On the Morgan-Shalen compactification of the SL(2,C) character varieties of surface groups

http://arxiv.org/PS_cache/math/pdf/9810/9810034v1.pdfA gauge theoretic description of the Morgan-Shalen compactification of the SL(2, C) character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map

Perturbed geodesics on the moduli space of flat connections and Yang-Mills theory

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang-Mills connections with the same Morse index.Comment: 58 pages, 3 figure

On the structure of the Yang-Mills-Higgs equations on R[sup]3

The Yang-Mills-Higgs theory has its origins in Physics. It describes particles with masses via the Higgs mechanism and predicts magnetic monopoles. We study here the mathematical aspects of the theory following an analytical and geometric approach. Our motivation comes from physics and we work all the time with the full Lagrangian of the theory. At the same time, we are interested in it from the variational point of view, as a functional on an infinite dimensional space and as a system of non-linear equations on a non-compact manifold with finite energy as the only constraint. We are concerned mainly with the configuration space of the theory, the existence of solutions and their behaviour at infinity

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.Comment: 89 pages, 3 figure

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

In this paper, we construct a Lagrangian submanifold of the moduli space associated to the fundamental group of a punctured Riemann surface (the space of representations of this fundamental group into a compact connected Lie group). This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on the moduli space. The notion of decomposable representation provides a geometric interpretation of this Lagrangian submanifold

The Conley Conjecture and Beyond

This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur

On the structure of the Yang-Mills-Higgs equations on IR"3

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Editor: R. de la Llave HOMOGENEOUS AND ISOTROPIC STATISTICAL SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

Abstract. Two constructions of homogeneous and isotropic statistical solutions of the 3D Navier-Stokes system are presented. First, homogeneous and isotropic probability measures supported by weak solutions of the Navier-Stokes system are produced by av-eraging over rotations the known homogeneous probability measures, supported by such solutions, of [VF1], [VF2]. It is then shown how to approximate (in the sense of con-vergence of characteristic functionals) any isotropic measure on a certain space of vector fields by isotropic measures supported by periodic vector fields and their rotations. This is achieved without loss of uniqueness for the Galerkin system, allowing for the Galerkin ap-proximations of homogeneous statistical Navier-Stokes solutions to be adopted to isotropic approximations. The construction of homogeneous measures in [VF1], [VF2] then applies to produce homogeneous and isotropic probability measures, supported by weak solutions of the Navier-Stokes equations. In both constructions, the restriction of the measures at t = 0 is well defined and coincides with the initial measure