global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

asymptotically simple solutions of the vacuum einstein equations in even dimensions we show that a set of conformally invariant equations derived from the fefferman graham tensor can be used to construct global solutions of vacuum einstein equations in all even dimensions this gives in particular a new simple proof of friedrich s result on the future hyperboloidal stability of minkowski space time and extends its validity to even dimensions

weighted pseudo almost periodic functions convolutions and abstract integral equations abstract this paper deals with a systematic study of the convolution operator k f f k defined on weighted pseudo almost periodic functions space pap x and with k l 1 r upon making several different assumptions on k f and we get five main results the first two main results establish sufficient conditions on k and such that the weighted ergodic space pap 0 x is invariant under the operator k the third result specifies a sufficient condition on all functions k f and such that the k f pap 0 x the fourth result is a sufficient condition on the weight function such that pap 0 x is invariant under k the hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of pap 0 x as a consequence of the fifth result we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of pap x with the supremum norm in addition the results on convolution are applied to general abstract integral and differential equations

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

asymptotic behaviors of governing equation of gauged sigma model for heisenberg ferromagnet abstract in this note we study weak solutions of equation 0 1 u 4 e u 1 e u 4 i 1 n p i 4 j 1 m q j in r 2 where p i i 1 n resp q j j 1 m are dirac masses concentrated at the points p i i 1 n resp q j j 1 m and n m 1 eq 0 1 represents a governing equation of gauged sigma model for heisenberg ferromagnet we show that it has a sequence of solutions u having behaviors as ln x o 1 at infinity with a free parameter 2 2 n m and our concern in this paper is to study the asymptotic behavior of b as approaching the extremal values 2 and 2 n m

does asymptotic simplicity allow for radiation near spatial infinity a representation of spatial infinity based on the properties of conformal geodesics is used to obtain asymptotic expansions of the gravitational field near the region where null infinity touches spatial infinity these expansions show that generic time symmetric initial data with an analytic conformal metric at infinity will give rise to developments with a certain type of logarithmic singularities at the points where null infinity and spatial infinity meet these logarithmic singularities produce a non smooth null infinity the sources of the logarithmic singularities are traced back down to the initial data it is shown that if the parts of the initial data responsible for the non regular behaviour of the solutions are not present then the initial data is static to a certain order on the basis of these results it is conjectured that the only time symmetric data sets with developments having a smooth null infinity are those which are static in a neighbourhood of infinity this conjecture generalises a previous conjecture regarding time symmetric conformally flat data the relation of these conjectures to penrose s proposal for the description of the asymptotic gravitational field of isolated bodies is discussed

almost periodic solutions to some semilinear non autonomous thermoelastic plate equations abstract the paper is concerned with the existence of almost periodic solutions to the so called semilinear thermoelastic plate systems for that the strategy consists of seeing these systems as a particular case of the semilinear parabolic evolution equations x t a t x t f t x t t r where a t for t r is a family of sectorial linear operators on a banach space x satisfying the so called acquistapace terreni conditions and f is a function defined on a real interpolation space x for 0 1 under some reasonable assumptions it will be shown that has a unique almost periodic solution we then make use of the previous result to obtain the existence and uniqueness of an almost periodic solution to the thermoelastic plate systems

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

decomposition of weighted pseudo almost periodic functions abstract first we show by constructing two counterexamples that the decomposition of weighted pseudo almost periodic functions is not unique in general then we prove that the decomposition of such functions is unique if p a p 0 x is translation invariant but not necessarily unique without the assumption moreover we give an example to show that the mean value under a certain weight may not exist for all almost periodic functions with these results we answer some fundamental questions on weighted pseudo almost periodic functions

on generalized m quasi einstein manifolds with constant scalar curvature abstract we study generalized m quasi einstein manifolds with constant scalar curvature first we establish a classification of generalized m quasi einstein manifolds with parallel ricci tensor second and particularly if an m quasi einstein manifold possesses constant scalar curvature then we determine the scalar curvature in explicit form and show that which values are attainable by presenting examples

conformal geodesics on vacuum space times we discuss properties of conformal geodesics on general vacuum and warped product space times and derive a system of conformal deviation equations it is then shown how to construct on the schwarzschild kruskal space time globally defined systems of conformal gauss coordinates which extend smoothly and without degeneracy to future and past null infinity

weakly einstein critical metrics of the volume functional on compact manifolds with boundary abstract the goal of this paper is to study weakly einstein critical metrics of the volume functional on a compact manifold m with smooth boundary m here we will give the complete classification for an n dimensional n 3 or 4 weakly einstein critical metric of the volume functional with nonnegative scalar curvature moreover in the higher dimensional case n 5 we will establish a similar result for weakly einstein critical metric under a suitable constraint on the weyl tensor

uniqueness of isometric immersions with the same mean curvature abstract motivated by the quasi local mass problem in general relativity we study the rigidity of isometric immersions with the same mean curvature into a warped product space as a corollary of our main result two star shaped hypersurfaces in a spatial schwarzschild or ads schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature we also prove similar results if the mean curvature condition is replaced by an 2 curvature condition

existence of periodic solutions of partial neutral functional differential equations with unbounded delay abstract in this article we establish a result of existence of periodic solutions for the class of quasi linear partial neutral functional differential equations with unbounded delay that can be described as the abstract functional differential equation d dt x t f t x t ax t g t x t where a is the infinitesimal generator of a strongly continuous semigroup of linear operators on a banach space and f and g are appropriate functions defined on a phase space

existence of weighted pseudo anti periodic solutions to some non autonomous differential equations abstract in this paper we first introduce two new concepts called respectively weighted pseudo periodicity and weighted pseudo anti periodicity which generalize respectively the well known notions of periodicity and that of anti periodicity basic properties of these new functions are discussed furthermore the existence of weighted pseudo anti periodic solutions to some damped non autonomous second order differential equations will be investigated to illustrate our abstract results the existence of weighted pseudo anti periodic solutions to a plate like equation is discussed

elliptic operators in the functional quantisation for gauge field theories given a gauge theory with gauge groupg acting on a path spacex g andx being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary it is shown that when the action ofg onx is smooth free and proper the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection x x g and the existence of the faddeev popov determinant arising in the functional quantisation of the gauge theory this holds for certain gauge theories with anomalies like bosonic closed string theory in non critical dimension and also holds for a class of gauge theories which includes yang mills theory

a remark on comparison results via symmetrization in this paper we study the converse of comparison results for solutions to linear second order elliptic equations namely in the inequalities proved by g talenti and others we study the case of equality and prove that equalities are achieved only in the spherical situation we also present some applications of these results to semilinear elliptic equations

gluing and wormholes for the einstein constraint equations we establish a general gluing theorem for constant mean curvature solutions of the vacuum einstein constraint equations this allows one to take connected sums of solutions or to glue a handle wormhole onto any given solution away from this handle region the initial data sets we produce can be made as close as desired to the original initial data sets these constructions can be made either when the initial manifold is compact or asymptotically euclidean or asymptotically hyperbolic with suitable corresponding conditions on the extrinsic curvature in the compact setting a mild nondegeneracy condition is required in the final section of the paper we list a number ways this construction may be used to produce new types of vacuum spacetimes

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

existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations this paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations upon making some suitable assumptions the existence and uniqueness of a weighted pseudo almost periodic solution is obtained to illustrate our main result we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional differential equations with feedback control

pseudo almost periodic solutions to some differential equations abstract this paper is concerned with the pseudo almost periodic solutions to the abstract differential equations of the form d d s w s aw s bw s f s where a b are densely defined closed linear operators acting in a hilbert space h and f r h is a h valued pseudo almost periodic function using the theory of the invariant subspaces for unbounded linear operators we show that every bounded solution to is a pseudo almost periodic function

existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional differential equations and applications the paper considers the existence and uniqueness of pseudo almost periodic mild solutions to some classes of first order partial neutral functional differential equations upon making some suitable assumptions existence and uniqueness results are obtained applications include both a partial integro differential equation arising in control systems and the scalar reaction diffusion equation with delay

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

composition of pseudo almost periodic functions and semilinear differential equations under the assumption of uniformly continuity we establish a composition theorem of pseudo almost periodic functions in general banach spaces as an application we give some existence theorems of pseudo almost periodic solutions and mild pseudo almost periodic solutions for some differential equations

scalar curvature and hammocks scalar curvature is the simplest generalization of gaussian curvature to higher dimensions however there are many questions open with regard to its relation to other geometric quantities and topology here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature namely the one sided affinity for curvature decreasing deformations the first one is concerned with some prescribed decrease of the scalar curvature scal g of some riemannian metric g on a given manifoldmn of dimension 3 we denote the e neighborhood of some set u with respect to g by ue

initial boundary value problems and optimal control for nonautonomous parabolic systems a large class of linear nonautonomous parabolic systems in bounded domains is considered with control acting on the boundary through dirichlet or neumann conditions from the point of view of semigroup theory the results from rend sem mat univ padova 78 1987 pp 47 107 on fundamental solutions for abstract parabolic equations lecture notes in math vol 1223 springer verlag berlin heidelberg 1986 pp 1 11 on abstract homogeneous parabolic cauchy problems allow operators with varying domains and holder continuous coefficients to be handled a representation formula for solutions corresponding to square integrable control functions is derived and used to solve a linear quadratic regulator problem over finite time horizon by a direct study of the associated integral riccati equation

initial data engineering we present a local gluing construction for general relativistic initial data sets the method applies to generic initial data in a sense which is made precise in particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points which was the case for previous such constructions no global conditions on the initial data sets such as compactness completeness or asymptotic conditions are imposed as an application we prove existence of spatially compact maximal globally hyperbolic vacuum space times without any closed constant mean curvature spacelike hypersurface

on the asymptotics for the vacuum einstein constraint equations in this paper we prove density of asymptotically flat solutions with special asymptotics in general classes of solutions of the vacuum constraint equations the first type of special asymptotic form we consider is called harmonic asymptotics this generalizes in a natural way the conformally flat asymptotics for the k 0 constraint equations we show that solutions with harmonic asymptotics form a dense subset in a suitable weighted sobolev topology of the full set of solutions an important feature of this construction is that the approximation allows large changes in the angular momentum the second density theorem we prove allows us to approximate asymptotically flat initial data on a three manifold m for the vacuum einstein field equation by solutions which agree with the original data inside a given domain and are identical to that of a suitable kerr slice or identical to a member of some other admissible family of solutions outside a large ball in a given end the construction generalizes work in c where the time symmetric k 0 case was studied

existence of the self graviting chern simons vortices we prove existence of multivortex solutions of the self dual einstein chern simons higgs system proposed by clement phys rev d 54 1844 1847 1996 we consider both the topological and the nontopological boundary conditions for open conformally flat manifolds for nontopological boundary conditions we use perturbation argument from a solution of the liouville equation combined with the implicit function theorem using this argument we have existence for arbitrary positive number for the gravitational constant for topological boundary condition we construct solutions for small gravitational constant by using the super subsolution method for sufficiently large gravitational constant we have a nonexistence result for the radially symmetric topological solutions we also obtain the decay estimates near infinity for both of the topological and the nontopological solutions

localizing solutions of the einstein constraint equations we perform an optimal localization of asymptotically flat initial data sets and construct data that have positive adm mass but are exactly trivial outside a cone of arbitrarily small aperture the gluing scheme that we develop allows to produce a new class of n body solutions for the einstein equation which patently exhibit the phenomenon of gravitational shielding for any large t we can engineer solutions where any two massive bodies do not interact at all for any time t in 0 t in striking contrast with the newtonian gravity scenario

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

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

on the rigidity of riemannian penrose inequality for asymptotically flat 3 manifolds with corners in this paper we prove a rigidity result for the equality case of the penrose inequality on 3 dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners our result also has deep connections with the equality cases of theorem 1 in miao commun math phys 292 1 271 284 2009 and theorem 1 1 in lu and miao minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature arxiv 1703 08164v2 2017

a family of asymptotically hyperbolic manifolds with arbitrary energy momentum vectors a family of non radial solutions to the yamabe equation modeled on the hyperbolic space is constructed using power series as a result we obtain a family of asymptotically hyperbolic metrics with spherical conformal infinity with scalar curvature greater than or equal to n n 1 but which are a priori not complete moreover any vector of rn 1 is performed by an energy momentun vector of one suitable metric of this family they can in particular provide counter examples to the positive energy momentum theorem when one removes the completeness assumption

