on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

global bifurcation of limit cycles in a family of polynomial systems in this paper we study the number of limit cycles in a family of polynomial systems using bifurcation methods and melnikov functions we obtain the maximal number of limit cycles in global bifurcation

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli abstract we study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations the cubic planar system x y y x 1 2 y 2 x x 1 2 y 2 has simultaneously a center at the origin and at infinity we study up to first order averaging analysis the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously this problem is a generalization of 24 to the piecewise systems class when the polynomial perturbation has degree n we prove that the inner and outer abelian integrals are rational functions and we provide an upper bound for the number of zeros when the perturbation is cubic the same degree as the unperturbed vector field the maximum number of limit cycles up to first order perturbation from the inner and outer annuli is 9 and 8 respectively when the simultaneous bifurcation problem is considered 12 limit cycles exist these limit cycles appear in three types of configurations 9 3 6 6 and 4 8 in the non piecewise scenario only 5 limit cycles were found

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

equivalence theorems and hopf galois extensions in this article we apply equivalence theorems for categories of modules d to the category m h of right d y a hopf modules to get a charactera ization of hopf galois extensions indeed let p q g mod r and t s end p and assume that p ber longs to the category s q of modules subgenerated by q in theorem r r w 2 3 by modifying a theorem formulated by dal pio and orsatti 5 x theorem 2 6 we study the situation where for every m belonging to the category s q the map r

limit cycles appearing from the perturbation of a system with a multiple line of critical points abstract consider planar ordinary differential equations of the form x y c x y y x c x y where c x y is an algebraic curve we are interested in knowing whether the existence of multiple factors for c is important or not when we study the maximum number of zeros of the abelian integral m that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field with this aim we study in detail the case c x y 1 y m where m is a positive integer number and prove that m has essentially no impact on the number of zeros of m this result improves the known studies on m one of the key points of our approach is that we obtain a simple expression of m based on some successive reductions of the integrals appearing during the procedure

moduli spaces of critical riemannian metrics with l n 2 norm curvature bounds abstract we consider the moduli space of a class of critical metrics on compact manifolds that includes extremal kahler metrics we show that under the conditions of volume and diameter bounds l n 2 curvature bounds and sobolev constant bounds this moduli space can be compactified by including orbifolds with finitely many orbifold points in dimension 4 these orbifold points are c riemannian orbifold points and in the 4 dimensional extremal kahler case they are also of class c

hopf galois extensions and an exact sequence for h picard groups let h be a hopf algebra and a an h galois extension we investigate h morita autoequivalences of a introduce the concept of h picard group and we establish an exact sequence linking the h picard group of a and the picard group of acoh

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

initial data sets of minimal energy the total mass of a gravitational system the adm mass is considered as a functional of the initial data sets for sources and gravitation from the condition that such a functional has a minimum under a restricted class of variations tensorial equations for the gravitational initial data are obtained the solutions to these equations whose existence for a large class of source fields is asserted represent initial data sets for gravitation satisfying the constraint equations and may be considered as having no gravitational radiation

subcritical approach to conformally invariant extension operators on the upper half space abstract in this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators this family includes among others the classical poisson extension operator and the extension operator with riesz kernel we show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions we also compute the limiting behavior at the boundary of the extensions of the extremal functions

asymptotically flat initial data with prescribed regularity at infinity we prove the existence of a large class of asymptotically flat initial data with non vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space like infinity in terms of powers of a radial coordinate

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

the number of limit cycles in perturbations of polynomial systems with multiple circles of critical points abstract this paper investigates the problem for limit cycle bifurcations of system x y f x y e p x y y x f x y e q x y where f x y consists of multiple circles and p x y q x y are polynomials of degree n the upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order melnikov function furthermore hopf bifurcation for a cubic system of this type is discussed

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points abstract this paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems whose unperturbed system has a center at the origin and may have invariant multiply lines by melnikov function method this paper provides the upper and lower bounds for the poincare cyclicity in particular the exact cyclicities of two specific cases are given using analyzing method one of which has been discussed in the paper 7

the number of limit cycles of a class of polynomial differential systems abstract in this paper we consider the bifurcation of limit cycles of a class of polynomial differential systems of the form x y 1 x 4 e p n x y y x 1 x 4 e q n x y where p n q n are arbitrary polynomials of degree n we prove that there is a system of the above form having at least 3 n 1 2 2 limit cycles in hopf bifurcation then applying the argument principle we obtain that up to first order in e the number of limit cycles that bifurcate from period annulus surrounding the origin of this system is at most 5 n 1 4 n 1 2 for n 5

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

upper bounds for the number of zeroes for some abelian integrals consider the vector field x yg x y y xg x y where the set of critical points g x y 0 is formed by k straight lines not passing through the origin and parallel to one or two orthogonal directions we perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of k and n our approach is based on the explicit computation of the abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions when we apply our results for k 4 we recover or improve some results obtained in several previous works

on the number of limit cycles of a perturbed cubic polynomial differential center abstract for e sufficiently small we consider the planar polynomial differential system x y y 2 a x 2 b x c e p x y y x y 2 a x 2 b x c e q x y where p x y and q x y are polynomials of degree n by using the averaging method of first order we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

rough solutions of the einstein constraints on closed manifolds without near cmc conditions we consider the conformal decomposition of einstein s constraint equations introduced by lichnerowicz and york on a closed manifold we establish existence of non cmc weak solutions using a combination of priori estimates for the individual hamiltonian and momentum constraints barrier constructions and fixed point techniques for the hamiltonian constraint riesz schauder theory for the momentum constraint together with a topological fixed point argument for the coupled system although we present general existence results for non cmc weak solutions when the rescaled background metric is in any of the three yamabe classes an important new feature of the results we present for the positive yamabe class is the absence of the near cmc assumption if the freely specifiable part of the data given by the traceless transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small and if the energy density of matter is not identically zero in this case the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives so that it can be arbitrarily far from constant giving what is apparently the first existence results for non cmc solutions without the near cmc assumption using a coupled topological fixed point argument that avoids near cmc conditions we establish existence of coupled non cmc weak solutions with positive conformal factor w s p where p 1 and s p 1 3 p in the cmc case the regularity can be reduced to p 1 and s p 3 p 1 in the case of s 2 we reproduce the cmc existence results of choquet bruhat 10 and in the case p 2 we reproduce the cmc existence results of maxwell 33 but with a proof that goes through the same analysis framework that we use to obtain the non cmc results the non cmc results on closed manifolds here extend the 1996 non cmc result of isenberg and moncrief in three ways 1 the near cmc assumption is removed in the case of the positive yamabe class 2 regularity is extended down to the maximum allowed by the background metric and the matter and 3 the result holds for all three yamabe classes this last extension was also accomplished recently by allen clausen and isenberg although their result is restricted to the near cmc case and to smoother background metrics and data

morita equivalences induced by bimodules over hopf galois extensions let h be a hopf algebra and a b be h galois extensions we investigate the category mbha of relative hopf bimodules and the morita equivalences between a and b induced by them

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

conformal structures of static vacuum data in the cauchy problem for asymptotically flat vacuum data the solution jets along the cylinder at space like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future past null infinity the tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity for the solution jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the cotton tensor of the initial three metric h satisfies a certain conformally invariant condition at space like infinity it is sufficient that h be asymptotically static at space like infinity the purpose of this article is to characterize the gap between these conditions we show that with the class of metrics which satisfy condition on the cotton tensor and a certain non degeneracy requirement is associated a one form with conformally invariant differential d we provide two criteria if h is real analytic is closed and one of its integrals satisfies a certain equation then h is conformal to static data near space like infinity if h is smooth is asymptotically closed and one of its integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space like infinity

the momentum constraints of general relativity and spatial conformal isometries transverse tracefree tt tensors on r3 gab withgab an asymptotically flat metric of fast decay at infinity are studied when the source tensor from which these tt tensors are constructed has fast fall off at infinity tt tensors allow a multipole type expansion whengab has no conformal killing vectors ckv s it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable tt tensor when ckv s exist there are obstructions certain combinations of moments have to vanish which we study