generic vanishing for semi abelian varieties and integral alexander modules we revisit generic vanishing results for perverse sheaves with any field coefficients on a complex semi abelian variety and indicate several topological applications in particular we obtain finiteness properties for the integral alexander modules of complex algebraic varieties mapping to semi abelian varieties similar results were recently derived by the authors by using morse theoretic arguments

local and pointwise error estimates of the local discontinuous galerkin method applied to the stokes problem we prove local and pointwise error estimates for the local discontinuous galerkin method applied to the stokes problem in two and three dimensions by using techniques originally developed by a schatz math comp 67 1998 877 899 to prove pointwise estimates for the laplace equation we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

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

a volume comparison theorem for asymptotically hyperbolic manifolds we define a notion of renormalized volume of an asymptotically hyperbolic manifold moreover we prove a sharp volume comparison theorem for metrics with scalar curvature at least 6 finally we show that the inequality is strict unless the metric is isometric to one of the anti desitter schwarzschild metrics

toric complexes and artin kernels abstract a simplicial complex l on n vertices determines a subcomplex t l of the n torus with fundamental group the right angled artin group g l given an epimorphism g l z let t l be the corresponding cover with fundamental group the artin kernel n we compute the cohomology jumping loci of the toric complex t l as well as the homology groups of t l with coefficients in a field k viewed as modules over the group algebra k z we give combinatorial conditions for h r t l k to have trivial z action allowing us to compute the truncated cohomology ring h r t l k we also determine several lie algebras associated to artin kernels under certain triviality assumptions on the monodromy z action and establish the 1 formality of these not necessarily finitely presentable groups

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

the penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass in the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature 6 and negative mass when the genus of the conformal boundary at infinity is positive using inverse mean curvature flow we prove a penrose inequality for these negative mass metrics the motivation comes from a previous result of p chru ciel and w simon which states that the penrose inequality we prove implies a static uniqueness theorem for negative mass kottler metrics

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

pure virtual braids resonance and formality we investigate the resonance varieties lower central series ranks and chen ranks of the pure virtual braid groups and their upper triangular subgroups as an application we give a complete answer to the 1 formality question for this class of groups in the process we explore various connections between the alexander type invariants of a finitely generated group and several of the graded lie algebras associated to it and discuss possible extensions of the resonance chen ranks formula in this context

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

on the riemannian penrose inequality in dimensions less than eight the positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the euclidean metric the riemannian penrose inequality provides a sharp lower bound for the mass when black holes are present more precisely this lower bound is given in terms of the area of an outermost minimal hypersurface and equality is achieved only for schwarzschild metrics the riemannian penrose inequality was first proved in three dimensions in 1997 by g huisken and t ilmanen for the case of a single black hole see hi in 1999 bray extended this result to the general case of multiple black holes using a different technique see br in this article we extend the technique of br to dimensions less than eight part of the argument is contained in a companion article by lee l the equality case of the theorem requires the added assumption that the manifold be spin

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

on the topology of initial data sets with higher genus ends in this note we study the topology of 3 dimensional initial data sets with horizons of a sort associated with asymptotically locally anti de sitter spacetimes we show that within this class those initial data sets that contain no immersed marginally outer trapped surfaces in their interior must have simple topology they are a product of a surface and an interval or a mild variation thereof depending on the connectedness of the horizon and on its genus relative to that of the end the results obtained here extend results in eichmair et al j differ geom 95 389 405 2013 to the case of higher genus ends

poincare duality and resonance varieties we explore the constraints imposed by poincare duality on the resonance varieties of a graded algebra for a three dimensional poincare duality algebra a we obtain a fairly precise geometric description of the resonance varieties

perturbations of forms and error estimates for the finite element method at a point with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point we first derive a variety of local error estimates for u uh at a point x0 where uh belongs to a finite element space shr and is an approximation to u satisfying the local equations a u u h varphi f varphi for all varphi in shr with compact support in a neighborhood of x0 here the a cdot cdot are bilinear forms associated with second order elliptic equations and the f are linear functionals in the case that f equiv 0 our results coincide with those of schatz siam j numer anal 38 2000 pp 1269 1293 but are improvements when f neq 0 we apply these results to improve the superconvergence error estimates obtained by schatz sloan and wahlbin siam j numer anal 33 1996 pp 505 521 at points x0 where the subspaces are symmetric with respect to x0

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds we construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3 manifold which does not have the convergence properties needed in order to prove a penrose type inequality this contrasts sharply with the asymptotically flat case the main idea consists in combining inverse mean curvature flow with work done by shi tam regarding boundary behavior of compact manifolds assuming the penrose inequality holds we also derive a nontrivial inequality for functions on s