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

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

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

the longest shortest fence and sharp poincare sobolev inequalities we prove a long standing conjecture concerning the fencing problem in the plane among planar convex sets of given area the disc and only the disc maximizes the length of the shortest area bisecting curve although it may look intuitive the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so called auerbach triangle

on the capacity of surfaces in manifolds with nonnegative scalar curvature given a surface in an asymptotically flat 3 manifold with nonnegative scalar curvature we derive an upper bound for the capacity of the surface in terms of the area of the surface and the willmore functional of the surface the capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at even in the special case of 3 this is a new estimate more generally equality holds precisely for a spherically symmetric sphere in a spatial schwarzschild 3 manifold as applications we obtain inequalities relating the capacity of the surface to the hawking mass of the surface and the total mass of the asymptotically flat manifold

implications of energy conditions on standard static space times abstract in the framework of standard static space times we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories we concentrate our study on questions about the sub harmonicity of the warping function the scalar curvature map conformal hyperbolicity conjugate points and the time like diameter of this class of space times

on the gauge orbit space stratification a review first we review the basic mathematical structures and results concerning the gauge orbit space stratification this includes general properties of the gauge group action fibre bundle structures induced by this action basic properties of the stratification and the natural riemannian structures of the strata in the second part we study the stratification for theories with gauge group su n in spacetime dimension 4 we develop a general method for determining the orbit types and their partial ordering based on the 1 1 correspondence between orbit types and holonomy induced howe subbundles of the underlying principal su n bundle we show that the orbit types are classified by certain cohomology elements of spacetime satisfying two relations and that the partial ordering is characterized by a system of algebraic equations moreover operations for generating direct successors and direct predecessors are formulated which allow one to construct the set of orbit types starting from the principal type finally we discuss an application to nodal configurations in yang mills chern simons theory

pseudo almost periodic solutions of some differential equations ii we first present some results about pseudo almost periodic functions then we use these results to show some theorems about existence and uniqueness of the pseudo almost periodic solutions of the linear and the quasi linear nonautonomous equations finally we set up some theorems of the existence of pseudo almost periodic solutions implying the existence of almost periodic solutions of general ordinary differential equation systems

the equality case of the penrose inequality for asymptotically flat graphs we prove the equality case of the penrose inequality in all dimensions for asymptotically flat hypersurfaces it was re cently proven by g lam 19 that the penrose inequality holds for asymptotically flat graphical hypersurfaces in euclidean space with non negative scalar curvature and with a minimal boundary our main theorem states that if the equality holds then the hy persurface is a schwarzschild solution as part of our proof we show that asymptotically flat graphical hypersurfaces with a min imal boundary and non negative scalar curvature must be mean convex using the argument that we developed in 15 this en ables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation

on a comparison theorem via symmetrisation talenti s theorem allows us to estimate the norms of the solution of an elliptic boundary value problem posed on a bounded open set in terms of those of the symmetrised problem posed on the ball centred at 0 and such that we are interested in the limiting case when equality is achieved between the norms of the two solutions we show then that and that the solution is radially symmetric and decreasing the novelty of this exposition is that the method used is very elementary compared to the usual arguments

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

deformation of scalar curvature and volume the stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the einstein metrics one may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant in this paper we localize a condition satisfied by such stationary points to smooth bounded domains the condition involves a generalization of the static equations and we interpret solutions and their boundary values of this equation variationally on domains carrying a metric that does not satisfy the condition we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric we apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved finally we note that starting from a counterexample of min oo s conjecture such as that of brendle marques neves counterexamples of arbitrarily large volume and different topological types can be constructed

existence and uniqueness of pseudo almost periodic solutions to some classes of partial evolution equations abstract some sufficient conditions for the existence and uniqueness of pseudo almost periodic mild solutions to some classes of partial evolution equations are given we then make use of our abstract results to discuss the existence of pseudo almost periodic solutions to some partial differential equations

construction of cauchy data of vacuum einstein field equations evolving to black holes we show the existence of complete asymptotically at cauchy initial data for the vacuum einstein eld equations free of trapped surfaces whose future development must admit a trapped surface moreover the datum is exactly a constant time slice in minkowski space time inside and exactly a constant time slice in kerr space time outside the proof makes use of the full strength of christodoulou s work on the dynamical formation of black holes and corvino schoen s work on the construction of initial data sets

a new class of obstructions to the smoothness of null infinity expansions of the gravitational field arising from the development of asymptotically euclidean time symmetric conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6 to this end a certain representation of spatial infinity as a cylinder is used this setup is based on the properties of conformal geodesics it is found that these expansions suggest that null infinity has to be non smooth unless the newman penrose constants of the spacetime and some other higher order quantities of the spacetime vanish as a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders furthermore the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity then the initial data must in fact be schwarzschildean around spatial infinity

on existence of static metric extensions in general relativity motivated by problems related to quasi local mass in general relativity we study the static metric extension conjecture proposed by r bartnik 4 we show that for any metric on b1 that is close enough to the euclidean metric and has reflection invariant boundary data there always exists an asymptotically flat and scalar flat static metric extension in m 3 b1 such that it satisfies bartnik s geometric boundary condition 4 on b1

existence of radial mixed type solutions in chern simons theories of rank 2 in mathbb r 2 we are concerned with chern simons gauge theories of rank 2 in mathbb r 2 such as su 3 so 5 and g 2 chern simons theory one can reduce the self dual equations of these models to a 2 times 2 system of elliptic equations there exist three types of solutions of these elliptic system classified according to their asymptotic behaviors namely topological nontopological and mixed type solutions in this paper we consider the existence of radial mixed type solutions of these models by counting the degree of the corresponding operator we show the existence under a reasonable condition our result on the su 3 chern simons theory is an improvement of the existence result in choe et al 2017 also our results for the so 5 and g 2 chern simons theories are entirely new to which the variational approach developed in choe et al 2017 cannot be applied

positive solutions to a fourth order elliptic problem by the lusternik schnirelmann category abstract in this paper we consider the fourth order problem 2 u u s 1 u u 2 2 u in u u 0 in u u 0 on where is a smooth bounded domain in r n n 5 and 2 2 n n 4 we assume 2 s 1 2 in case n 8 and 2 2 s 1 2 for the critical dimensions n 5 6 7 then we prove that if has a rich topology described by its lusternik schnirelmann category then the problem has multiple solutions at least as many as cat in case the parameter 0 is sufficiently small

on scalar curvature rigidity of vacuum static spaces in this paper we extend the local scalar curvature rigidity result in brendle and marques j differ geom 88 379 394 2011 to a small domain on general vacuum static spaces which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper corvino commun math phys 214 137 189 2000 we obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces we also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains where the lapse functions are positive on vacuum static spaces with positive scalar curvature and show such domains are maximal which generalizes the work in hang and wang commun anal geom 14 91 106 2006

on optimization problems with prescribed rearrangements on considere certains resultats concernant des fonctions avec des distributions imposees et des prroblemes d optimisation qui y sont relies

on elliptic systems in rn on etudie des systemes elliptiques k k d operateurs differentiels dans r n qui peuvent s ecrire sous la forme a a q ou a est un systeme elliptique d operateurs a coefficients constants et q est une perturbation a coefficient variable avec certaines proprietes de decroissance a x

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

gauge choice in witten s energy expression witten s equation not d psi 0 can be interpreted as a gauge fixing condition for classical supergravity we rigorously prove the existence of asymptotically constant solutions of the more general gauge condition not d psi a psi for almost all endomorphismsa of the spin bundle each gives an expression for the gravitational energy similar to witten s these include the choice a sqrt r which yields the particularly elegant energy expression first noticed by deser

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

the bounded l 2 curvature conjecture this is the main paper in a sequence in which we give a complete proof of the bounded l 2 curvature conjecture more precisely we show that the time of existence of a classical solution to the einstein vacuum equations depends only on the l 2 norm of the curvature and a lower bound on the volume radius of the corresponding initial data set we note that though the result is not optimal with respect to the scaling of the einstein equations it is nevertheless critical with respect to its causal geometry indeed l 2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries we note also that while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 1 based on strichartz estimates were obtained in bahouri and chemin am j math 121 1337 1777 1999 imrn 21 1141 1178 1999 tataru am j math 122 349 376 2000 jams 15 2 419 442 2002 klainerman and rodnianski duke math j 117 1 1 124 2003 and optimized in klainerman and rodnianski ann math 161 1143 1193 2005 smith and tataru ann math 162 291 366 2005 the result we present here is the first in which the full structure of the quasilinear hyperbolic system not just its principal part plays a crucial role to achieve our goals we recast the einstein vacuum equations as a quasilinear so 3 1 valued yang mills theory and introduce a coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear covariant nature of the equations to prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure these require a careful construction and control of parametrices including l 2 error bounds which is carried out in szeftel parametrix for wave equations on a rough background i regularity of the phase at initial time arxiv 1204 1768 2012 parametrix for wave equations on a rough background ii construction of the parametrix and control at initial time arxiv 1204 1769 2012 parametrix for wave equations on a rough background iii space time regularity of the phase arxiv 1204 1770 2012 parametrix for wave equations on a rough background iv control of the error term arxiv 1204 1771 2012 as well as a proof of sharp strichartz estimates for the wave equation on a rough background which is carried out in szeftel sharp strichartz estimates for the wave equation on a rough background arxiv 1301 0112 2013 it is at this level that our problem is critical indeed any known notion of a parametrix relies in an essential way on the eikonal equation and our space time possesses barely the minimal regularity needed to make sense of its solutions

existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications this paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to the class of semilinear differential equations of the form u t au t f t bu t where a is the infinitesimal generator of an analytic semigroup acting on a complex banach space x b d b x x is a densely defined closed linear operator and f r x x is a jointly continuous function under some suitable assumptions on a b and f the existence and uniqueness of a pseudoalmost periodic solution to is obtained

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

positive scalar curvature with skeleton singularities we study positive scalar curvature on the regular part of riemannian manifolds with singular uniformly euclidean l infty metrics that consolidate gromov s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of einstein manifolds we show that in all dimensions edge singularities with cone angles le 2 pi along codimension 2 submanifolds do not affect the yamabe type in three dimensions we prove the same for more general singular sets which are allowed to stratify along 1 skeletons exhibiting edge singularities angles le 2 pi and arbitrary l infty isolated point singularities we derive as an application of our techniques positive mass theorems for asymptotically flat manifolds with analogous singularities

existence results for partial neutral functional differential equations with unbounded delay abstract we prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the leray schauder alternative

perturbative solutions of the extended constraint equations in general relativity the extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface mathcal z in an asymptotically simple space time satisfying the vacuum conformal einstein equations developed by h friedrich the extended constraint equations consist of a quasi linear system of partial differential equations for the induced metric the second fundamental form and two other tensorial quantities defined on mathcal z and are equivalent to the usual constraint equations that mathcal z satisfies as a space like hypersurface in a space time satisfying einstein s vacuum equation this article develops a method for finding perturbative asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on euclidean space this method is fundamentally different from the classical method of lichnerowicz and york that is used to solve the usual constraint equations

gluing initial data sets for general relativity we establish an optimal gluing construction for general relativistic initial data sets the construction is optimal in two distinct ways first it applies to generic initial data sets and the required generically satisfied hypotheses are geometrically and physically natural second the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place using this construction we establish the existence of cosmological maximal globally hyperbolic vacuum space times with no constant mean curvature spacelike cauchy surfaces

existence of weighted pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations abstract in this paper under acquistapace terreni conditions we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations applications include the existence of weighted pseudo almost periodic solutions to a nonautonomous heat equation with gradient coefficients