test fields on compact space times in this paper some basic aspects of lorentzian field theory on compact lorentz manifolds are studied all compact space times are acausal i e possess closed timelike curves this makes them a useful testbed in analyzing some new notions of causality that will be introduced for more general acausal space times in addition studying compact space times in their own right raises a wide range of fascinating mathematical problems some of which will be explored it will be shown that it is reasonable to expect lorentzian field theory on a compact space time to provide information on the topology of the underlying manifold if this is true then this information is likely to be orthogonal or complementary to the information obtained through the study of euclidean field theory

some subordination properties for spirallike functions the main object of the present note is to investigate several subordination properties for spirallike functions we also pose an open subordination problem associated with the familiar alexander integral operator

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

on closed anti de sitter spacetimes this article deals with the structure of the fundamental group of compact anti de sitter spacetimes i e lorentz manifolds with constant negative curvature algebraically such a manifold is the quotient of the universal cover of the homogeneous space so 2 n so 1 n by a discrete group gamma acting properly and co compactly on it this exists if and only if n is even indeed as this was observed by kulkarni u 1 d is contained in so 2 2d and acts properly transitively on so 2 2d so 1 2d it then suffices to take gamma as a co compact lattice in u 1 d the results of the present article give evidence to the question in dimension 3 are all compact anti de sitter spacetimes constructed in this way

the uniqueness of polynomial crystallographic actions let gamma be a polycyclic by finite group it is proved in 8 that gamma admits a polynomial action of bounded degree on mathbb r n which is properly discontinuous and such that the quotient gamma backslash mathbb r n is compact we prove here that such an action is unique up to conjugation by a polynomial transformation of mathbb r n

autour de la conjecture de l markus sur les varietes affines for any subgroupg of n we introduce some integer discg n called thediscompacity ofg this number measures to what extent the closure ofg is not compact the markus conjecture says that a compact affinely flat unimodular manifold is complete our main result called the discompact theorem is that this conjecture is true under the assumption that the linear holonomy i e the parallel transport has discompacity 1 because discso n 1 1 1 this ensures that a compact flat lorentz manifoldm is geodesically complete hence by a previous result of w goldman and y kamishima gk such am is up to finite covering a solvmanifold this achieves the proof of a bieberbach s theorem for compact lorentz flat manifolds

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

totally real submanifolds in s 6 1 satisfying chen s equality in this paper we study 3 dimensional totally real submanifolds of s6 1 if this submanifold is contained in some 5 dimensional totally geodesic s5 1 then we classify such submanifolds in terms of complex curves in cp 2 4 lifted via the hopf fibration s5 1 cp 2 4 we also show that such submanifolds always satisfy chen s equality i e m 2 where m p p inf k p for every p m then we consider 3 dimensional totally real submanifolds which are linearly full in s6 1 and which satisfy chen s equality we classify such submanifolds as tubes of radius 2 in the direction of the second normal space over an almost complex curve in s6 1

lorentzian affine hyperspheres with constant affine sectional curvature we study ane hyperspheres m with constant sectional curvature with respect to the ane metric h a conjecture by m magid and p ryan states that every such ane hypersphere with nonzero pick invariant is anely equivalent to either where the dimension n satises n 2 m 1o rn 2 m u p to now this conjecture was proved if m is positive denite or if m is a 3 dimensional lorentz space in this paper we give an armative answer to this conjecture for arbitrary dimensional lorentzian ane hyperspheres

topological tameness of margulis spacetimes we show that margulis spacetimes without parabolic holonomy elements are topologically tame a margulis spacetime is the quotient of the 3 dimensional minkowski space by a free proper isometric action of the free group of rank geq 2 we will use our particular point of view that the margulis spacetime is a manifold with boundary with an bbb r sf p 3 structure in an essential way the basic tools are a bordification by a closed bbb r sf p 2 surface with free holonomy group and the work of goldman labourie and margulis on geodesics in the margulis spacetimes and 3 manifold topology

locally strongly convex affine hypersurfaces with parallel cubic form we give a complete classification of locally strongly convex affine hypersurfaces ofr n 1 with parallel cubic form with respect to the levi civita connection of the affine berwald blaschke metric it turns out that all such affine hypersurfaces are quadrics or can be obtained by applying repeatedly the calabi product construction of hyperbolic affine hyperspheres using as building blocks either the hyperboloid or the standard immersion of one of the symmetric spaces sl m r so m sl m c su m su 2m sp m or e6 26 f4

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

archimedean superrigidity of solvables arithmetic groups abstract let g be a connected solvable linear algebraic group over a number fieldk letsbe a finite set of places ofkthat contains all the infinite places and let o s be the ring ofs integers ofk we define a certain closed subgroup formula of g s v s g kvthat contains g o s and prove that g o s is a superrigid lattice in formula by which we mean that finite dimensional representations g o s gln r more or less extend to representations of formula the subgroup formula may be a proper subgroup of g sfor only two reasons first it is well known that g o s is not a lattice in g sif g has nontrivialk characters so one passes to a certain subgroup g s 1 second g o s may fail to be zariski dense in g s 1 in an appropriate sense in this sense the subgroup formula is the zariski closure of g o s in g s 1 furthermore we note that a superrigidity theorem for many nonsolvables arithmetic groups can be proved by combining our main theorem with the margulis superrigidity theorem

on the full holonomy group of lorentzian manifolds the classification of restricted holonomy groups of n dimensional lorentzian manifolds was obtained about ten years ago however up to now not much is known about the structure of the full holonomy group in this paper we study the full holonomy group of lorentzian manifolds with a parallel null line bundle based on the classification of the restricted holonomy groups of such manifolds we prove several structure results about the full holonomy we establish a construction method for manifolds with disconnected holonomy starting from a riemannian manifold and a properly discontinuous group of isometries this leads to a variety of examples most of them being quotients of pp waves with disconnected holonomy including a non flat lorentzian manifold with infinitely generated holonomy group furthermore we classify the full holonomy groups of solvable lorentzian symmetric spaces and of lorentzian manifolds with a parallel null spinor finally we construct examples of globally hyperbolic manifolds with complete spacelike cauchy hypersurfaces disconnected full holonomy and a parallel spinor

an affine analogue of the hartman nirenberg cylinder theorem let x be a smooth complete connected submanifold of dimension n n in a complex affine space a n mathbb c and r is the rank of its gauss map gamma gamma x t x x the authors prove that if 2 leq r leq n 1 n n geq 2 and in the pencil of the second fundamental forms of x there are two forms defining a regular pencil all eigenvalues of which are distinct then the submanifold x is a cylinder with n r dimensional plane generators erected over a smooth complete connected submanifold y of rank r and dimension r this result is an affine analogue of the hartman nirenberg cylinder theorem proved for x subset r n 1 and r 1 for n geq 4 and r n 1 there exist complete connected submanifolds x subset a n mathbb c that are not cylinders

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

affine manifolds and orbits of algebraic groups this paper is the sequel to the radiance obstruction and parallel forms on a ffne manifolds lyans amer math soc 286 1984 629 649 which introduced a new family of secondary characteristic classes for affine structures on manifolds the present paper utilizes the representation of these classes in lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold mn to the action on rn of the algebraic hull a r of the affine holonomy group r c aff rn a main technical result of the paper is that if m has a nonzero cohomology class represented by a parallel k form then every orbit of a r has dimension k when m is compact then a r acts transitively provided that m is complete or has parallel volume the converse holds when r is nilpotent a 4 dimensional subgroup of aff r3 is exhibited which does not contain the holonomy group of any compact affine 3 manifold when m has solvable holonomy and is complete then m must have parallel volume conversely if m has parallel volume and is of the homotopy type of a solvmanifold then m is complete if m is a compact homogeneous affine manifold or if m possesses a rational riemannian metric then it is shown that the conditions of parallel volume and completeness are equivalent this paper is the sequel to our previous paper 22 in that paper we exploited certain characteristic classes exterior powers of the radiance obstruction to obtain relationships between various properties of affine manifolds our results supported the conjecture first made by l markus 36 that a compact affine manifold is complete if and only if it has parallel volume we shall refer to this as the main conjecture let m be a compact affine manifold with developing map dev m e and affine holonomy representation h 7r h 7r r c aff e here 1r is the group of deck transformations of the universal cover m e is the vector space rn and aff e is the group of affine automorphisms of e the map dev is a locally affine immersion which is equivariant respecting h the linear holonomy homb morphism is the composition a 1r gl e of h with the natural homomorphism l aff e gl e the obstruction to r fixing a point in e is a 1 dimensional cohomology class cm e h1 m ex with coefficients in e twisted by a it comes from a universal class in the group cohomology h1 aff e el which contains the crossed homomorphism u aff e e g g 0 received by the editors december 12 1984 and in revised form may 29 1985 1980 mathematics subject classification 1985 revision primary 57r99 53c05 secondary 53c10 55r25 1 research partially supported by fellowships and grants from the national science foundation a 1986 american mathematical society 0002 9947 86 1 00 25 per page

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

coefficient estimates for the inverses of a certain general class of spirallike functions in the present paper the authors derive several sharp coefficient estimates for the function class s a b 1 consisting of the inverses of functions in a certain class s a b of spirallike functions in the open unit disk u which was introduced by libera they also obtain a number of sharp coefficient bounds of functions in the more general classes s a b and s a b 1 each of which is introduced here some of the results derived in this paper would generalize those in a recent work of kapoor and mishra

fuchsian affine actions of surface groups let q be the irreducible representation of sl 2 r i nsl q r define a fuchsian subgroup of sl q r to be a subgroup conjugate to a discrete subgroup of q sl 2 r we prove in this paper that the fundamental group of a compact surface does not act properly on the affine space by affine tranformations if its linear part is fuchsian

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

on the isolation phenomena of locally conformally flat manifolds with constant scalar curvature submanifolds versions abstract in this paper from the viewpoint of submanifold theory we study the isolation phenomena of riemannian manifolds with constant scalar curvature and vanishing weyl conformal curvature tensor firstly for any locally strongly convex affine hyperspheres in an n 1 dimensional equiaffine space r n 1 with constant scalar curvature we prove an inequality involving the traceless ricci tensor the pick invariant and the scalar curvature the inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality secondly and analogously for lagrangian minimal submanifolds of the complex projective space c p n equipped with the fubini study metric under the condition that the weyl conformal curvature tensor vanishes we establish a similar but reverse inequality involving the traceless ricci tensor the scalar curvature and the squared norm of the second fundamental form the inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality

affine 3 spheres with constant affine curvature we classify the affine hyperspheres in r 4 which have constant curvature in the affine metric h and whose pick invariant is nonzero in particular the metric h must be flat

affine action and margulis invariant abstract in this paper we show that two zariski dense subgroups consisting of hyperbolic elements in so n 1 n r 2n 1 with the same marked margulis invariant are conjugate we also consider in affine deformations an analogue of quasifuchsian deformation of fuchsian groups

existence and regularity of the solutions of some singular monge ampere equations abstract in this paper we investigate the following singular monge ampere equation 0 1 det d 2 u 1 h u n k 2 u k in r n u 0 on where k 0 h 0 are constants and u x u x u x is the legendre transformation of u equation 0 1 is related to proper affine hyperspheres we will show the existence of solutions of 0 1 u c c via regularization method using the technique in 10 12 we also obtain the optimal graph regularity of the solution of 0 1

some characterizations of space forms integral conditions on the traceless ricci tensor are used to characterize euclidean and hyperbolic spaces among complete locally conformally flat manifolds of constant scalar curvature the main tools are vanishing type results for l p solutions of a large class of differential inequalities further applications of the technique are also given

locally strongly convex affine hyperspheres realizing chen s equality abstract in affine differential geometry of hypersurface c scharlach et al found an inequality involving intrinsic and extrinsic curvatures and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2 dimensional distribution d 2 is integrable in this paper we continue to study affine hyperspheres realizing the equality including parabolic affine hyperspheres as main results firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant or d 2 is integrable next by introducing a well defined 3 dimensional distribution d 3 when d 2 is not integrable we complete the classification of locally strongly convex affine hyperspheres realizing the equality if d 3 is integrable finally we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality

complete affine hypersurfaces part i the completeness of affine metrics on etudie les proprietes des hypersurfaces invariantes sous le groupe des transformations affines unimodulaires

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

on the uniqueness of some affinely flat infra nilmanifolds i p irn irk u p irn irk since pi irn irk pi irn ir k each element of i 1 ir irk is completely determined by k polynomials of total degree at most i in n variables let g be a group and suppose we have a morphism 4 g g l k ir and a homomorphism p g aff ir we can provide a g module structure for p ir f k in the following way for g e g and pepi ir irk we define

a certain general subclass of analytic and bi univalent functions and associated coefficient estimate problems abstract motivated essentially by the recent works of srivastava et al 10 frasin and aouf 6 xu et al 15 and other authors the authors of the present sequel investigate the coefficient estimate problems associated with an interesting general subclass b h p of analytic and bi univalent functions in the open unit disk u which is introduced here in particular for functions belonging to this general class b h p the problems involving the estimates on the first two taylor maclaurin coefficients a 2 and a 3 are investigated the results presented in this paper generalize and improve the aforecited recent works of frasin and aouf 6 and xu et al 15 see also a closely related earlier investigation by srivastava et al 10

sharp coefficient estimates for certain subclasses of starlike functions of complex order in this paper the authors derive several sharp coefficient estimates for the subclassess l c a b andk l c a b m m of the class of normalized starlike functions of complex order email protected email protected 0 the results presented here would generalize as well as improve the recent work by srivastava et al 2011 20 some closely related open problems are also posed

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

the auslander conjecture for nil affine crystallographic groups let n be a simply connected connected real nilpotent lie group of finite dimension n we study subgroups in aff n n aut n acting properly discontinuously and cocompactly on n this situation is a natural generalization of the so called affine crystallographic groups we prove that for all dimensions 1 n 5 the generalized auslander conjecture holds i e that such subgroups are virtually polycyclic

on the existence of closed timelike geodesics in compact spacetimes in 19 tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact cauchy surfaces necessarily contains a closed timelike geodesic the restriction to compact spacetimes with just regular globally hyperbolic coverings i e the cauchy surfaces are not required to be compact is still an open question here we shall answer this question negatively by providing examples of compact flat lorentz space forms without closed timelike geodesics and shall give some criterion for the existence of such geodesics more generally we will show that in a compact spacetime having a regular globally hyperbolic covering each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic whether or not a compact flat spacetime contains closed nonspacelike geodesics is as far as we know an open question we shall answer this question affirmatively we shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

isotropic affine hypersurfaces of dimension 5 abstract we study affine hypersurfaces m which have isotropic difference tensor note that any surface always has isotropic difference tensor therefore we may assume that n 2 such hypersurfaces have previously been studied by the first author and m djoric in 1 under the additional assumption that m is an affine hypersphere here we study the general case as for affine spheres we first show that isotropic affine hypersurfaces which are not congruent to quadrics are necessarily 5 8 14 or 26 dimensional from this we also obtain a complete classification in dimension 5

affine transformations and parallel lightlike vector fields on compact lorentzian 3 manifolds we describe the compact lorentzian 3 manifolds admitting a parallel lightlike vector field the classification of compact lorentzian 3 manifolds admitting non isometric affine diffeomorphisms follows together with the complete description of these morphisms such a lorentzian manifold is in some sense an equivariant deformation of a flat one