rigidity of nonpositively curved graphmanifolds there are strong relations between the metric structure of a complete riemannian manifold m of nonpositive sectional curvature and the algebraic structure of its fundamental group in particular we have the following rigidity results for compact m the fundamental group i m determines whether the universal covering space x of m is a nontrivial riemannian product or a symmetric space of higher rank the product case follows from the work of lawson yau ly and gromollwolf gw combined with the results in e 1 the statement is also true i fm is complete and has finite volume is the case of symmetric spaces is stated in be it is based on the rigidity results of mostow m gromov bgs and eberlein e2 and the investigation of higher rank manifolds by ballmann brin burns eberlein and spatzier bbe bbs b this result also holds in the finite volume case these examples products and higher rank symmetric spaces have rank 2 in the sense of bbe on the other hand manifolds of higher rank are products or symmetric spaces b thus the above rigidity results reflect the rigidity properties of higher rank spaces in this paper we study the rigidity of a special class of rank 1 manifolds namely the 3 dimensional graphmanifolds of non positive curvature our main result says that the whole metric structure of nonpositively curved graphmanifolds is determined by the fundamental group see theorem 2 in sect 4 for a precise statement we should say a few words about the relevance of nonpositively curved graphmanifolds waldhausen w studied graphmanifolds from a topological viewpoint gromov g mentioned that a special class of graphmanifolds allow complete riemannian metrics of nonpositive curvature in this way he obtained the first examples of complete riemannian manifolds with bounded nonpositive curvature a 2 k 0 and finite volume with infinite topological type

development of a distributed dislocation dipole technique for the analysis of multiple straight kinked and branched cracks in an elastic half plane abstract a distributed dislocation dipole technique for the analysis of multiple straight kinked and branched cracks in an elastic half plane has been developed the dipole density distribution is represented with a weighted jacobi polynomial expansion where the weight function captures the asymptotic behaviour at each end of the crack to allow for opening and sliding at crack kinking and branching the dipole density representation contains conditional extra terms which fulfills the asymptotic behaviour at each endpoint several test cases involving straight kinked and branched cracks have been analysed and the results suggest that the accuracy of the method is within 1 provided that jacobi polynomial expansions up to at least the sixth order are used adopting even higher order jacobi polynomials yields improved accuracy the method is compared to a simplified procedure suggested in the literature where stress singularities associated with corners at kinking or branching are neglected in the representation for the dipole density distribution the comparison suggests that both procedures work but that the current procedure is superior in as much as the same accuracy is reached using substantially lower order polynomial expansions

the stretch of a foliation and geometric superrigidity we consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type such objects can be considered as compact models for the geometry of the symmetric space based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions to achieve this we consider the heat flow equation along the leaves of a foliation a bochner formula on foliations and a geometric invariant for foliations with leafwise riemannian metrics called the stretch we obtain as applications a metric rigidity theorem for foliations and a rigidity type result for riemannian manifolds whose geometry is only partially symmetric

kinked crack analysis by a hybridized boundary element boundary collocation method in this research a two dimensional displacement discontinuity method which is a kind of indirect boundary element method using higher order elements i e a source element with a cubic variation of displacement discontinuities having four sub elements is used to obtain the displacement discontinuities along each boundary element in this paper three kinds of the higher order boundary elements are used the ordinary elements the kink elements and the special crack tip elements the boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub elements again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis considering the two source elements each having four sub elements joined at a corner kink point the collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated the displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides the special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip some simple example problems are solved numerically by the proposed method the numerical results are compared with the corresponding results obtained by the previous methods cited in the literature this comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method

boundary manifolds of projective hypersurfaces abstract we study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface we show that under certain hodge theoretic conditions the cohomology ring of the complement of the hypersurface functorially determines that of the boundary when the hypersurface defines a hyperplane arrangement the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension we also study the ls category and topological complexity of the boundary manifold as well as the resonance varieties of its cohomology ring

rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

amenable groups and stabilizers of measures on the boundary of a hadamard manifold let x be a symmetric space of noncompact type or more generally a hadamard manifold i e a complete simply connected riemannian manifold of nonpositive sectional curvature we consider a finite positive borel measure on the ideal boundary x see sect 2 for the definitions let g be the isometry group of x and g the subgroup which stabilizes the measure in the case of a symmetric space we obtain the following result

