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

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

a positive mass theorem in three dimensional cauchy riemann geometry in this note we summarize the results from 6 on the positive mass problem in 3 dimensional cr cauchy riemann geometry

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

positivity of quasi local mass ii a spacetime is a four manifold with a pseudo metric of signature a hypersurface or a 2 surface in a spacetime is spacelike if the induced metric is pos itive definite a quasi local energy momentum vector is a vector in r3 1 associated to a spacelike 2 surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime the time component of the four vector is called quasi local energy mass similar to 10 8 we require the quasi local energy momentum vector to satisfy the following properties 1 it should be zero for the flat spacetime 2 the quasi local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi local mass is evaluated on the spheres 7 we say that two masses mi and m2 are equivalent if there is a universal constant c 0 such that c lm area e 167r 6 the quasi local energy momentum vector should be nonspacelike and the quasi local mass should be nonnegative

a penrose like inequality for the mass of riemannian asymptotically flat manifolds we prove an optimal penrose like inequality for the mass of any asymptotically flat riemannian 3 manifold having an inner minimal 2 sphere and nonnegative scalar curvature our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere as in the original penrose conjecture and some nomalized sobolev ratio as expected the equality case is achieved if and only if the metric is that of a standard spacelike slice in the schwarzschild space

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

cpmc automatic object segmentation using constrained parametric min cuts we present a novel framework to generate and rank plausible hypotheses for the spatial extent of objects in images using bottom up computational processes and mid level selection cues the object hypotheses are represented as figure ground segmentations and are extracted automatically without prior knowledge of the properties of individual object classes by solving a sequence of constrained parametric min cut problems cpmc on a regular image grid in a subsequent step we learn to rank the corresponding segments by training a continuous model to predict how likely they are to exhibit real world regularities expressed as putative overlap with ground truth based on their mid level region properties then diversify the estimated overlap score using maximum marginal relevance measures we show that this algorithm significantly outperforms the state of the art for low level segmentation in the voc 2009 and 2010 data sets in our companion papers 1 2 we show that the algorithm can be used successfully in a segmentation based visual object category recognition pipeline this architecture ranked first in the voc2009 and voc2010 image segmentation and labeling challenges

second order hamiltonian equations on t and almost periodic solutions abstract motivated by problems arising in nonlinear pde s with a hamiltonian structure and in high dimensional dynamical systems we study a suitable generalization to infinite dimensions of second order hamiltonian equations of the type xv x t n x x1 xn extending methods from quantitative perturbation theory kolmogorov arnold moser theory nash moser implicit function theorem etc we construct uncountably many almost periodic solutions for the infinite dimensional system i i x i z d x t z d endowed with the compact topology the hamiltonian structure is reflected by being a generalized gradient such a result is derived under suitable analyticity assumptions on i but without requiring any smallness conditions

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

monopoles and maps from s 2 to s 2 the topology of the configuration space the configuration space for the su 2 yang mills higgs equations on 3 is shown to be homotopic to the space of smooth maps froms 2 tos 2 this configuration space indexes a family of twisted dirac operators the dirac family is used to prove that the configuration space does not retract onto any subspace on which the su 2 yang mills higgs functional is bounded

equivalence of the spinor and tensor methods in the positive energy problem we prove that nontrivial solutions of the sen witten equation with asymptotically flat data set on maximal hypersurface does not equal zero at any point of this hypersurface on this basis we ascertain the equivalence of witten s spinor method and nester s tensor method in the positive energy problem in general relativity

existence theorem for solutions of witten s equation and nonnegativity of total mass we prove that given an asymptotically flat in a very weak sense initial data set there always exists a spinor field that satisfies witten s equation and that becomes constant at infinity thus we fill a gap in witten s arguments on the nonnegativity of the total mass of an isolated system when measured at spatial infinity we also include a review of witten s argument

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

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

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

on riemannian spaces with conformal symmetries or a tool for the study of generalized kaluza klein theories the riemann geometry of a space with conformal symmetries is written in terms of intrinsic objects defined from the action of the symmetries its application in the study of generalized kaluza klein theories is discussed

on the formulation of the positive energy theorem in kaluza klein theories the positive energy theorem is formulated in the context of kaluza klein theories different cases are considered including the situation in which no symmetry is assumed this work offers a new technique for stability considerations in kaluza klein theories

localized breather like solution in a discrete klein gordon model and application to dna abstract non propagating localized oscillating solutions are studied in the case of a discrete one dimensional system with the 2 3 power onsite potential a multiple scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider starting from an exact pulse like solution in the continuum we derive an approximate oscillating solution our numerical simulations reveal that whereas the excitations are strongly damped in the strong intersite coupling case they are long lived in the discrete cases an additional modulation phenomenon of the oscillation seen in the simulation is explained by combination of the breather oscillations with the zero group velocity phonons modes then applying the green s functions formalism in analogy with a previous work s takeno et al prog theor phys suppl 94 1988 we derive a very accurate expression for the discrete breather with a quasi infinite lifetime lastly we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of dna

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

a definition of total energy momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds i two sets of asymptotically hyperbolic initial data are defined which correspond to the spatial infinity in asymptotically ads spacetimes and to the null infinity in asymptotically minkowski spacetimes respectively the positive mass theorem involving the total energy the total linear momentum and the total angular momentum is established for these initial data sets

moduli spaces of self dual connections over asymptotically locally flat gravitational instantons we investigate yang mills instanton theory over four dimensional asymptotically locally flat alf geometries including gravitational instantons of this type by exploiting the existence of a natural smooth compactification of these spaces introduced by hausel hunsicker mazzeo first referring to the codimension 2 singularity removal theorem of sibner sibner and rade we prove that given a smooth finite energy self dual su 2 connection over a complete alf space its energy is congruent to a chern simons invariant of the boundary three manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly then we introduce framed moduli spaces of self dual connections over ricci flat alf spaces we prove that the moduli space of smooth irreducible rapidly decaying self dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold we calculate its dimension by a variant of the gromov lawson relative index theorem as an application we study yang mills instantons over the flat mathbb r 3 times s 1 the multi taub nut family and the riemannian schwarzschild space

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

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

quasi periodic breathers in hamiltonian networks of long range coupling this work is concerned with hamiltonian networks of weakly and long range coupled oscillators with either variable or constant on site frequencies we derive an infinite dimensional kam like theorem by which we establish that given any n sites of the lattice there is a positive measure set of small amplitude quasi periodic breathers solutions of the hamiltonian network that are quasi periodic in time and exponentially localized in space having n frequencies which are only slightly deformed from the on site frequencies

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

slic superpixels compared to state of the art superpixel methods computer vision applications have come to rely increasingly on superpixels in recent years but it is not always clear what constitutes a good superpixel algorithm in an effort to understand the benefits and drawbacks of existing methods we empirically compare five state of the art superpixel algorithms for their ability to adhere to image boundaries speed memory efficiency and their impact on segmentation performance we then introduce a new superpixel algorithm simple linear iterative clustering slic which adapts a k means clustering approach to efficiently generate superpixels despite its simplicity slic adheres to boundaries as well as or better than previous methods at the same time it is faster and more memory efficient improves segmentation performance and is straightforward to extend to supervoxel generation

positive mass conjecture for five dimensional lorentzian manifolds we extend witten s proof on positive mass conjecture to five dimensional lorentzian manifolds with a rigorous treatment in mathematics

on the positive energy theorem involving mass and electromagnetic charges a new theorem relating mass and charges is deduced which can be applied to more general physical systems than those covered by the theorem of gibbons and hull 1

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

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

killing vectors in asymptotically flat space times i asymptotically translational killing vectors and the rigid positive energy theorem we study killing vector fields in asymptotically flat space times we prove the following result implicitly assumed in the uniqueness theory of stationary black holes if the conditions of the rigidity part of the positive energy theorem are met then in such space times there are no asymptotically null killing vector fields except if the initial data set can be embedded in minkowski space time we also give a proof of the nonexistence of nonsingular in an appropriate sense asymptotically flat space times that satisfy an energy condition and that have a null adm four momentum under conditions weaker than previously considered

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

stereo object proposals object proposal detection is an effective way of accelerating object recognition existing proposal methods are mostly based on detecting object boundaries which may not be effective for cluttered backgrounds in this paper we leverage stereopsis as a robust and effective solution for generating object proposals we first obtain a set of candidate bounding boxes through adaptive transformation which fits the bounding boxes tightly to object boundaries detected by rough depth and color information a two level hierarchy composed of proposal and cluster levels is then constructed to estimate object locations in an efficient and accurate manner three stereo based cues exactness focus and distribution are proposed for objectness estimation two level hierarchical ranking is proposed to accurately obtain ranked object proposals a stereo data set with 400 labeled stereo image pairs is constructed to evaluate the performance of the proposed method in both indoor and outdoor scenes extensive experimental evaluations show that the proposed stereo based approach achieves a better performance than the state of the arts with either a small or a large number of object proposals as stereopsis can be a complement to the color information the proposed method can be integrated with existing proposal methods to obtain superior results

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

invariant manifolds of parabolic fixed points i existence and dependence on parameters abstract in this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases using the parametrization method the parametrization method relies on a suitable approximate solution of a functional equation in the case of parabolic points if the manifolds have dimension two or higher in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in 4 assuming a sufficiently good approximation is found here we provide an a posteriori result which gives a true invariant manifold close to the approximate one in the differentiable case in some cases there is a loss of regularity we also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters examples are provided we apply our method to prove that in several situations namely related to the parabolic infinity in the elliptic spatial three body problem these invariant manifolds exist and do have polynomial approximations

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

positive mass theorems for higher dimensional lorentzian manifolds we extend witten s proof a new proof of the positive energy theorem commun math phys 80 381 402 1981 of the positive mass theorem to a rigorous proof for lorentzian manifolds of any dimension this includes the original higher dimensional positive mass theorem proved by spinors parker t and taubes c on witten s proof of the positive energy theorem commun math phys 84 223 238 1982 for dimension 4 and zhang x positive mass conjecture for five dimensional lorentzian manifolds j math phys 40 3540 3552 1999 for dimension 5 stimulated by the article of zhang positive mass theorem for hypersurface in 5 dimensional lorentzian manifolds commun anal geom 8 635 652 2000 we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for lorentzian manifolds for dimensions 4 5 and 6

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

a simple proof of the positivity of the bondi mass using a nonsingular hypersurface that is not only asymptotically null the authors prove the existence of spinor fields that yield a positive expression for the bondi mass standard existence theorems based on compact hypersurfaces are applied

equality in the spacetime positive mass theorem we affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight namely if an asymptotically flat initial data set satisfies the dominant energy condition and has e p then e p 0 where e p is the adm energy momentum vector the dimensional restriction can be removed if we assume the positive mass inequality holds previously the result was only known for spin manifolds beig and chru ciel in j math phys37 4 1939 1961 1996 chru ciel and maerten in j math phys 47 2 2006

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

asymptotically flat conformal structures in the first part of this paper we revisit the theory of weighted spinors on conformal manifolds in the second part we introduce the notions of asymptotically flat weyl structures and of associated mass and we prove a conformal version of the positive mass theorem on conformal spin manifolds

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

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

the dirac witten operator on pseudo riemannian manifolds we study the dirac witten operator for spacelike spin submanifolds in pseudo riemannian manifolds when the normal bundles are spin and odd dimensional we derive new eigenvalue lower bounds we also prove the generalized positive mass theorem using this operator

remarks on positive mass theorem we establish a positive mass theorem for time symmetric initial data sets by relaxing the nonnegative scalar curvature to the case where the first neumann eigenvalue of conformal laplacian operators is eventually nonnegative

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space

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

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry