about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

complex earthquakes and teichmuller theory it is known that any two points in teichmuller space are joined by an earthquake path in this paper we show any earthquake path r t s extends to a proper holomorphic mapping of a simplyconnected domain d into teichmuller space where r d c these complex earthquakes relate weil petersson geometry projective structures pleated surfaces and quasifuchsian groups using complex earthquakes we prove grafting is a homeomorphism for all 1 dimensional teichmuller spaces and we construct bending coordinates on bers slices and their generalizations in the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold

conway spheres as ideal points of the character variety we prove that any bonahon siebenmann family of conway spheres for a hyperbolic link is associated to an ideal point of the character variety of the link

hyperbolic cone manifolds short geodesics and schwarzian derivatives given a geometrically finite hyperbolic cone manifold with the cone singularity sufficiently short we construct a one parameter family of cone manifolds decreasing the cone angle to zero we also control the geometry of this one parameter family via the schwarzian derivative of the projective boundary and the length of closed geodesics

regenerating singular hyperbolic structures from sol let m be a torus bundle over s 1 with an orientation preserving anosov monodromy the manifold m admits a geometric structure modeled on sol we prove that the sol structure can be deformed into singular hyperbolic cone structures whose singular locus m is the mapping torus of the fixed point of the monodromy the hyperbolic cone metrics are parametred by the cone angle in the interval 0 2 when 2 the cone manifolds collapse to the basis of the fibration s 1 and they can be rescaled in the direction of the fibers to converge to the sol manifold

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

laplace transforms of airy functions analytic expressions involving gamma incomplete gamma and degenerate hypergeometric functions are obtained for the laplace transforms of airy functions which arise in the theory of electron tunnelling from solids

plissage des varietes hyperboliques de dimension 3 we give a characterization of the measured geodesic laminations which can occur as the bending measured lamination of some geometrically finite metric on a 3 manifold when the 3 manifold has incompressible boundary such a characterization has already been given by f bonahon and j p otal here we deal with the general case

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

collisions of particles in locally ads spacetimes ii moduli of globally hyperbolic spaces we investigate globally hyperbolic 3 dimensional ads manifolds containing particles i e cone singularities of angles less than 2 along a time like graph to each such space equipped with a time like vector field satisfying some additional properties we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities we show that this data is sufficient to recover the space locally i e in the neighborhood of a fixed metric this is a partial extension of a result of mess for non singular globally hyperbolic ads manifolds

hyperbolic manifolds with convex boundary let m m be a 3 manifold which carries a hyperbolic metric with convex boundary we consider the hyperbolic metrics on m such that the boundary is smooth and strictly convex we show that the induced metrics on the boundary are exactly the metrics with curvature k 1 and that the third fundamental forms of m are exactly the metrics with curvature k 2 each is obtained exactly once

fixed points of compositions of earthquakes let s be a closed surface of genus at least 2 and consider two measured geodesic laminations that fill s right earthquakes along these laminations are diffeomorphisms of the teichmuller space of s we prove that the composition of these earthquakes has a fixed point in the teichmuller space another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic ads manifold the proof uses some estimates from the geometry of those ads manifolds

laplace transform of a class of g functions the laplace transform of meijer s g function with an argument proportional to an integer power of the integration variable is obtained in closed form in particular this allows a straightforward evaluation of the laplace transform of the airy functions ai or x and bi x

local rigidity of hyperbolic 3 manifolds after dehn surgery it is well known that some lattices inso n 1 can be nontrivially deformed when included in so n 1 1 e g via bending on a totally geodesic hypersurface this contrasts with the super rigidity of higher rank lattices m kapovich recently gave the first examples of lattices in so 3 1 which are locally rigid inso 4 1 by considering closed hyperbolic3 manifolds obtained by dehn filling on hyperbolic two bridge knots we generalize this result to dehn filling on a more general class of one cusped finite volume hyperbolic 3 manifolds allowing us to produce the first examples of closed hyperbolic 3 manifolds which contain embedded quasi fuchsian surfaces but are locally rigid in so 4 1

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

projective structures with degenerate holonomy and the bers density conjecture we prove the bers density conjecture for singly degenerate kleinian surface groups without parabolics

