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

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

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

universal positive mass theorems in this paper we develop a general study of contributions at infinity of bochner weitzenbock type formulas on asymptotically flat manifolds inspired by witten s proof of the positive mass theorem as an application we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied

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

a large class of non constant mean curvature solutions of the einstein constraint equations on an asymptotically hyperbolic manifold we construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method our approach consists in decreasing a certain exponent appearing in the equations constructing solutions of these sub critical equations and then in letting the exponent tend to its true value we prove that the solutions of the sub critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non trivial solution finally we give conditions which ensure that the limit equation admits no non trivial solution

metrics of negative ricci curvature one of the most natural and important topics in riemannian geometry is the relation between curvature and global structure of the underlying manifold for instance complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic furthermore it seemed to be a natural principle that a closed manifold cannot carry two metrics of different signed curvatures as it is a basic fact that this is true for sectional curvature but it turned out to be wrong much later and from a strongly analytic argument for the scalar curvature s since each manifold m n 3 admits a complete metric with s 1 cf aubin a and bland kalka bik hence the situation for ricci curvature ric lying between sectional and scalar curvature seemed to be quite delicate up to now the most general results concerning ric 0 were proved by gao yau gy and brooks br using thurston s theory of hyperbolic threemanifolds viz each closed three manifold admits a metric with ric 0 this is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities gao and yau resp brooks smoothed these singularities to get a regular metric with ric 0 these methods extend to three manifolds of finite type and certain hyperbolic orbifolds in any case the arguments rely on exploiting some extraordinary metric structures whose existence is neither obvious nor conceptually related to the ricci curvature problem indeed the existence depends on the assumption that the manifold is three dimensional and compact moreover this approach does not provide insight into the typical behaviour of metrics with ric 0 since one is led to very special metrics in this article we approach negative ricci curvature using a completely different and new concept which will become even more significant in l2 as we deliberately produce ric 0 actually we will prove the following results in these notes ric g resp r g denotes the ricci tensor resp curvature of a smooth metric g

a note on positive energy theorem for spaces with asymptotic susy compactification we extend the higher dimensional positive mass theorem in dai x commun math phys 244 335 345 2004 to the lorentzian setting this includes the original higher dimensional positive energy theorem whose spinor proof is given in witten e commun math phys 80 381 402 1981 and parker t and taubes c commun math phys 84 223 238 1982 for dimension 4 and in zhang x j math phys 40 3540 3552 1999 for dimension 5

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

on a conformally invariant variational problem on differential forms abstract we consider a conformally invariant constrained variational problem on p differential forms for p dim m 2 2 coming from conformal geometry and the associated yamabe type system of partial differential equations it is shown that the infimum p m of the functional is not bigger than p r n and if p m p r n then p m is achieved by a c 2 p form for some 0

elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation abstract we derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems this result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation in particular we characterize the bifurcation values from infinity to positive steady states the stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed

a spinorial analogue of aubin s inequality let m g be a compact riemannian spin manifold of dimension 2 for any metric tilde g conformal to g we denote by tilde lambda the first positive eigenvalue of the dirac operator on m tilde g sigma we show that rm inf tilde g in g tilde lambda rm vol m tilde g 1 n leq n 2 rm vol s n 1 n this inequality is a spinorial analogue of aubin s inequality an important inequality in the solution of the yamabe problem the inequality is already known in the case n 3 and in the case n 2 ker d 0 our proof also works in the remaining case n 2 ker d 0 with the same method we also prove that any conformal class on a riemann surface contains a metric with 2 tilde lambda 2 leq tilde mu where tilde mu denotes the first positive eigenvalue of the laplace operator

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

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

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 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

fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines a method is presented for fitting a function defined on a general smooth spherelike surfaces given measurements on the function at a set of scattered points lying ons the approximating surface is constructed by mapping the surface onto a rectangle and using a tensor product of polynomial splines with periodic trigonometric splines the use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points ons two alternative algorithms for computing the coefficients of the tensor fit are presented one based on global least squares and the other on the use of local quasi interpolators the approximation order of the method is established and the numerical performance of the two algorithms is compared

quadratic spline wavelets with arbitrary simple knots on the sphere in this paper we extend the method for fitting functions on the sphere described in lyche and schumaker siam j sci comput 22 2 2000 724 to the case of nonuniform knots we present a multiresolution method leading to functions on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences we determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces we describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces we give the different steps of computer implementation and finally we present a test example by using two knot sequences a uniform one and a sequence of chebyshev points

quantum existence testing and its application for finding extreme values in unsorted databases many information processing and computing problems can be traced back to find the extreme value of a database or a function unfortunately classical solutions suffer from high computational complexity if the database is unsorted or equivalently the function has many local minimum maximum points proposed quantum computing based solutions involve the repeated application of grower s searching algorithm in this paper we introduce a new technique exploiting the parallel processing capabilities of quantum computing in a different way we derive a special case of quantum counting we call it quantum existence testing which allows adapting the classical logarithmic search algorithm so that it is suitable for structured databases to unstructured ones the paper analyzes the required number of database queries the corresponding computational complexity and the probability of error and their relationship

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

a compactness theorem for scalar flat metrics on 3 manifolds with boundary abstract let m g be a compact riemannian three dimensional manifold with boundary we prove the compactness of the set of scalar flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface this involves a blow up analysis of a yamabe type equation with critical sobolev exponent on the boundary

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

yamabe flow and adm mass on asymptotically flat manifolds in this paper we investigate the behavior of adm mass and einstein hilbert functional under the yamabe flow through studying the yamabe flow by weighted spaces for parabolic operators we show that the asymptotically flat property is preserved under the yamabe flow we also obtain that adm mass is invariant under the yamabe flow and yamabe flow is the gradient flow of einstein hilbert functional on n dimensional n 3 asymptotically flat manifolds with order n 22 for n 3 4 or n 3 for n 4 moreover we show that adm mass and einstein hilbert functional are non increasing under the yamabe flow on n dimensional asymptotically flat manifolds if we only assume the order n 22 for n 4

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

growing up positive solutions of semilinear elliptic boundary value problems this paper is devoted to the study of the existence uniqueness and asymptotic behavior of positive solutions of a class of degenerate boundary value problems for semilinear second order elliptic differential operators which originates from the so called yamabe problem in riemannian geometry our approach is based on the super sub solution method adapted to the degenerate case

prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature let m n g be a complete noncompact riemannian manifold with the curvature bounded between two negative constants given a function k on m n in terms of the behaviors of k at infinite we give a fairly complete answer to when the k can be the scalar curvature function of a complete metric g 1 which is conformal to g

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

maskit combinations of poincare einstein metrics we establish a boundary connected sum theorem for asymptotically hyperbolic einstein metrics and also show that if the two metrics have scalar positive conformal infinities then the same is true for this boundary join this construction is also extended to spaces with a finite number of interior conic singularities and as a result we show that any 3 manifold which is a finite connected sum of quotients of s3 and s2 s1 bounds such a space with conic singularities putatively any 3 manifold admitting a metric of positive scalar curvature is of this form

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

a class of fully nonlinear equations arising from conformal geometry in this paper we study the long time existence of solutions for a class of fully nonlinear parabolic equations arising from conformal geometry in particular we prove that every smooth compact n dimensional manifold n ge3 admits a riemannian metric g with its ricci curvature ric and scalar curvature r satisfying det ric r cdot g const

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

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

uniqueness of the functional determinant the functional determinant of the conformal laplacian and the square of the dirac operator are known to be extremized at the standard round metric of the four sphere among all conformal metrics up to gauge equivalence in this article we show that this is the unique critical point thus extending the work of onofri and osgood phillips and sarnak for the functional determinant on s 2 which characterized the constant curvature metric as the unique critical point of the determinant in addition we introduce a new symmetric two tensor field which is defined on any conformally flat four manifold and can be viewed as a fourth order generalization of the einstein gravitational tensor as a consequence we prove a pohozaev identity for manifolds with boundary which admit conformal killing vector fields

an obata singular theorem for stratified spaces consider a stratified space with a positive ricci lower bound on the regular set and no cone angle larger than 2 for such stratified space we know that the first non zero eigenvalue of the laplacian is larger than or equal to the dimension we prove here an obata rigidity result when the equality is attained the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension moreover we show that the diameter is at most equal to and it is equivalent for the diameter to be equal to and for the first non zero eigenvalue of the laplacian to be equal to the dimension we finally give a consequence of these results related to the yamabe problem consider an einstein stratified space without cone angles larger than 2 if there is a metric conformal to the einstein metric and with constant scalar curvature then it is an einstein metric as well furthermore if its conformal factor is not a constant then the space is isometric to a spherical suspension

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

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

on the positive solutions of semilinear equations δ ℎ 0 on the compact manifolds in this paper we study the existence nonexistence and uniqueness of positive solutions of semilinear equations u u hu p 0 on compact riemannian manifolds as well as on bounded smooth domains in r n with homogeneous dirichlet or neumann boundary conditions

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

a multiresolution tensor spline method for fitting functions on the sphere we present the details of a multiresolution method which we proposed at the taormina wavelet conference in 1993 see l spline wavelets in wavelets theory algorithms and applications c chui l montefusco and l puccio eds academic press new york pp 197 212 which is suitable for fitting functions or data on the sphere the method is based on tensor products of polynomial splines and trigonometric splines and can be adapted to produce surfaces that are either tangent plane continuous or almost tangent plane continuous the result is a convenient compression algorithm for dealing with large amounts of data on the sphere we give full details of a computer implementation that is highly efficient with respect to both storage and computational cost we also demonstrate the performance of the method on several test examples

compactness for a class of yamabe type problems on manifolds with boundary abstract in this paper we are interested in studying the set of positive solutions to the following yamabe type problem g u a x u 0 in m u g b x u n 2 u n n 2 on m where m n g is a compact riemannian n manifold n 3 with boundary g is the laplace beltrami operator for g g is the pointing unit outward normal to m a c m and b c m assuming suitable hypotheses on the functions a n 2 4 n 1 r g and b n 2 2 h g where r g denotes the scalar curvature of the metric g and h g is the mean curvature of m we prove some results of compactness for the set of positive solutions to the problem