metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

comparison of relative cohomology theories with respect to semidualizing modules we compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules we prove a general balance result for relative cohomology over a cohen macaulay ring with a dualizing module and we demonstrate the failure of the naive version of balance one might expect for these functors we prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases and we provide a yoneda type description of the first relative ext functor finally we show by example that each distinct relative cohomology construction does in fact result in a different functor

asymptotic behaviors of governing equation of gauged sigma model for heisenberg ferromagnet abstract in this note we study weak solutions of equation 0 1 u 4 e u 1 e u 4 i 1 n p i 4 j 1 m q j in r 2 where p i i 1 n resp q j j 1 m are dirac masses concentrated at the points p i i 1 n resp q j j 1 m and n m 1 eq 0 1 represents a governing equation of gauged sigma model for heisenberg ferromagnet we show that it has a sequence of solutions u having behaviors as ln x o 1 at infinity with a free parameter 2 2 n m and our concern in this paper is to study the asymptotic behavior of b as approaching the extremal values 2 and 2 n m

two dimensional solutions of a mean field equation on flat tori abstract we study the mean field equation on the flat torus t c z z u e u t e u 1 t 0 where is a real parameter for a general flat torus we obtain the existence of two dimensional solutions bifurcating from the trivial solution at each eigenvalue up to a multiplicative constant t of laplace operator on the torus in the space of even symmetric functions we further characterize the subset of all eigenvalues through which only one bifurcating curve passes finally local convexity near bifurcating points of the solution curves are obtained

on anisotropic singularities for semi linear elliptic equations in r2 abstract in this note we study weak solutions of 1 u g u 8 0 x 2 in r 2 g u x 0 as x where g s 2 e s 1 e s 1 is an odd increasing and bounded function and 0 x 2 is defined in the sense of distribution 0 x 2 0 x 2 for c c 1 r 2 we obtain that problem 1 admits a unique x 2 odd weak solution which has anisotropic singularity 4 x 2 x 2 near the origin

on the conformal gaussian curvature equation inr2 abstract in this paper we consider the equation u k x e 2u 0 in r 2 0 1 wherek x k x inr2andk x does not decay at x large from the geometric viewpoint it is quite interesting to ask what the best possible range is of total curvature of all solutions of 0 1 in this paper we study this problem for radial solutions we also construct some particularkto demonstrate the rich phenomenon in particular we show by examples that whenk x is negative for x large eq 0 1 possess a branch of solutions which satisfies some monotonicity property

relative cohomology of complexes abstract let r be an arbitrary ring and c a complex with finite gorenstein projective dimension that is the supremum of gorenstein projective dimension of all r modules in c is finite for each complex d we define the nth relative cohomology group ext gp n c d by the equality ext gp n c d h n h om g d where g c is a strict gorenstein projective precover of c if d is a complex with finite gorenstein injective dimension that is the supremum of gorenstein injective dimension of all r modules in d is finite then one can use a dual argument to define a notion of relative cohomology group ext gi n c d we show that if c is a complex with finite gorenstein projective dimension and d a complex with finite gorenstein injective dimension then for each n z there exists an isomorphism ext gp n c d ext gi n c d this shows that the relative cohomology functor of complexes is balanced some induced exact sequences concerning relative cohomology groups are considered

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

classification of solutions of elliptic equations arising from a gravitational o 3 gauge field model abstract in this paper we study an elliptic equation arising from the self dual maxwell gauged o 3 sigma model coupled with gravity when the parameter equals 1 and there is only one singular source we consider radially symmetric solutions there appear three important constants a positive parameter a representing a scaled gravitational constant a nonnegative integer n 1 representing the total string number and a nonnegative integer n 2 representing the total anti string number the values of the products a n 1 a n 2 0 play a crucial role in classifying radial solutions by using the decay rates of solutions at infinity we provide a complete classification of solutions for all possible values of a n 1 and a n 2 this improves previously known results

conformal metrics with prescribed gaussian curvature on s2 we consider on s 2 the problem of which functions k can be the scalar curvature of a metric conformal to the standard metric on s 2 we assume that k is a function of one variable and we obtain a necessary and sufficient condition for the problem to be solvable we also obtain several new sufficient conditions on k which are easy to check in order that the problem be solvable

symmetric patterns of geodesics and automorphisms of surface groups we prove a non equivariant version of mostow rigidity for symmetric patterns of geodesics in hyperbolic space this result allows for a classification of pseudo anosov surface group automorphisms based on the large scale geometric structure of their orbits

gorenstein injective complexes of modules over noetherian rings abstract a complex c is called gorenstein injective if there exists an exact sequence of complexes i 1 i 0 i 1 such that each i i is injective c ker i 0 i 1 and the sequence remains exact when hom e is applied to it for any injective complex e we show that over a left noetherian ring r a complex c of left r modules is gorenstein injective if and only if c m is gorenstein injective in r mod for all m z also gorenstein injective dimensions of complexes are considered

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

evaluating solutions on an elliptic problem in a gravitational gauge field theory abstract an elliptic equation arising from the study of static solutions with prescribed zeros and poles of the einstein equations coupled with the classical sigma model and an abelian gauge field is considered we classify the solutions and establish the uniqueness of radially symmetric solutions we also complete a classification of symmetric solutions of an elliptic equation on the sphere

moving planes moving spheres and a priori estimates abstract in this paper we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi linear elliptic equations using the method of moving planes we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of lin applying the method of moving spheres we give a different proof for a well known sup inf inequality established by brezis li and shafrir

multiplicity results for the assigned gauss curvature problem in r2 to study the problem of the assigned gauss curvature with conical singularities on riemanian manifolds we consider the liouville equation with a single dirac measure on the two dimensional sphere by a stereographic projection we reduce the problem to a liouville equation on the euclidean plane we prove new multiplicity results for bounded radial solutions which improve on earlier results of c s lin and his collaborators based on numerical computations we also present various conjectures on the number of unbounded solutions using symmetries some multiplicity results for non radial solutions are also stated

existence of topological multi string solutions in abelian gauge field theories in this paper we consider a general form of self dual equations arising from abelian gauge field theories coupled with the einstein equations by applying the super subsolution method we prove that topological multi string solutions exist for any coupling constant which improves previously known results we provide two examples for application the self dual einstein maxwell higgs model and the gravitational maxwell gauged o 3 sigma model

gorenstein projective dimension for complexes we define and study a notion of gorenstein projective dimension for complexes of left modules over associative rings for complexes of finite gorenstein projective dimension we define and study a tate cohomology theory tate cohomology groups have a natural transformation to classical ext groups in the case of module arguments we show that these maps fit into a long exact sequence where every third term is a relative cohomology group defined for left modules of finite gorenstein projective dimension

cohomology theories based on gorenstein injective modules in this paper we study relative and tate cohomology of modules of finite gorenstein injective dimension using these cohomology theories we present variations of grothendieck local cohomology modules namely gorenstein and tate local cohomology modules by applying a sort of avramov martsinkovsky exact sequence we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by j herzog we discuss some properties of these modules and give some results concerning their vanishing and non vanishing

on the structure of the conformal gaussian curvature equation on r2 ii in this paper we continue our investigation initiated in ecn on the structure of the set of all possible solutions of the equation 1 1 au ke2 o in r 2 where a is the laplace operator and k is a given nonpositive locally hslder continuous function in r 2 equation 1 1 arises in the problem of finding a riemannian metric which realizes the given function k as its gaussian curvature and is conformal to the standard euclidean metric in r 2 we refer the reader to cn for a brief description of the background and the history of this problem the study of the equation 1 1 dates back to at least 1930 s but it seems that a complete classification of all solutions of 1 1 was obtained only very recently in cn for the case k x ix t near oo where l 2 is a constant here we continue to use the notation g near to denote that there exist two positive constants cl c2 such that c l f g c2f near oo as we did in cn more precisely the following result was established in cn theorem suppose that k 2 then the following conclusions hold i for each 0 1 1 possesses a unique solution u such that

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

the conformal plate buckling equation the linear equation 2 u 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over r 2 has the particularly interesting nonlinear generalization g 2 u 1 where ag e 2u is the laplace beltrami operator for the metric g e 2u g 0 with g 0 the standard euclidean metric on r 2 this conformal elliptic pde of fourth order is equivalent to the nonlinear system of elliptic pdes of second order u x k g x exp 2u x 0 and k g x exp 2u x 0 with x r 2 describing a conformally flat surface with a gauss curvature function k g that is generated self consistently through the metric s conformal factor we study this conformal plate buckling equation under the hypotheses of finite integral curvature f kg exp 2u dx k finite area exp 2u dx a and the mild compactness condition k l 1 b 1 y uniformly with respect to y r 2 we show that asymptotically for x all solutions behave like u x 2 ln x c o 1 and k x 2 ln x c o 1 with 2 4 and 2 4 we also show that for each k 2 4 there exists a k and a radially symmetric solution pair u k satisfying k u k and max k k which is unique modulo translation of the origin and scaling of x coupled with a translation of u

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations we prove the existence and uniqueness of fast decay solutions and clarify the structure of positive radial solutions of the quasilinear elliptic equation mu f u 0 in rn where m denotes the m laplace operator and f has a supercritical growth for small u 0 and a subcritical growth for large u our proofs use only elementary arguments based on several variational identities and a maximum principle of peletier and serrin

the solution structure of the o 3 sigma model in a maxwell chern simons theory in this paper a system of semilinear elliptic equations arising from a relativistic self dual maxwell chern simons o 3 sigma model is considered we reveal the uniqueness aspect of the topological solutions for the model the uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case we locate each solution set denoted by a planar diagram