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

on the laplace transform of the g function it is pointed out that the main result of a recent letter on the laplace transform of meijer s g function is actually a special case of an already established transform

on the relation between rates of relaxation and convergence of wild sums for solutions of the kac equation abstract in the case of maxwellian molecules the wild summation formula gives an expression for the solution of the spatially homogeneous boltzmann equation in terms of its initial data f as a sum f v t n 1 e t 1 e t n 1 q n f v here q n f is an average over n fold iterated wild convolutions of f if m denotes the maxwellian equilibrium corresponding to f then it is of interest to determine bounds on the rate at which q n f m l 1 r tends to zero in the case of the kac model we prove that for every e 0 if f has moments of every order and finite fisher information there is a constant c so that for all n q n f m l 1 r cn e where is the least negative eigenvalue for the linearized collision operator we show that is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f t to m a key role in the analysis is played by a decomposition of q n f into a smooth part and a small part this depends in an essential way on a probabilistic construction of mckean it allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

laplace transforms of airy functions via their integral definitions the laplace transforms of the functions ai or eta and bi eta are obtained using the integral definitions of these functions and a correction made to a recent derivation of the results of ai eta and bi eta

the grafting map of teichmuller space one of the underlying principles in the study of kleinian groups is that aspects of the complex projective geometry of quotients of by these groups reflect properties of the three dimensional hyperbolic geometry of the quotients ofh by these groups yet even though it has been over thirty five years since lipman bers exhibited a holomorphic embedding of the teichmuller space of riemann surfaces in terms of the projective geometry of a teichmuller space of quasi fuchsian manifolds no corresponding embedding in terms of the three dimensional hyperbolic geometry has been presented one of the goals of this paper is to give such an embedding this embedding is straightforward and has been expected for some time ta97 mc98 to each member of a bers slice of the space qf of quasi fuchsian 3manifolds we associate the bending measured lamination of the convex hull facing the fixed conformal end the geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting an operation on projective structures on surfaces that traces its roots back at least to klein kl33 p 230 50 with a modern history developed by many authors ma69 he75 fa83 st83 go87 gkm00 ta97 mc98 the main technical tool in our proof that bending measures give coordinates for bers slices and the second major goal of this paper is the completion of the proof of the grafting conjecture this conjecture states that for a fixed measured lamination the self map of teichmuller space induced by grafting a surface along is a homeomorphism of teichmuller space our contribution to this argument is a proof of the injectivity of the grafting map while the principal application of this result that we give is to geometric coordinates on the bers slice of qf one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of kleinian groups see 5 2 thus we feel that this result is of interest in its own right we now state our results and methods more precisely throughout s will denote a fixed differentiable surface which is closed orientable and of genus g 2 let tg be the corresponding teichmuller space of marked conformal structures on s and let pg denote the deformation space of complex projective structures on s see 2 for definitions

on the optimal choice of coefficients in a truncated wild sum and approximate solutions for the kac equation we study an approximate solution of the boltzmann problem for kac s caricature of a maxwellian gas by using a truncated and modified expansion of wild type we choose the coefficients in the wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes

limits of quasi fuchsian groups with small bending we study limits of quasi fuchsian groups for which the bending measures on the convex hull boundary tend to zero giving necessary and sufficient conditions for the limit group to exist and be fuchsian as an application we complete the proof of a conjecture made in 24 conjecture 6 5 that the closures of pleating varieties for quasi fuchsian groups meet fuchsian space exactly in kerckhoff s lines of minima of length functions doubling our examples gives rise to a large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach 2

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

quasi isometric classification of graph manifold groups we show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi isometric we also classify the quasi isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs a corollary is the quasi isometric classification of artin groups whose presentation graphs are trees in particular any two right angled artin groups whose presentation graphs are trees of diameter greater than 2 are quasi isometric further this quasi isometry class does not include any other right angled artin groups a finitely generated group can be considered geometrically when endowed with a word metric up to quasi isometric equivalence such metrics are unique in this article only finitely generated groups are considered given a collection of groups g gromov 11 proposed the fundamental questions of identifying which groups are quasi isometric to those in g rigidity and which groups in g are quasi isometric to each other classification in this article we focus on the classification question for graph manifold groups and right angled artin groups a compact 3 manifold m is called geometric if m m admits a geometric structure in the sense of thurston i e a complete locally homogeneous riemannian metric of finite volume thurston s geometrization conjecture see 29 17 21 23 22 provides that every irreducible 3 manifold of zero euler characteristic i e with boundary consisting only of tori and klein bottles admits a decomposition along tori and klein bottles into geometric pieces the minimal such decomposition being called the geometric decomposition there is a considerable literature on quasi isometric rigidity and classification of 3 manifold groups the rigidity results can be briefly summarized in the following form

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

volume et entropie minimale des espaces localement symetriques m gromov introduit la notion de volume minimal d une variete riemannienne min vol m inf vol m g courbure sectionnelle de g 1 les questions naturelles sont alors i quelles varietes m verifient min vol m 0 min vol m 0 ii dans le second cas existe t il des metriques realisant le minimum lesquelles en guise de reponse a la seconde question on peut au mieux esperer que si m supporte une metrique naturelle elle realisera le volume minimal ceci conduit a la conjecture suivante posee par m gromov si m est une variete admettant une metrique hyperbolique le volume fini notee hyp alors min vol m vol m hyp le but de l article est de donner une reponse partielle a cette conjecture

lipschitz volume rigidity in alexandrov geometry abstract we prove a lipschitz volume rigidity theorem in alexandrov geometry that is if a 1 lipschitz map f x x l y between alexandrov spaces preserves volume then it is a path isometry and an isometry when restricted to the interior of x we furthermore characterize the metric structure on y with respect to x when f is also onto this implies the converse of petrunin s gluing theorem if a gluing of two alexandrov spaces via a bijection between their boundaries produces an alexandrov space then the bijection must be an isometry

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

convergence of freely decomposable kleinian groups we consider a compact orientable hyperbolic 3 manifold with a compressible boundary suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination we prove that if this limit projective lamination is doubly incompressible then the sequence has compact closure in the deformation space as a consequence we generalise thurston s double limit theorem and solve his conjecture on convergence of function groups affirmatively

universal bounds for hyperbolic dehn surgery this paper gives a quantitative version of thurston s hyperbolic dehn surgery theorem applications include the first universal bounds on the number of nonhyperbolic dehn fillings on a cusped hyperbolic 3 manifold and estimates on the changes in volume and core geodesic length during hyperbolic dehn filling the proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

collisions of particles in locally ads spacetimes i local description and global examples we investigate 3 dimensional globally hyperbolic ads manifolds or more generally constant curvature lorentz manifolds containing particles i e cone singularities along a graph we impose physically relevant conditions on the cone singularities e g positivity of mass angle less than 2 on time like singular segments we construct examples of such manifolds describe the cone singularities that can arise and the way they can interact the local geometry near the vertices of we then adapt to this setting some notions like global hyperbolicity which are natural for lorentz manifolds and construct some examples of globally hyperbolic ads manifolds with interacting particles

cannon thurston maps for surface groups we prove the existence of cannon thurston maps for simply and doubly degenerate surface kleinian groups as a consequence we prove that connected limit sets of nitely generated kleinian groups are locally connected

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

quasi isometries preserve the geometric decomposition of haken manifolds we prove quasi isometry invariance of the canonical decomposition for fundamental groups of haken 3 manifolds with zero euler characteristic we show that groups quasi isometric to haken manifold groups with nontrivial canonical decomposition are finite extensions of haken orbifold groups as a by product we describe all 2 dimensional quasi flats in the universal covers of non geometric haken manifolds

convexity of the renormalized volume of hyperbolic 3 manifolds the hessian of the renormalized volume of geometrically finite hyperbolic 3 manifolds without rank 1 cusps computed at the hyperbolic metric g rm geod with totally geodesic boundary of the convex core is shown to be a strictly positive bilinear form on the tangent space to teichm uller space the metric g rm geod is known from results of bonahon and storm to be an absolute minimum for the volume of the convex core we deduce the strict convexity of the functional volume of the convex core at its minimum point

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

infinitesimal rigidity of cone manifolds and the stoker problem for hyperbolic and euclidean polyhedra the deformation theory of hyperbolic and euclidean cone manifolds with all cone angles less than 2 plays an important role in many problems in low dimensional topology and in the geometrization of 3 manifolds furthermore various old conjectures dating back to stoker about the moduli space of convex hyperbolic and euclidean polyhedra can be reduced to the study of deformations of cone manifolds by doubling a polyhedron across its faces this deformation theory has been understood by hodgson and kerckhoff 7 when the singular set has no vertices and by weis 32 when the cone angles are less than we prove here an infinitesimal rigidity result valid for cone angles less than 2 stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case and reduce to some simple deformations in the euclidean case the method is to treat this as a problem concerning the deformation theory of singular einstein metrics and to apply analytic methods about elliptic operators on stratified spaces this work is an important ingredient in the local deformation theory of cone manifolds by the second author 22 see also the concurrent work by weis 33

cusp shapes under cone deformation a horospherical torus about a cusp of a hyperbolic manifold inherits a euclidean similarity structure called a cusp shape we bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp the bounds are in terms of the change in structure in a neighborhood of the singular locus alone we then apply this result to provide information on the cusp shape of many hyperbolic knots given only a diagram of the knot specifically we show there is a universal constant c such that if a knot admits a prime twist reduced diagram with at least c crossings per twist region then the length of the second shortest curve on the cusp torus is bounded the bound is linear in the number of twist regions of the diagram

local rigidity of 3 dimensional cone manifolds we investigate the local deformation space of 3 dimensional conemanifold structures of constant curvature 1 0 1 and coneangles under this assumption on the cone angles the singular locus will be a trivalent graph in the hyperbolic and the spherical case our main result is a vanishing theorem for the first l 2 cohomology group of the smooth part of the cone manifold with coefficients in the flat bundle of infinitesimal isometries weconclude local rigidity from this in the euclidean case we prove that the first l 2 cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms

examples of relative deformation spaces that are not locally connected we provide an infinite family of pared manifolds with the property that the relative deformation spaces of hyperbolic structures on these manifolds are not locally connected this is a natural extension of the recent result of bromberg that shows the space of kleinian punctured torus groups is not locally connected