gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

cone metrics on the sphere and livne s lattices we give an explicit construction of a family of lattices in pu 1 2 originally constructed by livne following thurston we construct these lattices as the modular group of certain euclidean cone metrics on the sphere we give connections between these groups and other groups of complex hyperbolic isometries

the bms group and generalized gravitational instantons the ordinary bondi metzner sachs bms group b is the best candidate for the fundamental symmetry group of general relativity it has been shown that b admits generalizations to real space times of any signature and also to complex space times it has been suggested that certain continuous unitary irreducible representations irs of b and of its generalizations correspond to gravitational instantons here i make this correspondence more precise and i take this suggestion one step further by arguing that a subclass of irs of b and of its generalizations correspond to generalized gravitational instantons some of these generalized gravitational instantons involve in their definition certain subgroups of the cartesian product group cn cm where cr is the cyclic group of order r with this motivation i give the subgroups of cn cm explicitly

compact riemannian 7 manifolds with holonomy g sb 2 ii this is the second of two papers about metrics of holonomy g2 on compact 7manifolds in our first paper 15 we established the existence of a family of metrics of holonomy g2 on a single compact simply connected 7 manifold m using three general results theorems a b and c our purpose in this paper is to explore the theory of compact riemannian 7 manifolds with holonomy g2 in greater detail by relying on theorems a c we will be able to avoid the emphasis on analysis that characterized 15 so that this sequel will have a more topological flavour the paper has four chapters the first chapter consists of introductory material section 1 1 gives some elementary geometric and topological material on compact 7 manifolds with torsion free g2structures then 1 2 describes the holonomy groups su 2 and su 3 and 1 3 explains the concept of asymptotically locally euclidean riemannian manifolds shortened to ale spaces with special holonomy recall that in 15 a compact 7 manifold m was defined by desingularizing a quotient t of the 7 torus by a finite group of isometries z2 the subject of chapters 2 and 3 is a generalization of this idea chapter 2 defines a general construction for compact 7 manifolds with torsion free g2structures which works by desingularizing quotients t for finite groups the ale spaces with holonomy su 2 and su 3 discussed in 1 3 are an essential ingredient in performing this desingularization the central result of chapter 2 is theorem 2 2 3 which states that given a suitable finite group and certain other data one may construct a compact 7manifold m from t that admits torsion free g2structures this result is proved using theorems a c of 15 chapter 3 is devoted entirely to examples of this construction we give many examples of compact 7 manifolds with holonomy g2 and determine their basic topological invariants the betti numbers and fundamental group finally in chapter 4 we discuss some areas of interest and give a number of open problems this paper is not written to be read independently of 15 the language and results of 15 will be used freely in particular the introductory material in 15 1 1 for reference we reproduce here the model 3and 4forms defining the flat g2structure on r as given in 15 1 1

a torelli type theorem for gravitational instantons on montre que chaque 4 variete hyper kahler asymptotiquement localement euclidienne et isometrique a un membre d une des familles obtenues dans un article precedent

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

new non arithmetic complex hyperbolic lattices we produce a family of new non arithmetic lattices in mathrm pu 2 1 all previously known examples were commensurable with lattices constructed by picard mostow and deligne mostow and fell into nine commensurability classes our groups produce five new distinct commensurability classes most of the techniques are completely general and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

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

real hyperbolic on the outside complex hyperbolic on the inside the rank one symmetric spaces of negative curvature come in three infinite families real hyperbolic space h complex hyperbolic space ch and quaternionic hyperbolic space qh the cayley plane is the remaining example aside from the obvious embeddings h ch qh the three geometries seem fairly unrelated to each other for instance h admits non arithmetic lattices in all dimensions grp while qh only admits arithmetic lattices grs see c for a related result the question of non arithmetic lattices in ch is a basic unsolved problem dm for a representation theoretic comparison of discrete subgroups in the different rank one spaces see sh in this paper we make a new connection between h and ch we construct a closed hyperbolic 3 manifold which as a diffeomorphic copy is the ideal boundary of a complex hyperbolic 4 manifold up to index 2 the isometry group of ch is pu 2 1 the group of complex projective automorphisms of the unit ball in c the ideal boundary of ch is the unit 3 sphere s a spherical cr structure on a 3 manifold is a system of coordinate charts into s whose transition functions are restrictions of elements of pu 2 1 while plenty of closed seifert fibered manifolds admit spherical cr structures kt our example gives the only known spherical cr structure on a closed hyperbolic 3 manifold

stability analysis of networked control systems a sum of squares approach this paper presents a sum of squares sos approach to the stability analysis of networked control systems ncss incorporating bounded time varying delays bounded time varying transmission intervals and a shared communication medium a shared communication medium imposes that per transmission only one node which consists of several actuators or sensors can access the network and transmit its corresponding data which node obtains access is determined by a network protocol we will provide mathematical models that describe these ncss and transform them into suitable hybrid systems formulations based on these hybrid systems formulations we construct lyapunov functions using sos techniques that can be solved using lmi based computations this leads to several beneficial features i we can deal with plants and controllers which are described by nonlinear piecewise polynomial differential equations ii we can allow for non zero lower bounds on the delays and transmission intervals in contrast with various existing approaches iii we allow more flexibility in the lyapunov functions thereby obtaining less conservative estimates of the maximal allowable transmission intervals mati and maximal allowable delay mad and finally iv it provides an automated method to address stability analysis problems in ncs several numerical examples illustrate the strengths of our approach

complex hyperbolic 3 n triangle groups abstract let n 5 be a positive integer a complex hyperbolic 3 n triangle group is a representation from the hyperbolic 3 n triangle group into the holomorphic isometry group of complex hyperbolic plane which maps the generators to complex reflections fixing complex lines in this paper we show that a complex hyperbolic 3 n triangle group i 1 i 2 i 3 is discrete and faithful if and only if i 1 i 3 i 2 i 3 is not elliptic our result answers a conjecture of schwartz for complex hyperbolic 3 n triangle groups

ideal triangle groups dented tori and numerical analysis we prove the goldman parker conjecture a complex hyperbolic ideal triangle group is discretely embedded in pu 2 1 if and only if the product of its three standard generators is not elliptic we also prove that such a group is indiscrete if the product of its three standard generators is elliptic a novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates

construction of the irreducibles of b 2 2 the ordinary bondi metzner sachs bms group b is the common asymptotic symmetry group of all radiating asymptotically flat lorentzian spacetimes as such b is the best candidate for the universal symmetry group of general relativity however in studying quantum gravity spacetimes with signatures other than the usual lorentzian one and complex spacetimes are frequently considered generalizations of b appropriate to these other signatures have been defined earlier in particular the generalization b 2 2 appropriate to the ultrahyperbolic signature has been described in detail and the study of its irreducible unitary representations irs of b 2 2 has been initiated the infinite little groups have been given explicitly but the finite little groups have only been partially described this study is completed by describing in detail the finite little groups and by giving all the necessary information in order to construct the irs of b 2 2 in all cases

gravitational instantons of type d k and a generalization of the gibbons hawking ansatz we describe a quaternionic based ansatz generalizing the gibbons hawking ansatz to a class of hyperkahler metrics with hidden symmetries we then apply it to obtain explicit expressions for gravitational instanton metrics of type d k

rigidity for multi taub nut metrics this paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity it proves that a complete hyperkahler manifold asymptotic to a circle fibration over the euclidean three space is either the standard r 3 s 1 or a multi taub nut manifold in par ticular the underlying complex manifold is either c c z or a minimal resolution of a cyclic kleinian singularity