on the geometry and mass of static asymptotically ads spacetimes and the uniqueness of the ads soliton we prove two theorems announced in 6 for static spacetimes that solve einstein s equation with negative cosmological constant the first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity analogous to the structure theorems of cheeger and gromoll for manifolds of non negative ricci curvature for spacetimes with ricci flat conformal boundary the convexity condition is associated with negative mass the second theorem is a uniqueness theorem for the negative mass ads soliton spacetime this result lends support to the new positive mass conjecture due to horowitz and myers which states that the unique lowest mass solution which asymptotes to the ads soliton is the soliton itself this conjecture was motivated by a nonsupersymmetric version of the ads cft correspondence our results add to the growing body of rigorous mathematical results inspired by the ads cft correspondence conjecture our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ground state spacetimes such as minkowski spacetime namely the presence of a null line or complete achronal null geodesic and the totally geodesic null hypersurface that it determines en route we provide an analysis of the boundary data at conformal infinity for the lorentzian signature static einstein equations in the spirit of the fefferman graham analysis for the riemannian signature case this leads us to generalize to arbitrary dimension a mass definition for static asymptotically ads spacetimes given by chru ciel and simon we prove equivalence of this mass definition with those of ashtekar magnon and hawking horowitz

topology and geometry of cohomology jump loci we elucidate the key role played by formality in the theory of characteristic and resonance varieties we define relative characteristic and resonance varieties v k and r k related to twisted group cohomology with coefficients of arbitrary rank we show that the germs at the origin of v k and r k are analytically isomorphic if the group is 1 formal in particular the tangent cone to v k at 1 equals r k these new obstructions to 1 formality lead to a striking rationality property of the usual resonance varieties a detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth complex quasi projective algebraic variety this sheds new light on a classical problem of j p serre applications to arrangements configuration spaces coproducts of groups and artin groups are given

eine aussage zurl stabilitat und zur genauen konvergenzordnung derh 0 0 projektionen in this paper we settle a question concerning the necessity of the logh factor appearing in error estimates of linear finite element solutions let 2 be a bounded convex polygon and t h a regular family of subdivisions of into triangles be furthers 0 h h 0 1 the corresponding linear finite element spaces andp h h 0 1 s 0 h theh 0 1 projections with respect to the uniform norm asymptotically estimates of the form parallel u p h u parallel infty leqq c logh dist u s 0 h u in c bar omega cap h 0 1 omega hold as pointed out by schatz 1980 for functionsu with bounded second derivatives this implies parallel u p h u parallel infty leqq ch 2 log h u 2 infty a result which was proved by nitsche 1975 and scott 1976 with different methods it was an open question if the factor logh is necessary or only due to the methods of proof in this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if logh is replaced by a termo logh

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

characteristic varieties of hypersurface complements abstract we give divisibility results for the global characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface as an application we recast old and obtain new finiteness and divisibility results for the classical infinite cyclic alexander modules of complex hypersurface complements moreover for the special case of hyperplane arrangements we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

the mass of asymptotically hyperbolic manifolds motivated by certain problems in general relativity and riemannian geometry we study manifolds which are asymptotic to the hyperbolic space in a certain sense it is shown that an invariant the so called total mass can be defined unambiguously a positive mass theorem is established by using the spinor method

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

arithmetic group symmetry and finiteness properties of torelli groups we examine groups whose resonance varieties characteristic varieties and sigma invariants have a natural arithmetic group symmetry and we explore implications on various niteness properties of subgroups we compute resonance varieties characteristic varieties and alexander polynomials of torelli groups and we show that all subgroups containing the johnson kernel have nite rst betti number when the genus is at least 4 we also prove that in this range the i adic completion of the alexander invariant is nite dimensional and the kahler property for the torelli group implies the nite generation of the johnson kernel

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

on the mass of static metrics with positive cosmological constant ii this is the second of two works in which we discuss the definition of an appropriate notion of mass for static metrics in the case where the cosmological constant is positive and the model solutions are compact in the first part we have established a positive mass statement characterising the de sitter solution as the only static vacuum metric with zero mass in this second part we prove optimal area bounds for horizons of black hole type and of cosmological type corresponding to riemannian penrose inequalities and to cosmological area bounds a la boucher gibbons horowitz respectively building on the related rigidity statements we also deduce a uniqueness result for the schwarzschild de sitter spacetime

flat connections and resonance varieties from rank one to higher ranks given a finitely generated group and a linear algebraic group g the representation variety homp gq has a natural filtration by the characteristic varieties associated to a rational representation of g its algebraic counterpart the space of g valued flat connections on a commutative differential graded algebra pa dq admits a filtration by the resonance varieties associated to a representation of g we establish here a number of results concerning the structure and qualitative properties of these embedded resonance varieties with particular attention to the case when the rank 1 resonance variety decomposes as a finite union of linear subspaces the general theory is illustrated in detail in the case when is either an artin group or the fundamental group of a smooth quasi projective variety

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

some error estimates for the lumped mass finite element method for a parabolic problem we study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous dirichlet boundary conditions improving earlier results we show that known optimal order smooth initial data error estimates for the standard galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation we also discuss the application to time discretization by the backward euler and crank nicolson methods

multinets parallel connections and milnor fibrations of arrangements the characteristic varieties of a space are the jump loci for homology of rank 1 local systems the way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers we exploit this phenomenon to detect torsion in the homology of milnor fibers of projective hypersurfaces one tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid another tool is a polarization construction based on the parallel connection operad for matroids our main result gives a combinatorial machine for producing arrangements whose milnor fibers have torsion in homology in particular this shows that milnor fibers of hyperplane arrangements do not necessarily have a minimal cell structure

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives