scalar curvature of a metric with unit volume the problem of finding riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of kazdan and warner in 1970 s 10 and references cited in it in this paper we shall consider the same problem under a constraint on the volume for this purpose it is useful to introduce an invariant p m of a smooth closed manifold m which will be called the yamabe number of m defined as the supremum of m c of all conformal classes c of riemannian metrics on m

a morse theoretical approach for the boundary mean curvature problem on b4 abstract in this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4 dimensional ball due to the existence of critical point at infinity the standard variational methods cannot be applied to overcome this difficulty we prove that in a neighborhood of critical points at infinity a morse lemmas at infinity reduction holds then develop a whole morse theory of this noncompact variational problem in particular we establish under generic boundary condition morse inequalities at infinity which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated euler lagrange functional as further application of this morse theoretical approach we prove more existence results and extend a topological invariant introduced by a bahri

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

the geometry of the first non zero stekloff eigenvalue let mn g be a compact riemannian manifold with boundary and dimensionn 2 in this paper we discuss the first non zero eigenvalue problem begin align delta varphi 0 qquad on quad m frac partial varphi partial eta u 1 varphi qquad on quad partial m end align eqno 1 problem 1 is known as the stekloff problem because it was introduced by him in 1902 for bounded domains of the plane we discuss estimates of the eigenvalue 1in terms of the geometry of the manifold mn g in the two dimensional case we generalize payne s theorem p for bounded domains in the plane to non negative curvature manifolds in this case we show that 1 k0 wherekg k0andkgrepresents the geodesic curvature of the boundary in higher dimensionsn 3 for non negative ricci curvature manifolds we show that 1 k0 2 wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary we introduce an isoperimetric constant and prove a cheeger s type inequality for the stekloff eigenvalue

on a spin conformal invariant on manifolds with boundary let m be an n dimensional connected compact manifold with non empty boundary equipped with a riemannian metric g a spin structure and a chirality operator we define and study some properties of a spin conformal invariant given by lambda rm min m partial m underset overline g in g rm inf lambda 1 pm overline g rm vol m overline g frac 1 n where lambda 1 pm overline g is the smallest eigenvalue of the dirac operator under the chiral bag boundary condition mathbb b pm overline g more precisely we show that if n 2 then lambda rm min m partial m leq lambda rm min mathbb s n partial mathbb s n

the prescribed scalar curvature problem for metrics with unit total volume we solve the modified kazdan warner problem of finding metrics with prescribed scalar curvature and unit total volume

lyapunov functions for strong exponential contractions abstract our main aim is to give a complete characterization of an exponential contraction in terms of lyapunov sequences for discrete time or lyapunov functions for continuous time in particular we obtain inverse theorems giving explicitly a quadratic lyapunov function for each exponential contraction we consider the general cases of 1 nonautonomous dynamics either obtained from a product of linear operators or from a nonautonomous differential equation respectively for discrete and continuous time 2 nonuniform exponential contractions in which the uniform stability is replaced by a nonuniform stability 3 strong exponential behavior in the sense that we have simultaneously lower and upper contraction bounds we emphasize that the two last properties are ubiquitous in the context of ergodic theory and in particular for almost all trajectories of any measure preserving flow which justifies completely the study of this general type of exponential behavior as a nontrivial application of our work we establish the robustness of any strong nonuniform exponential contraction that is the persistence of the asymptotic stability of a strong nonuniform exponential contraction under sufficiently small linear perturbations

yamabe equations on half spaces where n 2 n 2 is the critical sobolev exponent n n 2 is the critical sobolev exponent on the boundary is the unit outer normal of and h x is the mean curvature this problem is related to the yamabe problem on manifolds with boundary see 7 8 it was proved in 7 8 that if either k x 1 h x 0 or k x 0 h x 1 then eq 1 1 has a classical solution which minimizes the yamabe ratio when k x 6 0 h x 6 0 it is natural to seek for the nehari minimizers this idea works when is not conformal to a ball see 9 however when is a ball the situation is very di erent in fact for certain functions k x and h x eq 1 1 has no positive solution at all therefore solvability conditions on k x and h x are needed to make it more clear we consider the yamabe equation

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

information distance in multiples information distance is a parameter free similarity measure based on compression used in pattern recognition data mining phylogeny clustering and classification the notion of information distance is extended from pairs to multiples finite lists we study maximal overlap metricity universality minimal overlap additivity and normalized information distance in multiples we use the theoretical notion of kolmogorov complexity which for practical purposes is approximated by the length of the compressed version of the file involved using a real world compression program

metrics for labelled markov processes the notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions thus a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent this instability is due to the quantitative nature of probabilistic processes in a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence this paper studies a metric between labelled markov processes this metric has the property that processes are at zero distance if and only if they are bisimilar the metric is inspired by earlier work on logics for characterizing bisimulation and is related in spirit to the hutchinson metric

the k yamabe flow on manifolds with boundary abstract we consider the problem of conformally deforming a metric by a fully nonlinear flow on a compact manifold with totally geodesic boundary we prove that the k curvature defined by the modified schouten tensor always can be deformed in a natural way to a constant at exponential rate and the boundary preserves totally geodesic

infinitesimal lyapunov functions for singular flows we present an extension of the notion of infinitesimal lyapunov function to singular flows and from this technique we deduce a characterization of partial sectional hyperbolic sets in absence of singularities we can also characterize uniform hyperbolicity these conditions can be expressed using the space derivative dx of the vector field x together with a field of infinitesimal lyapunov functions only and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region

differential privacy in probabilistic systems abstract ever since proposed by dwork differential privacy has been a hot topic in academia however few attempts have been made on reasoning about differential privacy at a system level in this paper we propose a formal framework to verify differential privacy in probabilistic systems with a metric on the states of a system we formalize differential privacy by the ratio of the probabilities in the distributions after the same labeled transitions of relevant states we explain how traditional differential privacy can be embedded in our framework and raise an infimum metric the least distance between two states while not violating differential privacy it is proven that the infimum metric is also a metric instance of differential privacy itself furthermore we propose a two level logic a privacy variant of the familiar hennessy milner logic to characterize differential privacy in our framework our results have close relations to probabilistic bisimilarity as well

a refined complexity analysis of degree anonymization in graphs motivated by a strongly growing interest in graph anonymization we study the np hard degree anonymity problem asking whether a graph can be made k anonymous by adding at most a given number of edges herein a graph is k anonymous if for every vertex in the graph there are at least k 1 other vertices of the same degree our algorithmic results shed light on the performance quality of a popular heuristic due to liu and terzi acm sigmod 2008 in particular we show that the heuristic provides optimal solutions if many edges need to be added based on this we develop a polynomial time data reduction yielding a polynomial size problem kernel for degree anonymity parameterized by the maximum vertex degree in terms of parameterized complexity analysis this result is in a sense tight since we also show that the problem is already np hard for h index three implying np hardness for smaller parameters such as average degree and degeneracy

manifolds of positive scalar curvature and conformal cobordism theory for a supergoup we study closed manifolds with positive conformal classes we use the relative yamabe invariant from 2 to define the conformal cobordism relation on the category of such manifolds we prove that the corresponding conformal cobordism groups are isomorphic to the cobordism groups defined by stolz in 19 as a corollary we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics our main technical tools come from analysis and conformal geometry

classification of prime 3 manifolds with σ invariant greater than rp 3 in this paper we compute the invariants sometimes also called the smooth yamabe invariants of rp 3 and rp 2 s 1 which are equal and show that the only prime 3 manifolds with larger invariants are s 3 s 2 s 1 and s 2 s 1 the nonorientable s 2 bundle over s 1 more generally we show that any 3 manifold with invariant greater than rp 3 is either s 3 a connect sum with an s 2 bundle over s 1 or has more than one nonorientable prime compo

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

minimal hypersurfaces and bordism of positive scalar curvature metrics let y g be a compact riemannian manifold of positive scalar curvature psc it is well known due to schoen yau that any closed stable minimal hypersurface of y also admits a psc metric we establish an analogous result for stable minimal hypersurfaces with free boundary furthermore we combine this result with tools from geometric measure theory and conformal geometry to study psc bordism for instance assume y 0 g 0 and y 1 g 1 are closed psc manifolds equipped with stable minimal hypersurfaces x 0 subset y 0 and x 1 subset y 1 under natural topological conditions we show that a psc bordism z bar g y 0 g 0 rightsquigarrow y 1 g 1 gives rise to a psc bordism between x 0 and x 1 equipped with the psc metrics given by the schoen yau construction

escobar yamabe compactifications for poincare einstein manifolds and rigidity theorems abstract let x n g n 3 be a poincare einstein manifold which is c 3 conformally compact with conformal infinity x g on the conformal compactification x g 2 g via a boundary defining function there are two types of yamabe constants y x x g and q x x g see definitions 1 and 2 in 13 gursky and han obtained an inequality between y x x g and y x g in this paper we first show that the equality holds in gursky han s inequality if and only if x n g is isometric to the standard hyperbolic space h n g h secondly we derive a conformal invariant inequality between q x x g and y x g and show the equality holds if and only if x n g is isometric to h n g h based on this we give a simple proof of the rigidity theorem for poincare einstein manifolds with conformal infinity being conformally equivalent to the standard sphere

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

a non compactness result on the fractional yamabe problem in large dimensions abstract let x n 1 g be an n 1 dimensional asymptotically hyperbolic manifold with conformal infinity m n h the fractional yamabe problem addresses to solve p g h u c u n 2 n 2 u 0 on m where c r and p g h is the fractional conformal laplacian whose principal symbol is the laplace beltrami operator on m in this paper we construct a metric on the half space x r n 1 which is conformally equivalent to the unit ball for which the solution set of the fractional yamabe equation is non compact provided that n 24 for 0 and n 25 for 1 where 0 1 is a certain transition exponent the value of turns out to be approximately 0 940197

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

uniqueness and non uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary abstract let m n g be a compact manifold with boundary with n 2 in this paper we discuss uniqueness and non uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of m

partial hyperbolicity lyapunov exponents and stable ergodicity we present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non zero lyapunov exponents the main tool is local ergodicity theory for non uniformly hyperbolic systems

multiplicity results for the yamabe equation by lusternik schnirelmann theory abstract let m g be any closed riemannianan manifold and n h be a riemannian manifold of constant positive scalar curvature we prove that the yamabe equation on the riemannian product m n g h has at least cat m 1 solutions for small enough where cat m denotes the lusternik schnirelmann category of m the solutions obtained are functions of m and c a t m of them have energy arbitrarily close to the minimum

a hierarchy of probabilistic system types we arrange various classes of probabilistic systems studied in the literature in an expressiveness hierarchy our expressiveness criterion is the existence of a system translation from the less expressive type into the more expressive type that preserves and reflects probabilistic bisimilarity we model the different system types as coalgebras of suitable behaviour functors and argue that the corresponding coalgebraic bisimilarity coincides with probabilistic bisimilarity for the classes for which the latter notion has been proposed in the literature the theory of coalgebras provides a unified framework for the presentation of the different classes and the system translations we needed to establish the hierarchy all these translations arise in a standard way from natural transformations between the two behaviour functors involved such a translation generally preserves coalgebraic bisimilarity we exploit a new result that under mild assumptions on the behaviour functors a system translation induced by a natural transformation with injective components also reflects bisimilarity

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

an obstruction to the positivity of relative yamabe invariants for a supergroup gamma we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n dimensional gamma manifold w with nonempty boundary m 3 leq n leq 8 in terms of the bordism class m in the stolz obstruction group r n 1 gamma associated to gamma st2 in par ticular when w is a 5 dimensional spin manifold and the alpha invariant of a connected component of m is nonzero we prove that w does not admit a positive scalar curvature metric with minimal boundary condition

eigenvalue estimate of the dirac operator and rigidity of poincare einstein metrics we re visit the eigenvalue estimate of the dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal laplace operator as well as the conformal mean curvature operator these problems were studied earlier by hijazi montiel zhang and raulot and we re prove them under weaker assumption that a boundary chirality operator exists moreover on these spin manifolds with boundary we show that any c 3 alpha conformal compactification of some poincare einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the dirac operator achieves its lowest value and any c 3 alpha conformal compactification of some poincare einstein metric must be the flat ball in euclidean space when the first positive eigenvalue of the boundary dirac operator achieves certain value relating to the second yamabe invariant in two cases the poincare einstein metrics are standard hyperbolic metric

on a weak solution for a doubly critical fourth order semilinear elliptic equation in a compact manifold abstract the purpose of the article is to provide sufficient conditions for having a positive weak solution for a fourth order semilinear elliptic equation in a compact riemannian manifold also we furnish the palais smale property for a suitable functional

the spinorial tau invariant and 0 dimensional surgery let m be a compact manifold with a metric g and with a fixed spin structure chi let lambda 1 g be the first non negative eigenvalue of the dirac operator on m g chi we set tau m chi sup inf lambda 1 g where the infimum runs over all metrics g of volume 1 in a conformal class g 0 on m and where the supremum runs over all conformal classes g 0 on m let m chi be obtained from m chi by 0 dimensional surgery we prove that tau m chi geq tau m chi as a corollary we can calculate tau m chi for any riemann surface m

sobolev trace inequalities of order four we establish sharp trace sobolev inequalities of order four on euclidean d balls for d 4 when d 4 our inequality generalizes the classical second order lebedev milin inequality on euclidean 2 balls our method relies on the use of scattering theory on hyperbolic d balls as an application we characterize the extremal metric of the main term in the log determinant formula corresponding to the conformal laplacian coupled with the boundary robin operator on euclidean 4 balls which surprisingly is not the flat metric on the ball

the han li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds abstract the han li conjecture states that let m g 0 be an n dimensional n 3 smooth compact riemannian manifold with boundary having positive generalized yamabe constant and c be any real number then there exists a conformal metric of g 0 with scalar curvature 1 and boundary mean curvature c combining with z c han and y y li s results we answer this conjecture affirmatively except for the case that n 8 the boundary is umbilic the weyl tensor of m vanishes on the boundary and has an interior non zero point

lyapunov functions and cone families we describe systematically the relation between lyapunov functions and nonvanishing lyapunov exponents both for maps and flows this includes a brief survey of the existing results in the area in particular we consider separately the cases of nonpositive and arbitrary lyapunov functions thus yielding optimal criteria for negativity and positivity of the lyapunov exponents of linear cocycles over measure preserving transformations moreover we describe converse results of these criteria with the explicit construction of eventually strict lyapunov functions for any map or flow with nonzero lyapunov exponents we also construct examples showing that in general the existence of an eventually strict invariant cone family does not imply the existence of an eventually strict lyapunov function

conformal transformation of metrics on the n ball abstract in this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of r n n 3 under the assumption that the prescribed function is flat near its critical point we give precise estimates on the losses of the compactness and we provide new existence and multiplicity results of bahri coron type

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

reliability of mobile processes with noisy channels to model the behavior of channels in real world mobile systems ying introduced an extension of the calculus by taking channel noise into account unfortunately this extension is not faithful in the sense that its semantics does not coincide with the standard one for the calculus in the noise free case in this paper we consider a simple variant of the calculus the asynchronous calculus a which has been used for modeling some concurrent systems with asynchronous communication to model these systems with noisy channels we propose a faithful extension of a called the a n calculus after giving a probabilistic transitional semantics of a n we introduce bisimilarity in a n and show that it is a partial input congruence if a specification of a system is described as a process p in a and we view the behavior of p in a n as an implementation of the specification then it is interesting to measure how far the behavior in a n is from that in a we thus introduce the notion of reliability degree which is based upon a new approximate bisimulation we find that bisimilar agents may have different reliability degrees and even the agent with the greatest reliability degree may not be satisfactory we thus appeal to shannon s noisy channel coding theorem and show that reliability degrees can be improved by employing coding techniques

on fractional gjms operators we describe a new interpretation of the fractional gjms operators as generalized dirichlet to neumann operators associated to weighted gjms operators on naturally associated smooth metric measure spaces this gives a geometric interpretation of the caffarelli silvestre extension for when 0 1 and both a geometric interpretation and a curved analogue of the higher order extension found by r yang for when 1 we give three applications of this correspondence first we exhibit some energy identities for the fractional gjms operators in terms of energies in the compactified poincare einstein manifold including an interpretation as a renormalized energy second for 1 2 we show that if the scalar curvature and the fractional q curvature q2 of the boundary are nonnegative then the fractional gjms operator p2 is nonnegative third by assuming additionally that q2 is not identically zero we show that p2 satisfies a strong maximum principle 2014 wiley periodicals inc

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

solutions of the einstein constraint equations with apparent horizon boundaries we construct asymptotically euclidean solutions of the vacuum einstein constraint equations with an apparent horizon boundary condition specifically we give sufficient conditions for the constant mean curvature conformal method to generate such solutions the method of proof is based on the barrier method used by isenberg for compact manifolds without boundary suitably extended to accommodate semilinear boundary conditions and low regularity metrics as a consequence of our results for manifolds with boundary we also obtain improvements to the theory of the constraint equations on asymptotically euclidean manifolds without boundary

convergence of scalar flat metrics on manifolds with boundary under a yamabe type flow abstract we study a conformal flow for compact riemannian manifolds of dimension greater than two with boundary convergence to a scalar flat metric with constant mean curvature on the boundary is established in dimensions up to seven and in any dimensions if the manifold is spin or if it satisfies a generic condition

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

blow up phenomena for linearly perturbed yamabe problem on manifolds with umbilic boundary abstract we build blowing up solutions for linear perturbation of the yamabe problem on manifolds with umbilic boundary provided the weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n 11

rigidite conforme des hemispheres rm s 4_ et rm s 6_ conformal rigidity of the hemispheres rm s 4_ and rm s 6_ let m g be a four or six dimensional compact riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical in this note we prove that if the euler characteristic of m is equal to 1 and if its yamabe invariant is positive then m g is conformally isometric to the standard hemisphere as an application and using a result of hang and wang comm anal geom 14 1 91 106 2006 we prove a rigidity result for these hemispheres regarding the min oo conjecture

non cmc solutions of the einstein constraint equations on compact manifolds with apparent horizon boundaries in this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the einstein constraint equations on compact manifolds with boundary by building in a natural way on our recent work in holst and tsogtgerel class quantum gravity 30 205011 2013 and holst et al phys rev lett 100 16 161101 2008 commun math phys 288 2 547 613 2009 and also on the work of maxwell j hyperbolic differ eqs 2 2 521 546 2005a commun math phys 253 3 561 583 2005b math res lett 16 4 627 645 2009 and dain class quantum gravity 21 2 555 573 2004 under reasonable assumptions on the data we prove existence of both near and far from constant mean curvature cmc solutions for a class of robin boundary conditions commonly used in the literature for modeling black holes with a third existence result for cmc appearing as a special case dain and maxwell addressed initial data engineering for space times that evolve to contain black holes determining solutions to the conformal formulation on an asymptotically euclidean manifold in the cmc setting with interior boundary conditions representing excised interior black hole regions holst and tsogtgerel compiled the interior boundary results covered by dain and maxwell and then developed general interior conditions to model the apparent horizon boundary conditions of dainand maxwell for compact manifolds with boundary and subsequently proved existence of solutions to the lichnerowicz equation on compact manifolds with such boundary conditions this paper picks up where holst and tsogtgerel left off addressing the general non cmc case for compact manifolds with boundary as in our previous articles our focus here is again on low regularity data and on the interaction between different types of boundary conditions while our work here serves primarily to extend the solution theory for the compact with boundary case we also develop several technical tools that have potential for use for other cases

the sharp affine l 2 sobolev trace inequality and variants we establish a sharp affine l p sobolev trace inequality by using the l p busemann petty centroid inequality for p 2 our affine version is stronger than the famous sharp l 2 sobolev trace inequality proved independently by escobar and beckner our approach allows also to characterize all extremizers in this case for this new inequality no euclidean geometric structure is needed

lichnerowicz type equations with sign changing nonlinearities on complete manifolds with boundary abstract we prove an existence theorem for positive solutions to lichnerowicz type equations on complete manifolds with boundary m m and nonlinear neumann conditions this kind of nonlinear problems arise quite naturally in the study of solutions for the einstein scalar field equations of general relativity in the framework of the so called conformal method

nonuniform exponential contractions and lyapunov sequences abstract for a nonautonomous dynamics with discrete time obtained from the product of linear operators we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict lyapunov sequences we note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics we also obtain inverse theorems that give explicitly strict lyapunov sequences for each nonuniform exponential contraction essentially the lyapunov sequences are obtained in terms of what are usually called lyapunov norms that is norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform we also show how the characterization of nonuniform exponential contractions in terms of quadratic lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations moreover we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict lyapunov function implies that all lyapunov exponents are negative almost everywhere

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