stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

instanton approximation periodic asd connections and mean dimension abstract we study a moduli space of asd connections over s 3 r we consider not only finite energy asd connections but also infinite energy ones so the moduli space is infinite dimensional in general we study the local mean dimension of this infinite dimensional moduli space we show the upper bound on the mean dimension by using a runge approximation for asd connections and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic asd connections

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

a bilinear estimate and its application to a quadratic nonlinear klein gordon equation in two space dimensions abstract we study the existence of the wave operators for the nonlinear klein gordon equation with quadratic nonlinearity in two space dimensions t 2 1 u u 2 t x r r 2 we prove existence of wave operators in lower order sobolev spaces by using estimates of bilinear operators associated with klein gordon equation

conformal fields and stability lagrangiens naturels conformes deformations conformes theoremes d instabilite pour des spheres theoremes d annulation sur des espaces non compacts

a direct method for minimizing the yang mills functional over 4 manifolds a direct method is employed to minimize the yang mills functional over a 4 dimensional manifold the limiting connection is shown to be yang mills but in a possibly new bundle we show that a topological invariant of the bundle is preserved by the minimizing process this implies the existence of an absolute minimum of the yang mills functional in a wide class of bundles

the atiyah jones conjecture for rational surfaces we show that if the atiyah jones conjecture holds for a surface x then it also holds for the blow up of x at a point since the conjecture is known to hold for and for ruled surfaces it follows that the conjecture is true for all rational surfaces

the topology of instanton moduli spaces i the atiyah jones conjecture in this paper we study the global geometry and topology of the moduli spaces of based su 2 instantons over the 4 sphere s4 these instanton moduli spaces have a rich history and have been analyzed from many points of view originally these spaces which we denote by mk were defined as solution spaces modulo gauge equivalence to certain partial differential equations namely the self duality equations associated to the yang mills functional in su 2 gauge theory they have been successfully studied from this point of view by taubes t1 t2 uhlenbeck u and others an important alternative approach was initiated by ward w who related instantons to certain holomorphic bundles on cp3 and was continued by atiyah and ward in aw this allowed the classification of instantons on 4 in terms of quaternionic linear algebra by atiyah drinfeld hitchin and manin adhm this holomorphic approach was further extended by donaldson d who showed that these bundles were determined by their restriction to a cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in c2r2 hurtubise hul then exploited this fact to study the moduli spaces mk as did atiyah a to show that mk arise naturally in the theory of holomorphic maps into loop groups this latter approach was continued by gravesen g atiyah and jones aj obtained the first results and formulated the foundational questions on the global topology of these moduli spaces recall that an element of mk is a based gauge equivalence class of a connection on the principal su 2 bundle over 4 denoted by pk with second chern class k satisfying the self duality equations there is a natural forgetful map to the based equivalence classes of all connections in pk atiyah and jones aj showed that this target space which we denote by bk is homotopy equivalent

small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in four dimensions ii in this paper we complete the proof of the existence of multiple solutions and in particular non minimal ones to the e dirichlet problem obtained as a variational problem for the su 2 e yang mills functional this is equivalent to proving the existence of multiple solutions to the dirichlet problem for the su 2 yang mills functional with small boundary data in the first paper of this series this non compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function je q which is essentially the yang mills functional evaluated on the approximate solutions constructed via a gluing technique in the present paper we establish a morse theory for je q by means of ljusternik schnirelmann theory thus complete the proofs of theorems 1 3 given by isobe and marini small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in 4 dimensions part i j math phys 53 063706 2012 10 1063 1 4728211

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

convergence of yang mills higgs fields in this paper we study the convergence of yang mills higgs ymh fields defined on fiber bundles over riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary we show that away from the nodes the ymh fields converges up to gauge to a smooth ymh field modulo finitely many harmonic spheres while near the nodes where the conformal structure degenerates the ymh fields converges to a pair consisting of a flat connection and a twisted geodesic with potential in particular we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general ymh fields

yang mills fields which are not self dual the purpose of this paper is to prove the existence of a new family of non self dual finite energy solutions to the yang mills equations on euclidean four space withsu 2 as a gauge group the approach is that of equivariant geometry attention is restricted to a special class of fields those that satisfy a certain kind of rotational symmetry for which it is proved that 1 a solution to the yang mills equations exists among them and 2 no solution to the self duality equations exists among them the first assertion is proved by an application of the direct method of the calculus of variations existence and regularity of minimizers and the second assertion by studying the symmetry properties of the linearized self duality equations the same technique yields a new family of non self dual solutions on the complex projective plane

remark on the global existence and large time asymptotics of solutions for the quadratic nls abstract we study the global in time existence of small solutions to the nonlinear schrodinger equation with quadratic interactions 0 1 i t u 1 2 u u 2 t 0 x r 4 u 0 x u 0 x x r 4 we prove that if the initial data u 0 satisfy smallness conditions in the weighted sobolev norm then the solution of the cauchy problem 0 1 exists globally in time furthermore we prove the existence of the usual scattering states and find the large time asymptotics of the solutions

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

optimal small data scattering for the generalized derivative nonlinear schrodinger equations abstract in this work we consider the following generalized derivative nonlinear schrodinger equation i t u x x u i u 2 x u 0 t x r r we prove that when 2 the solution is global and scattering when the initial data is small in h s r 1 2 s 1 moreover we show that when 0 2 there exist a class of solitary wave solutions c satisfying c h 1 r 0 when c tends to some endpoint which is against the small data scattering statement therefore in this model the exponent 2 is optimal for small data scattering we remark that this exponent is larger than the short range exponent and the strauss exponent

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

energy gap for yang mills connections i four dimensional closed riemannian manifolds we extend an l2l2 energy gap result due to min oo 28 theorem 2 and parker 30 proposition 2 2 for yang mills connections on principal g bundles p over closed connected four dimensional oriented smooth manifolds x from the case of positive riemannian metrics to the more general case of good riemannian metrics including metrics that are generic and where the topologies of p and x obey certain mild conditions and the compact lie group g is su 2 su 2 or so 3 so 3

a framework for morse theory for the yang mills functional on explique comment la theorie de morse pour la fonctionnelle de yang mills peut se retrouver en examinant la restriction de la fonctionnelle a un ensemble denombrable de varietes non compactes de dimension finie

response of a three degree of freedom system with cubic non linearities to harmonic excitation an investigation of the response of a three degree of freedom system with cubic non linearities and the autoparametric resonances w 3 3 w 2 and w 2 3 w 1 to a harmonic excitation of the third mode where the w n are the linear natural frequencies of the system is presented the method of multiple scales is used to determine six first order non linear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes the fixed points of these equations are obtained and their stability is determined numerical solutions are presented discussion of the figures is given

a quadratic nonlinear schrodinger equation in one space dimension abstract in this paper we study the cauchy problem for the quadratic derivative nonlinear schrodinger equation iu t u xx u x 2 t x r 2 u 0 x u 0 x r we suppose that the initial data are small in the weighted sobolev space h 3 1 r then we prove that there exist global in time small solutions to the cauchy problem we study the large time behavior of solutions and construct the modified asymptotics for large time values

symmetric instantons and the adhm construction we construct allsu 2 yang mills instantons ons 4 that admit a certain symmetry quadrupole symmetry this is accomplished by an equivariant version of the adhm monad classification of instantons this work is part of an attempt to better understand the structure of non self dual yang mills connections with the same symmetry

modal interaction in circular plates abstract the dynamic analogue of the von karman equations in polar form is used to study the symmetric response of a circular plate to a harmonic external excitation the method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the phases and amplitudes of oscillation for the case of an internal or autoparametric resonance the results show that the internal resonance is responsible for coupling of the modes involved and that the excited mode is not necessarily the dominant one furthermore the results show that for a choice of parameters the multi mode response loses its stability through a hopf bifurcation resulting in periodically or chaotically modulated motions of the plate

the palais smale conditions for the yang mills functional on considere la fonctionnelle de yang mills definie sur des connexions sur un fibre principal sur une variete de riemann compacte de dimension 2 ou 3 on montre que si on considere la fonctionnelle de yang mills comme etant definie sur une variete de hilbert appropriee d orbites des connexions sous l action du groupe des automorphismes du fibre principal alors la fonctionnelle satisfait la condition de palais smale

non self dual yang mills connections with quadrupole symmetry we prove the existence of non self dual yang mills connections onsu 2 bundles over the four sphere specifically on all bundles with second chern number not equal 1 we study connections equivariant under ansu 2 symmetry group to reduce the effective dimensionality from four to one and then use variational techniques the existence of non self dualsu 2 ym connections on the trivial bundle second chern number equals zero has already been established by sibner sibner and uhlenbeck via different methods

solutions to yang mills equations that are not self dual abstract the yang mills functional for connections on principle su 2 bundles over s4 is studied critical points of the functional satisfy a system of second order partial differential equations the yang mills equations if in particular the critical point is a minimum it satisfies a first order system the self dual or anti self dual equations here we exhibit an infinite number of finite action nonminimal unstable critical points they are obtained by constructing a topologically nontrivial loop of connections to which min max theory is applied the construction exploits the fundamental relationship between certain invariant instantons on s4 and magnetic monopoles on h3 this result settles a question in gauge field theory that has been open for many years

heat flow of yang mills higgs functionals in dimension two abstract we consider the heat flow of yang mills higgs functional where the base manifold is a riemannian surface and the fiber is a compact symplectic manifold we show that the corresponding cauchy problem admits a global weak solution for any h 1 initial data moreover the solution is smooth except for finitely many singularities we prove an energy identity at finite time singularities and give a description of the asymptotic behavior at time infinity

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations abstract for singular perturbation problems the renormalization group rg method of chen goldenfeld and oono phys rev e 49 1994 4502 4511 has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system it has been applied to a variety of problems traditionally analyzed using disparate methods including the method of multiple scales boundary layer theory the wkbj method the poincare lindstedt method the method of averaging and others in this article we show how the rg method may be used to generate normal forms for large classes of ordinary differential equations first we apply the rg method to systems with autonomous perturbations and we show that the reduced or amplitude equations generated by the rg method are equivalent to the classical poincare birkhoff normal forms for these systems up to and including terms of o 2 where is the perturbation parameter this analysis establishes our approach and generalizes to higher order second we apply the rg method to systems with nonautonomous perturbations and we show that the reduced or amplitude equations so generated constitute time asymptotic normal forms which are based on kbm averages moreover for both classes of problems we show that the main coordinate changes are equivalent up to translations between the spaces in which they are defined in this manner our results show that the rg method offers a new approach for deriving normal forms for nonautonomous systems and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves finally we establish how well the solution to the rg equations approximates the solution of the original equations on time scales of o 1

classification of singular sobolev connections by their holonomy for a connection on a principalsu 2 bundle over a base space with a codimension two singular set a limit holonomy condition is stated in dimension four finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy if there is no holonomy then a codimension two removable singularity theorem is obtained

the atiyah jones conjecture the purpose of this note is to announce our proof of the atiyah jones conjecture concerning the topology of the moduli spaces of based su 2 instantons over s 4 full details and proofs appear in our paper bhmm1

periodic and non periodic combination resonance in kinematically excited system of rods a theoretical investigation has been made of periodic and non periodic combination resonances of the system of rods under vertical and horizontal kinematic excitations the elements of the system are connected with articulated points the coupling of the elements of the system through internal longitudinal forces which are transverse forces at the ends of neighboring rods are taken into account the equations of motion are obtained from the lagrange equations the mathematical analysis of the equations of motion is accomplished by using tondl s technique the steady state solutions and their stability are determined resonance curves for the stationary states have been determined numerically

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

on the internal resonance of multi degree of freedom systems with cubic non linearity abstract a new method of multiple scales procedure is used to analyze the forced responses of multiple degrees of freedom systems with cubic non linearity in which the internal resonance takes place simultaneously together with the forced resonances attention is focused on the superharmonic and subharmonic resonance caused by internal resonance as is near 1 or 2 where is the forcing frequency and 1 and 2 are the first and the second natural frequencies it is found that the responses are usually excitation level dependent and there is a critical excitation amplitude c in the case of near 1 when c the out of phase superharmonic resonance disappears in the case of near 2 when 0 c there are two branches of response when c one branch of responses vanishes the critical excitation amplitudes in different forced vibration are determined analytically in this paper

morse theory for the yang mills functional via equivariant homotopy theory in this paper we show the existence of non minimal critical points of the yang mills functional over a certain family of 4 manifolds m29 g 0 1 2 with generic su 2 invariant metrics using morse and homotopy theoretic methods these manifolds are acted on fixed point freely by the lie group su 2 with quotient a compact riemann surface of even genus we use a version of invariant morse theory for the yang mills functional used by parker in a morse theory for equivariant yang mills duke math j 66 2 1992 337 356 and rade in compactness theorems for invariant connections submitted for publication

moduli spaces of anti self dual connections on ale gravitational instantons we study the framed moduli space m of anti self dual connections on a principal bundle over an ale hyperkahler 4 manifold it has the natural riemannian metric g m and the hyperkahler structure i m j m k m induced from those on the base manifold