on the asymptotic geometry of nonpositively curved graphmanifolds in this paper we study the tits geometry of a 3 dimensional graphmanifold of nonpositive curvature in particular we give an optimal upper bound for the length of nonstandard components of the tits metric in the special case of a 2 metric we determine the whole length spectrum of the nonstandard components

eigenvalues of the drifting laplacian on complete noncompact riemannian manifolds abstract in this paper we investigate eigenvalues of the eigenvalue problem with dirichlet boundary condition of the drifting laplacian on an n dimensional complete noncompact riemannian manifold some estimates for eigenvalues are obtained by utilizing cheng and yang recursion formula we give a sharp upper bound of the k th eigenvalue as we know product riemannian manifolds ricci solitons and self shrinkers are some important riemannian manifolds therefore we investigate the eigenvalues of the drifting laplacian on those riemannian manifolds in particular by some theorems of classification for ricci solitons we can obtain some eigenvalue inequalities of drifting laplacian on the ricci solitons with certain conditions

martin points on open manifolds of non positive curvature the martin boundary of a cartan hadamard manifold describes a fine geometric structure at infinity which is a sub space of positive harmonic functions we describe conditions which ensure that some points of the sphere at infinity belong to the martin boundary as well in the case of the universal cover of a compact manifold with ballmann rank one we show that martin points are generic and of full harmonic measure the result of this paper provides a partial answer to an open problem of s t yau

boundary manifolds of line arrangements abstract while the boundary 3 manifold of a line arrangement in the complex plane depends only on the incidence correspondence of the line arrangement the homotopy type of the complement depends on the relative positions of incidences in this paper we describe the homotopy type of line arrangement complements in terms of an associated plumbed graph for pseudo real line arrangements this method provides an explicit description of the homotopy type and fundamental group of the complement in terms of its ordered incidence graph the method also extends to a larger class of unknotted line arrangements

crack paths in plane situations ii detailed form of the expansion of the stress intensity factors abstract in a previous paper we established the general form of the expansion of the stress intensity factors in powers of the crack extension length for a crack propagating in a two dimensional body along an arbitrary kinked and curved path the aim of the present paper is to calculate precisely the various functions of the geometric and mechanical parameters which appear in this expansion the functions involved in the case of a straight extension are identified by considering the special case of a crack composed of two straight branches placed in an infinite body loaded by uniform forces at infinity the problem is solved with the aid of muskhelishvili s formalism and conformal mapping the functions describing the effect of the curvature of the crack extension are determined by studying another special case identical to the first one except that the crack extension is curved the method of solution consists of using a perturbative procedure with respect to the curvature parameters to reduce the original problem to a simpler one involving a crack with two straight branches and solving again the latter problem by conformal mapping a numerical strategy using these results for the prediction of crack paths over arbitrary long distances is discussed in conclusion

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

acylindrical hyperbolicity of groups acting on trees we provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees in particular we consider amalgamated products and hnn extensions one relator groups automorphism groups of polynomial algebras 3 manifold groups and graph products acylindrical hyperbolicity is then used to obtain some results about the algebraic structure analytic properties and measure equivalence rigidity of groups from these classes

cusp closing in rank one symmetric spaces we investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature we show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or cayley hyperbolic type cannot be closed for cusps of real hyperbolic type we derive necessary and sufficient closing conditions in this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the gromov hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats by the same methods we get an explicit resolution of the singularities in the baily borel resp siu yau compactification of finite volume quotients of the complex hyperbolic space

amenable isometry groups of hadamard spaces one of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology since the universal covering space of a complete space of nonpositive curvature is contractible this amounts to relations between the geometry of the space or its universal covering space and the algebraic structure of its fundamental group a paradigm is the well known result of avez av that the fundamental group of a compact riemannian manifold m of nonpositive sectional curvature has exponential growth if and only if m is not flat gromov observed that the proof of avez yields a somewhat stronger statement the fundamental group of m is amenable if and only if m is flat see g1 p 93 zimmer generalized this to the case where m is complete and of finite volume zi further generalizations were obtained by anderson an and burger and schroeder bs the objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are alexandrov spaces of nonpositive curvature for the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature it is convenient to study the isometric action of the fundamental group on the universal covering space in many situations it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature this is the case in most of our arguments below let x be a hadamard space that is a complete simply connected geodesic space of nonpositive curvature in the sense of alexandrov a flat in x is a

local rigidity of symmetric spaces we show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds then the new metric is isometric to the old one we also prove an analytic continuation property for symmetric spaces of rank 3 the aim of this paper is to show that a symmetric space is rigid under a local modification of the metric which preserves suitable curvature bounds our results were motivated by the following theorem of gromov bgs 5 theorem 1 gromov let x be a simply connected symmetric space of rank 2 and with nonpositive sectional curvature and q the complement of a compact set in x if f q y is a totally geodesic isometric embedding where y is a complete simply connected riemannian manifold with nonpositive sectional curvature then there exists a unique extension of f to a totally geodesic isometric embedding tf x y we will give a slightly different proof of this theorem which will then enable us to also prove the following dual result theorem 2 let x be a simply connected symmetric space of rank 2 and with sectional curvature 0 0 and dim y dim x and if q is the complement of a ball of radius 7r 30 in x then any isometric embedding f q y extends uniquely to an isometry 7 x y if in the above two theorems we choose y x and f id we obtain the desired local rigidity of symmetric spaces theorem 2 applies in particular to s2 x s and shows that the hopf conjecture is at least locally correct the optimal radius for the ball in theorem 2 would be the radius of a convex ball 7r 2 under those assumptions indeed in the case of nonpositive curvature gromov bgs 5 proved the following local version of theorem 1 received by the editors november 4 1987 and in revised form september 12 1988 1980 mathematics subject classification 1985 revision primary 53c35 secondary 53c20 the first author is supported by the schweizerischer nationalfond the second author is partially supported by a grant from the national science foundation and would like to thank the ihes and the max planck institut for its hospitality 1990 american mathematical society 0002 9947 90 1 00 25 per page

a concept of nonlinear block diagonal dominance we introduce a concept of block diagonal dominance for nonlinear functions and discuss the relations to strictly diagonally dominant functions and m functions some sufficient conditions for the new kind of functions are given the global convergence of block asynchronous sor methods for finding zeros of block diagonal dominant nonlinear functions is proved

three manifolds with many flat planes we discuss the rigidity or lack thereof imposed by dierent no tions of having an abundance of zero curvature planes on a complete riemann ian 3 manifold we prove a rank rigidity theorem for complete 3 manifolds showing that having higher rank is equivalent to having reducible universal covering we also study 3 manifolds such that every tangent vector is con tained in a at plane including examples with irreducible universal covering and discuss the eect of nite volume and real analiticity assumptions

symmetric galerkin bem simulation of fracture with frictional contact a symmetric galerkin boundary element framework for fracture analysis with frictional contact crack friction on the crack surfaces is presented the algorithm employs a continuous interpolation on the crack surface utilizing quadratic boundary elements and enables the determination of two important quantities for the problem namely the local normal tractions and sliding displacements on the crack surfaces an effective iterative scheme for solving this non linear boundary value problem is proposed the results of test examples are compared with available analytical solutions or with those obtained from the displacement discontinuity method ddm using linear elements and internal collocation the results demonstrate that the method works well for difficult kinked junction crack problems copyright 2003 john wiley sons ltd

compact kahler manifolds with nonpositive bisectional curvature in this article we prove that for any compact kahler manifold mn with real analytic metric and nonpositive bisectional curvature there exists a finite cover m of m such that m is a holomorphic and metric fiber bundle over a compact kahler manifold n with nonpositive bisectional curvature and c1 n 0 and the fiber is a flat complex torus this partially confirms a conjecture of yau

fundamental groups of manifolds of nonpositive curvature introduction the universal covering space h of a complete riemannian manifold m of nonpositive sectional curvature is diffeomorphic to r n dim f hence the homotopy type of m is completely determined by the isomorphism class of the fundamental group of m it is therefore only natural to expect strong relations between the geometric structure of m and the algebraic structure of in this paper we obtain several such relations a general assumption in the results we state below is that 1 the sectional curvature is nonpositive and bounded from below by some constant a and 2 the volume of m is finite we define the rank of a unit tangent vector of m rank u to be the dimension of the space of all parallel jacobi fields along the geodesic yv which has initial velocity v the minimum of rank over all v e sm is called the rank of m this agrees with the usual rank if m is a locally symmetric space manifolds of rank one resemble manifolds of strictly negative curvature see 2 3 and 2 below manifolds of higher rank are studied in 5 6 7 and 3 and the conclusive result is that h is a space of rank one or a symmetric space or a riemannian product of such spaces this is the basic ingredient in the proofs of our results it allows us more or less to consider only the cases that h is of rank one or a symmetric space as a first example of this principle we indicate in our preliminary section the proof of the following theorem theorem a either m is flat or contains a nonabelian free subgroup this is an improvement of the result of avez 1 that has exponential growth if m is compact and not flat

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

geometric properties of rank one asymptotically harmonic manifolds in this article we consider asymptotically harmonic manifolds which are simply connected complete riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h we prove the following equivalences for asymptotically harmonic manifolds x under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded a x has rank one b x has anosov geodesic flow c x is gromov hyperbolic d x has purely exponential volume growth with volume entropy equals h this generalizes earlier results by g knieper for noncompact harmonic manifolds and by a zimmer for asymptotically harmonic manifolds admitting compact quotients

riemannian manifolds with flat ends soit m une variete de riemann complete non compacte de dimension n 2 une extremite de m est une application e qui affecte a chaque sousensemble compact de m une composante connexe e de m telle que e e si une extremite e est dite plate si e est plate pour un sousensemble de m on determine le type d isometrie de e pour un bon ensemble

a hilbert mumford criterion for polystability in kaehler geometry consider a hamiltonian action of a compact lie group k on a kaehler manifold x with moment map x assume that the action of k extends to a holomorphic action of the complexification g of k we characterize which g orbits in x intersect 1 0 in terms of the maximal weights lim t e its x s where s we do not impose any a priori restriction on the stabilizer of x under some mild restrictions on the action k x we view the maximal weights as defining a collection of maps for each x x x k g where k g is the boundary at infinity of the symmetric space k g we prove that g x 1 0 0 if 1 x is everywhere nonnegative 2 any boundary point y such that x y 0 can be connected with a geodesic in k g to another boundary point y satisfying x y 0 we also prove that the maximal weight functions are g equivariant for any g g and any y k g we have g x y x y g

on nonpositively curved compact riemannian manifolds with degenerate ricci tensor in this article we prove that if the ricci tensor of a compact nonpositively curved manifold m n is nowhere negative definite then it admits local flat factors to be more precise let u c m be the open subset where the ricci tensor has maximum rank r then for any connected component u a of u its universal cover u a is isometric to r n r n r a where n r a is a nonpositively curved manifold with negative ricci curvature in particular if m n is compact nonpositively curved without euclidean de rham factor and both the manifold and the metric are real analytic then its ricci tensor must be negative definite somewhere

superrigidity for irreducible lattices and geometric splitting we prove general superrigidity results for actions of irreducible lattices on cat 0 spaces first in terms of the ideal boundary and then for the intrinsic geometry including for infinite dimensional spaces in particular one obtains a new and self contained proof of margulis superrigidity theorem for uniform irreducible lattices in non simple groups the proofs rely on simple geometric arguments including a splitting theorem which can be viewed as an infinite dimensional and singular generalization of the lawson yau gromoll wolf theorem appendix a gives a very elementary proof of commensurator superrigidity appendix b proves that all our results also hold for certain non uniform lattices

invariant measures on g γ for split simple lie groups g we study the left action of a cartan subgroup on the space x g where is a lattice in a simple split connected lie group g of rank n 1 let be an invariant measure on x we give several conditions using entropy and conditional measures each of which characterizes the haar measure on x furthermore we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure the main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures 2003 wiley periodicals inc

