on the invariant mass conjecture in general relativity an asymptotic symmetries theorem is proved under certain hypotheses on the behaviour of the metric at spatial infinity this implies that the einstein von freud adm mass can be invariantly assigned to an asymptotically flat four dimensional end of an asymptotically empty solution of einstein equations if the metric is a no radiation metric or if the end is defined in terms of a collection of boost type domains

dynamic feedback over commutative rings abstract we call a commutative ring r a ca n ring if for each n dimensional reachable system f g over r the system augmented by rank n projective modules with f augmented by a zero map and g augmented by an identity map is coefficient assignable we show that if r is a dedekind domain then r is a ca n 1 ring in particular a principal ideal domain is a ca n 1 ring we also show that if r is a ring with the gcs property and the 2 generator property then r is a ca 2 n 2 ring

partitions of elements in a monoid and its applications to systems theory abstract the feedback class of a locally brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants 2 in this paper we attack the problem of enumerating all feedback classes of locally brunovsky systems over an n dimensional state space and translate to the combinatorial problem of enumerating all the partitions of integer n in some abelian semigroup the problem of computing the number n k of all the partitions of integer n into k different summands is pointed out

on feedback invariants for linear dynamical systems abstract the goal of this paper is to prove that if r is a commutative ring containing a non zero finitely generated maximal ideal m containing its annihilator such that every unit of r m lifts to a unit of r then the category of reachable systems over r is wild in the sense of classical representation theory from this it follows that a canonical form for a reachable system over a pid is not likely to be found more specifically canonical forms are unlikely to be found for systems over z and k t when k is a field

on the radius pinching estimate and uniqueness of the cmc foliation in asymptotically flat 3 manifolds abstract in this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3 manifolds we require the metric to have the form g i j i j h i j with h i j o 4 r 1 and r o r 3 0 we do not require the metric to be close to schwarzschild metric in any sense or to satisfy rt conditions we prove that when the mass is not 0 stable cmc spheres that separate a certain compact part from the infinity satisfy the radius pinching estimate r 1 c r 0 which in many cases is critical to prove the uniqueness of the cmc spheres as applications of this estimate we remove the radius conditions of the uniqueness result in 8 and 15 in some special cases

enumeration of classes of linear systems via equations and via partitions in an ordered abelian monoid abstract we prove that the feedback classes of locally brunovsky linear systems are given by the decomposition of state space into direct summands hence the problem of enumerating feedback classes of systems is equivalent to the problem of enumerating all solutions of a linear equation over a commutative zerosumfree monoid if this monoid is also cancelative then the problem is solved by all the partitions

isoparametric functions on riemannian manifolds i on etablit les proprietes de base des fonctions transnormales et on demontre le lemme du tube on demontre que soit m une variete de riemann lisse complete connexe et f une fonction transnormale sur m alors a les varietes focales de f sont des sous varietes lisses de m b chaque ensemble de niveau regulier de f est un tube sur d autres des varietes focales

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

geometric characterizations of asymptotic flatness and linear momentum in general relativity abstract in 1996 huisken yau proved that every three dimensional riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature cmc if it is asymptotically equal to the spatial schwarzschild solution their decay assumptions were weakened by metzger huang eichmair metzger and the author at a later stage in this work we prove the reverse implication i e any three dimensional riemannian manifold is asymptotically flat if it possesses a cmc cover satisfying certain geometric curvature estimates a uniqueness property a weak foliation property while each surface has weakly controlled instability with the author s previous result that every asymptotically flat manifold possesses a cmc foliation we conclude that asymptotic flatness is characterized by existence of such a cmc cover additionally we use this characterization to provide a geometric i e coordinate free definition of a cmc linear momentum and prove its compatibility with the linear momentum defined by arnowitt deser misner

new results on the pompeiu problem let pn w enko a w w e c n e n be a polynomial with complex coefficients in this paper we prove that if dc c2 is a simplyconnected bounded open set whose boundary is a closed simple curve parametrized by x s xl s ix2 s pn els s e xt 7t then d has the pompeiu property unless n 1 and p1 w a1w a2 in which case d is a disk this result supports the conjecture that modulo sets of zero twodimensional lebesgue measure the disk is the only simply connected bounded open set which fails to have the pompeiu property

foliations by spheres with constant expansion for isolated systems without asymptotic symmetry motivated by the foliation by stable spheres with constant mean curvature constructed by huisken yau metzger proved that every initial data set can be foliated by spheres with constant expansion ce if the manifold is asymptotically equal to the standard t 0 timeslice of the schwarzschild solution in this paper we generalize his result to asymptotically flat initial data sets and weaken additional smallness assumptions made by metzger furthermore we prove that the ce surfaces are in a well defined sense asymptotically independent of time if the linear momentum vanishes

heat flow heat content and the isoparametric property let m be a riemannian manifold and omega a compact domain of m with smooth boundary we study the solution of the heat equation on omega having constant unit initial conditions and dirichlet boundary conditions the purpose of this paper is to study the geometry of domains for which at any fixed value of time the normal derivative of the solution heat flow is a constant function on the boundary we express this fact by saying that such domains have the constant flow property in constant curvature spaces known examples of such domains are given by geodesic balls and more generally by domains whose boundary is connected and isoparametric the question is are they all like that this problem is the analogous for the heat equation of the classical serrin s problem for harmonic domains in this paper we give an affirmative answer to the above question in fact we prove more generally that if a domain in an analytic riemannian manifold has the constant flow property then every component of its boundary is an isoparametric hypersurface for space forms we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r mean curvatures of the boundary and with the isoparametric property finally we discuss the constant flow property in relation to other well known overdetermined problems involving the laplace operator like the serrin problem or the schiffer problem

the cauchy problem in for linear second order partial differential equations with data on a quadric surface by means of a method developed essentially by leray some global existence results are obtained for the problem referred to in the title the partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic the cauchy data are assumed to be given by entire functions under these conditions the location of all possible singularities of solutions are determined the sets of singularities can be divided into two types kand l singularities k the set of k singularities is the global version of the characteristic tangent defined by leray the l sets are here quadric surfaces which in contrast to the assets allow unbounded singularities the l sets are in turn divided into three types initial asymptotic and latent singularities the initial singularities appear when the characteristic points of the initial surface are exceptional according to leray s local theory these sets of singularity intersect the initial surface at characteristic points the asymptotic case where the set of singularities does not cut the initial surface can be viewed as projectively equivalent to the initial case the intersection taking place at infinite characteristic points finally the latent singularities are sets which intersect the initial surface but where the solutions do not develop singularities initially in the case of the laplace equation with data on a real quadric surface it is shown that the singularities and the asymptotic singularities occur on the classical focal sets defined by poncelet pliicker darboux et al there are also latent singularities appearing in coordinate subspaces of r as a corollary a new proof is given of the fact that ellipsoids have the pompeiu property

rellich type identities for eigenvalue problems and application to the pompeiu problem for classical neumann eigenvalue buckling eigenvalue and clamped plate eigenvalue we give the corresponding rellich type identities as an application of these results then we obtain a new necessary and sufficient condition for a domain without the pompeiu property

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

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symmetry theorems for the overdetermined eigenvalue problems abstract the well known schiffer conjecture saying that for a smooth bounded domain r n if there exists a positive neumann eigenvalue such that the corresponding neumann eigenfunction u is constant on the boundary of then is a ball in this paper we shall prove that the schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding neumann eigenfunction is constant on the boundary we also prove a similar result to the berenstein conjecture for the overdetermined dirichlet eigenvalue problem

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

a new result on the pompeiu problem a nonernpty bounded open set q c r n 2 is said to have the pompeiu property if and only if the only continuous function f on rn for which the integral of f over oa is zero for all rigid motions a of rn is f 0 o we consider a nonempty bounded open set q c r t n 2 with lipschitz boundary and we assume that the complement of q is connected we show that the failure of the pomnpeiu property for q implies some geometric conditions using these conditions we prove that a special kind of solid tori in r n 3 has the pompeiu property so far the result was proved only for solid tori in jr we also examine the case of planar domains finally we extend the example of solid tori to domains in rn bounded by hypersurfaces of revolution

some results on the schiffer conjecture in r 2 abstract let be an open bounded domain in r 2 with connected and c boundary and a solution of 0 1 0 2 n 0 0 3 const 0 for some 0 denoting by 0 1 2 the set of all neumann eigenvalues for the laplacian on we show that 1 if 8 or 2 if is strictly convex and centrally symmetric 13 then must be a disk

foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics we prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the regge teitelboim condition near infinity it is known that the hamiltonian center of mass is well defined for manifolds satisfying this condition we also show that the foliation is asymptotically concentric and its geometric center is the center of mass the construction of the foliation generalizes the results of huisken yau ye and metzger where strongly asymptotically flat manifolds and their small perturbations were studied

foliations of asymptotically flat 3 manifolds by 2 surfaces of prescribed mean curvature we construct 2 surfaces of prescribed mean curvature in 3manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions the surfaces in question form a regular foliation of the asymptotic region of such a manifold we recover physically relevant data especially the adm momentum from the geometry of the foliation for a given set of data m g k with a three dimensional manifold m its riemannian metric g and the second fundamental form k in the surrounding four dimensional lorentz space time manifold the equation we solve is h p const or h p const here h is the mean curvature and p trk is the 2 trace of k along the solution surface this is a degenerate elliptic equation for the position of the surface it prescribes the mean curvature anisotropically since p depends on the direction of the normal

symmetry results for overdetermined boundary value problems of nonlinear elliptic equations abstract we prove the radial symmetry of the solutions of second order nonlinear elliptic equations for overdetermined dirichlet and neumann boundary value problems in addition a global uniqueness theorem of holmgren type is given for nonlinear elliptic equations

on the k theory of feedback actions on linear systems abstract a categorical approach to linear control systems is introduced feedback actions on linear control systems arise as symmetric monodical category s r stable feedback isomorphisms generalize dynamic enlargement of pairs of matrices subcategory of locally brunovsky linear systems b r is studied we prove that the stable feedback isomorphisms of locally brunovsky linear systems are characterized by the grothendieck group k 0 b r hence a link from linear dynamical systems to algebraic k theory is established

on the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3 manifolds in this note we study constant mean curvature surfaces in asymptotically flat 3 manifolds we prove that outside a given compact subset in an asymptotically flat 3 manifold with positive mass stable spheres of given constant mean curvature are unique therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3 manifold with positive mass outside a given compact subset is unique

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

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

foliations of asymptotically flat manifolds by surfaces of willmore type the goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the willmore functional subject to an area constraint equivalently these surfaces are critical points of the geroch hawking mass thus our result has applications in the theory of general relativity

a note on the pompeiu problem for convex domains as the pompeiu problem named after the late rumanian mathematician dimitric pompeiu we shall denote the following question let s denote the group of all rigid motions of the plane given a bounded measurable set d in the xy plane suppose that f is a continuous function satisfying f x y dx dy o re s does o this imply f 0 if it does we say that d is a pompeiu set the purpose of this note is to give a geometric condition on a set d which implies that it is a pompeiu set early attempts at the solution settled the question for such regions as discs no parallelograms yes and triangles yes in 2 brown et al have obtained via the theory of distributions and mean periodic functions a general theorem on spectral synthesis in c r which gives a solution to this problem in terms of the fourier laplace transform of the lebesque area measure on d using this result and asymptotic estimates of the growth of certain fourier laplace transforms developed for this purpose it was then shown in particular that any polygonal region ellipse or convex set with at least one corner is a pompeiu set while any rotationally invariant set is not studying the pompeiu problem leads to another problem of a similar nature which we shall refer to as the morera problem because of its relationship to the classical theorem of morera let f denote a rectifiable closed curve in the plane

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

large isoperimetric regions in asymptotically hyperbolic manifolds we show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds furthermore we show that large coordinate spheres are uniquely isoperimetric in manifolds that are schwarzschild anti desitter at infinity these results have important repercussions for our understanding of spacelike hypersurfaces in lorentzian space times which are asymptotic to null infinity in fact as an application of our results we verify the asymptotically hyperbolic penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes

when are dynamic and static feedback equivalent an open question in control theory over commutative rings is when does dynamic feedback equivalence imply static feedback equivalence a second open problem is given a linear system a b when does there exist a matrix f such that a bf is invertible in this paper we solve both questions obtaining two characterizations of stable rings

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

the dynamic feedback equivalence over principal ideal domains we show that over a principal ideal domain the dynamic feedback equivalence for not necessarily reachable linear systems is reduced to the feedback equivalence for one augmented systems we also obtain the dynamic feedback classification for two dimensional linear systems

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

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