–Mills–Higgs system using harmonic and Lorentz gauges.     

Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Local well-posedness for the (n + 1)-dimensional Yang–Mills and Yang–Mills–Higgs system in temporal gauge

The Yang–Mills and Yang–Mills–Higgs equations in temporal gauge are locally well-posed for small and rough initial data, which can be shown using the null structure of the critical bilinear terms. This carries over a similar result by Tao for the Yang–Mills equations in the (3+1)-dimensional case to the more general Yang–Mills–Higgs system and to general dimensions.     

Regularity and vanishing theorems for Yang–Mills–Higgs pairs

Archiv der Mathematik - We obtain a vanishing theorem for Yang–Mills–Higgs pairs on Euclidean and hyperbolic spaces in dimensions greater than 4, as well as a regularity theorem more...     

Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang?CMills?CHiggs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.     

On the Boyd–Kadomtsev system for the three-wave coupling problem

The three-wave coupling system is widely used in plasma physics, specially for the Brillouin instability simulations. We study here a related system obtained with an infinite speed of light. After showing that it is well posed, we propose a numerical method which is based on an implicit time discretization. This method is illustrated on test cases and an extension to the problem with finite speed of light is proposed. To cite this article: R. Sentis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Morse homology for the Yang–Mills gradient flow

We use the Yang–Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse chain complex. The chain groups are generated by Yang–Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang–Mills gradient flow lines that converge asymptotically to Yang–Mills connections.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.     

On the Cauchy problem for a two-component Degasperis–Procesi system

This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented.     

Langevin dynamic for the 2D Yang–Mills measure

Publications mathématiques de l'IHÉS - It is well known that a real analytic symplectic diffeomorphism of the $2d$ -dimensional disk ( $dgeq 1$ ) admitting the origin as a...     

Universal Invariant Renormalization for the Supersymmetric Yang–Mills Theory

We propose a manifestly invariant renormalization scheme for N=1 non-Abelian supersymmetric gauge theories.     

Cauchy problem on a characteristic cone for the Einstein–Vlasov system: (I) The initial data constraints

In this paper, one considers a Cauchy problem with data on a characteristic cone for the Einstein–Vlasov system in temporal gauge. One highlights gauge-dependent constraints that, supplemented by the standard constraints i.e. the Hamiltonian and the momentum constraints, define the full set of constraints for the considered setting. One studies their global resolution from a suitable choice of some free data, the behavior of the deduced initial data at the vertex of the cone, and the preservation of the gauge.     

On the Monge–Kantorovich problem with additional linear constraints

The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for c-monotonicity.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

We review the notions of (weak) Hermitian–Yang–Mills structure and approximate Hermitian–Yang–Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K?hler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin–Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian–Yang–Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian–Yang–Mills structure is in fact the differential- geometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object.     

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.     

The Cauchy Problem for the Massive Yang－Mills－Higgs Equation in Minkowski Space

ＴｈｅＣａｕｃｈｙＰｒｏｂｌｅｍｆｏｒｔｈｅＭａｓｓｉｖｅＹａｎｇ－Ｍｉｌｌｓ－ＨｉｇｇｓＥｑｕａｔｉｏｎｉｎＭｉｎｋｏｗｓｋｉＳｐａｃｅＧｕｏＢｏｌｉｎｇ（郭柏灵），ＹｕａｎＧｕａｎｇｗｅｉ（袁光伟）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｙｓｉ...     

Hamiltonian structure of the Yang–Mills functional

Hamilton equations based upon a general Lepagean equivalent of the Yang–Mills Lagrangian are investigated. A regularization of the Yang–Mills Lagrangian which is singular with respect to the standard regularity conditions is derived.