axial isometries of manifolds of non positive curvature let m be a complete c riemannian manifold of non positive sectional curvature we say that a geodesic 9 ir m bounds a fiat strip of width c 0 a fiat half plane if there is a totally geodesic isometric immersion i 0 c x ir m i 0 oo x ir m such that i 0 t 9 0 a 9eodesic without fiat strip without fiat half plane is a geodesic which does not bound a flat strip a flat half plane we will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of m in fact many of the properties of a manifold of strictly negative curvature resp of a visibility manifold still remain true if one assumes only the existence of a closed geodesic without flat half plane we will discuss the existence of free non abelian subgroups of gl m the existence of infinitely many closed geodesics the density of closed geodesics and a transitivity property of the geodesic flow it is therefore interesting to give conditions which ensure the existence of a closed geodesic without flat half plane we will prove that m has a closed geodesic without flat half plane if vol m oo and if m contains a geodesic without flat half plane note that a geodesic is not boundary of a flat strip and a fortiori not boundary of a flat half plane if it passes through a point p e m such that the sectional curvature of all tangent planes at p is negative in the proofs of our results we investigate the action of rtl m as group of isometries on the universal covering space h of m in the proofs of many of our results we do not use the fact that this action is properly discontinuous and free we therefore formulate these results for arbitrary groups d of isometries of h the paper is organized as follows in sect 1 we fix some definitions and notations and quote some standard results of non positive curvature section 2 is the central section of this paper we investigate the properties of those isometries of h which correspond to closed geodesics in m we also prove

4 dimensional locally cat 0 manifolds with no riemannian smoothings we construct examples of 4 dimensional manifolds m supporting a locally cat 0 metric whose universal covers q m satisfy hruska s isolated flats condition and contain 2 dimensional flats f with the property that 1f s s s s 1 q m are nontrivial knots as a consequence we obtain that the group 1 m cannot be isomorphic to the fundamental group of any compact riemannian manifold of nonpositive sectional curvature in particular if k is any compact locally cat 0 manifold then m k is a locally cat 0 manifold which does not support any riemannian metric of nonpositive sectional curvature

slightly curved or kinked cracks in anisotropic elastic solids abstract slightly curved or kinked cracks in anisotropic elastic solids are studied via a perturbation analysis valid for the second order accuracy in the deviation of the crack surfaces from a straight line the analysis is based on complex variable representations in the stroh formalism and known solutions for a perfectly straight reference crack first and second order perturbation solutions are given for the stress intensity factors at the tip of a finite curved crack under remote stresses in a solid with arbitrary anisotropy and are used to establish an approximate relationship between the apparent and local stress intensities at a crack tip in view of the possible shielding effects of the crack surface morphology near the tip the role of anisotropy is examined for a number of problems including circular arc cracks slightly jogged or kinked cracks cosine wavy cracks etc in several cases further insight into the anisotropic effect on crack curving or kinking is provided by specializing the general perturbation formulae to cracks in materials with orthotropic symmetry in the limiting case of a crack with an infinitesimal branch length comparison with numerical results reported in the literature indicates that our perturbation solutions are accurate over the full range of practically important branch angles up to nearly 150 for anisotropic elastic solids various fracture criteria such as those based on energy release rate g and stress intensity factors k 1 and k 11 cease to give consistent predictions on crack behaviour by investigating the effects of mode mixity material anisotropy and non singular t stress on the behaviour of a nearly symmetric crack path in an orthotropic solid we show that the k based fracture criteria lead to very peculiar perhaps even physically unreasonable predictions such as i branch angle at a mixed mode crack tip becomes infinite as the compliances perpendicular and parallel to the crack s vv s vv reach a four fold difference ii the sign of the branch angle is reversed when s vv s vv 4 with the consequence of reversing the role of the non singular t stress on the stability of a symmetric fracture path namely the path becomes stable for t 0 and unstable for t k t becomes a local minimum with respect to crack angle once s vv s vv 4 in contrast the g based criterion gives reasonable predictions such as i cracks under mixed mode loading always tend to branch toward symmetric orientations ii material and loading asymmetries play an equivalent role in affecting crack branching near a symmetric orientation iii a compressive t stress always tends to stabilize a symmetric fracture path while a tensile t stress destabilizes such a path for general crack path stability the role of anisotropy can be manifested through i the value of t stress at the crack tip and ii the variation of fracture resistance with respect to crack orientation

manifolds with quadratic curvature decay and fast volume growth we give sufficient conditions for a noncompact riemannian manifold which has quadratic curvature decay to have finite topological type with ends that are cones over spherical space forms

on splitting theorems for cat 0 spaces and compact geodesic spaces of non positive curvature in this paper we show some splitting theorems for cat 0 spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non positive curvature a cat 0 group is said to be rigid if determines its boundary up to homeomorphisms of a cat 0 space on which acts geometrically c croke and b kleiner have constructed a non rigid cat 0 group as an application of the splitting theorems for cat 0 spaces we obtain that if 1 and 2 are rigid cat 0 groups then so is 1 2

on the geometric rank of homogeneous spaces of nonpositive curvature in this paper we investigate differential geometric properties of homogeneous hadamard manifolds and prove a rigidity result in this class of spaces a hadamard manifold h is a simply connected complete riemannian manifold of nonpositive curvature if in addition h is homogeneous then by the work of all aw1 aw2 and hei we know that h is isometric to a solvable lie group s a n with lie algebra s of type nc see aw1 and a left invariant metric here a exp s s denotes an abelian complement to the nilpotent commutator subgroup n exp s s assuming that h has a trivial flat de rham factor the number k dim a only depends on h whereas s need not be uniquely determined and is called the aloebraic rank of h k is an upper bound for the oeometric rank r rk h dr3 i e the maximal integer such that every geodesic is contained in a r fiat a totally geodesic copy of in h the geometric meaning of rk m is well understood for a huge class of hadamard manifolds m thanks to the following remarkable result by ballmann and burns spatzier we quote a generalisation from e3 and eh

higher rank hyperbolicity the large scale geometry of hyperbolic metric spaces exhibits many distinctive features such as the stability of quasi geodesics the morse lemma the visibility property and the homeomorphism between visual boundaries induced by a quasi isometry we prove a number of closely analogous results for spaces of rank n ge 2 in an asymptotic sense under some weak assumptions reminiscent of nonpositive curvature for this purpose we replace quasi geodesic lines with quasi minimizing locally finite n cycles of r n volume growth prime examples include n cycles associated with n quasiflats solving an asymptotic plateau problem and producing unique tangent cones at infinity for such cycles we show in particular that every quasi isometry between two proper text cat 0 spaces of asymptotic rank n extends to a class of n 1 cycles in the tits boundaries

k theory and dynamics i un theoreme de h cobordisme avec commande feuillettee la variete associee le transfert asymptotique dynamique du flot geodesique completion de la demonstration du theoreme principal revue de la theorie de la commande structures de longues cellules minces

a note on generalized nonlinear diagonal dominance in this paper an open problem proposed by a frommer about nonlinear generalized diagonal dominance is solved on some weak restriction a counterexample is presented if such a restriction is omitted and some new properties of nonlinear generalized diagonally dominant functions are investigated

a geometric proof of the berger holonomy theorem we give a geometric proof of the berger holonomy theorem the proof uses euclidean submanifold geometry of orbits and gives a link between riemannian holonomy groups and normal holonomy groups

the spectrum of the laplacian on forms over flat manifolds in this article we prove that the spectrum of the laplacian on k forms over a non compact flat manifold is always a connected closed interval of the non negative real line the proof is based on a detailed decomposition of the structure of flat manifolds

local convexity and nonnegative curvature gromov s proof of the sphere theorem an immersed hypersurface s in a riemannian manifold m will be called e convex for some e 0 if all principal curvatures have the same sign and absolute value at least e can one characterize all compact e convex immersed hypersurfaces of a complete manifold m if m is euclidean n space n 3 this problem is solved by a theorem of hadamard 14 generalized by hopf 15 s is embedded and bounds a convex n disk see 6 6 6 3 if m is a flat space form s is no longer necessarily embedded but it still bounds an immersed convex n disk the aim of this paper is to show the same fact for a manifold m of curvature k 0

the uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds in this paper we prove a conjecture of a katok stating that the geodesic flow on a compact rank 1 manifold admits a uniquely determined invariant measure of maximal entropy this generalizes previous work of r bowen and g margulis as an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy

on tori without conjugate points in this paper we consider riemannian metrics without conjugate points on an n torus recent work of j heber established that the gradient vector fields of busemann functions on the universal cover of such a manifold induce a natural foliation akin to the weak stable foliation for a riemannian manifold with negative sectional curvature on the unit tangent bundle the main result in the paper is that the metric is flat if this foliation is lipschitz we also prove that this foliation is lipschitz if and only if the metric has bounded asymptotes this confirms a conjecture of e hopf in this case

visibility and rank one in homogeneous spaces of 0 in this paper we study relationships between the visibility axiom and rank one in homogeneous spaces of nonpositive curvature we obtain a complete classification in terms of rank of simply connected homogeneous spaces of nonpositive curvature and dimension 4 of simply connected irreducible homogeneous spaces k 0 which are neither visibility manifolds nor symmetric spaces introduction this paper was motived by the following question what relations are there between the visibility axiom and rank one in homogeneous spaces of nonpositive curvature we give a complete answer to this problem in dimension 4 which led us to obtain a complete classification in terms of rank of simply connected homogeneous spaces of nonpositive curvature and dimension 4 we were also interested in knowing if a simply connected and irreducible homogeneous space of nonpositye curvature was necessarily a visibility manifold or a symmetric space of noncompact type we provide examples showing that the above is not true in general since a riemannian homogeneous manifold h of sectional curvature k 0 admits a simply transitive and solvable lie group of isometries it can be represented as a solvable lie group g with a left invariant metric of nonpositive curvature and the relations and examples mentioned above are investigated in this context we recall that the lie groups g endowed with left invariant metrics of nonpositive curvature satisfying visibility were characterized in 4 theorem 2 3 and corollary 2 3j where it is proved that visibility is equivalent to the existence on g of a left invariant metric of negative curvature the organization of the paper is as follows in 1 we describe the space of parallel jacobi fields along a one parameter subgroup of g which is a geodesic in terms of algebraic conditions on the lie algebra g of g theorem 1 4 in 2 relationships between visibility and rank one are obtained if g is abelian and has codimension one in g visibility and rank one are equivalent in dimension 4 visibility implies rank one theorems 2 4 and 2 6 we improve a result obtained in 4 theorem 2 3 related to the characterization of the lie groups rcceived by the editors november 8 1985 and in revised form august 14 1986 1980 muthenluties su1 ject cl lssific ltion 1985 revision primary 53c30 secondary 53c25 ke usords lnd phr lses homogeneous spaces visibility manifolds rank t 1987 american mathematical society 0002 9947 87 1 00 25 per page

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

on the farrell jones conjecture for higher algebraic k theory here b is the classifying space of the group and we denote by k r the non connective algebraic k theory spectrum of the ring r the homotopy groups of this spectrum are denoted kn r and coincide with quillen s algebraick groups of r qui73 in positive dimensions and with the negative k groups of bass bas68 in negative dimensions the homotopy groups of the spectrum x k r are denotedhn x k r they yield a generalized homology theory and in particular standard computational tools such as the atiyah hirzebruch spectral sequence apply to the left hand side of the assembly map above as a corollary of the main result of this paper we prove conjecture 1 1 in the case where is the fundamental group of a closed riemannian manifold with strictly negative sectional curvature in fact our result is more general and applies to group rings r where r is a completely arbitrary coefficient ring note that if one replaces in conjecture 1 1 the coefficient ring z by an arbitrary coefficient ringr the corresponding conjecture would be false already in the simplest non trivial case if c is the infinite cyclic group the bass heller swan formula bhs64 gra76 p 236 for kn rc kn r t 1 yields that kn rc kn 1 r kn r nkn r nkn r where nkn r is defined as the cokernel of the split inclusion kn r kn r t and does not vanish in general but since s is a model for bc one obtains on the left hand side of the assembly map only hn bc k r kn r kn 1 r

alexander invariants of complex hyperplane arrangements leta be an arrangement of n complex hyperplanes the funda mental group of the complement of a is determined by a braid monodromy homomorphism fi fs pn using the gassner representation of the pure braid group we flnd an explicit presentation for the alexander invariant ofa from this presentation we obtain combinatorial lower bounds for the ranks of the chen groups ofa we also provide a combinatorial criterion for when these lower bounds are attained

generalized nonlinear diagonal dominance and applications to asynchronous iterative methods abstract we introduce a concept of generalized diagonal dominance for nonlinear functions as in the linear case this brings together several apparently different classes of nonlinear functions such as strictly diagonally dominant functions and certain m functions with our concept we easily obtain a quite far reaching result on the global convergence of asynchronous iterative methods for finding zeros of nonlinear functions special cases include some known and several new convergence results for special iterative methods such as the nonlinear jor sor and ssor method

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2

harmonic measures on covers of compact surfaces of nonpositive curvature let m be the universal cover of a compact nonflat surface n of nonpositive curvature we show that on the average the brownian motion on m behaves similarly to the brownian motion on negatively curved manifolds we use this to prove that harmonic measures on the sphere at infinity have positive hausdorff dimension and if the geodesic flow on n is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant