the topology of deformation spaces of kleinian groups let m be a compact hyperbolizable 3 manifold with nonempty incompressible boundary and let ah 1 m denote the space of conjugacy classes of discrete faithful representations of 1 m i nto psl 2 c the components of the interior mp 1 m of ah 1 m as a subset of the appropriate representation variety are enumerated by the spacea m of marked homeomorphism types of oriented compact irreducible 3 manifolds homotopy equivalent to m in this paper we give a topological enumeration of the components of the closure of mp 1 m and hence a conjectural topological enumeration of the components of ah 1 m we do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable kleinian groups we use this enumeration to exhibit manifolds m for which ah 1 m has inflnitely many components in this paper we begin a study of the global topology of deformation spaces of kleinian groups the basic object of study is the space ah 1 m of marked hyperbolic 3 manifolds homotopy equivalent to a flxed compact 3manifold m the interior mp 1 m of ah 1 m is very well understood due to work of ahlfors bers kra marden maskit sullivan and thurston in particular the components ofmp 1 m are enumerated by topological data namely the set a m of marked compact oriented irreducible 3 manifolds homotopy equivalent to m while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3 manifolds thurston s ending lamination conjecture provides a conjectural classiflcation for elements of ah 1 m by data which are partially topologi

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

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

on efficiency and duality for multiobjective variational control problems with f ρ convexity abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational control problems wolfe and mond weir type duals are formulated under the generalized f convexity on the functions involved weak and strong duality theorems are proved

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

uniform perfectness of the limit sets of kleinian groups in this note we show in a quantitative fashion that the limit set of a non elementary kleinian group is uniformly perfect if the quotient orbifold is of lehner type i e if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of hyperbolically bounded ones this result covers the known case when the group is analytically finite as applications we present estimates of the hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a kleinian group several examples will also be given

on efficiency and duality for multiobjective variational problems abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational problems wolfe and mond weir type duals are formulated under generalized f convexity assumptions on the functions involved weak and strong duality theorems are proved

tameness persists in weakly type preserving strong limits we show that if a weakly type preserving sequence of tame hyperbolic 3 manifolds converges strongly then the limit is tame as a first corollary we observe that we can replace the assumption of strong convergence with algebraic convergence in most cases as a second corollary we observe that given a finitely generated geometrically finite kleinian group tame groups are dense in the boundary of its quasiconformal deformation space in most cases

framed holomorphic bundles on rational surfaces we study the moduli space of framed holomorphic bundles of any rank over the blow up of cp at q points for c2 1 2 we introduce an open cover of the moduli space and describe its nerve in the limit when r we use this result to obtain the homotopy type of the moduli spaces in particular we compute the cohomology of the moduli spaces

dirac operators coupled to instantons on positive definite four manifolds we study the moduli space of instantons on a simply connected positive definite closed four manifold by analyzing the classifying map of the index bundle of a family of dirac operators parameterized by the moduli space as an application we compute the cohomology rings for the charge 1 and 2 moduli spaces in the rank stable limit

multiobjective control problem with generalized invexity abstract a multiobjective control problem is introduced the definition of invexity for continuous functions is extended to invexity pseudoinvexity and quasi invexity duality results are established for wolfe as well as mond wier type duals

a conformally invariant metric on riemann surfaces associated with integrable holomorphic quadratic differentials in this paper we define a conformally invariant pseudo metric on all riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it this metric is closely related to an extremal problem on the surface as a result we have a kind of reproducing formula for integrable quadratic differentials furthermore we establish a new characterization of uniform thickness of hyperbolic riemann surfaces in terms of invariant metrics

volumes of hyperbolic haken manifolds ii in 14 a program was initiated for using the topological theory of 3 manifolds to obtain lower bounds for volumes of hyperbolic 3 manifolds in 1 by a combination of new geometric ideas with relatively standard but specifically 3 dimensional topological techniques we showed that every closed orientable hyperbolic 3 manifold whose first betti number is at least 3 has volume exceeding 0 92 by contrast the best known lower bound 10 5 for the volume of an arbitrary closed hyperbolic 3 manifold is approximately 0 0012 in 3 we showed that every closed orientable hyperbolic 3 manifold whose first betti number is 2 has volume exceeding 0 34 the proof depended on supplementing the results and techniques of 1 with ingenious elementary arguments due to zagier 11 and numerical computations in the present paper we shall show that if one excludes certain special manifolds such as fiber bundles over s 1 then the lower bound of 0 34 also holds for hyperbolic 3 manifolds with betti number 1 the proof depends heavily on the results of 1 and 3 but it involves much deeper topological ideas than these papers the new topological results needed for the proof occupy most of the present paper to some extent these results have the flavor of general topology but the proofs make use of such specifically low dimensional techniques as the characteristic submanifold theory 9 8 the interaction between trees and incompressible surfaces and scott s theorem 12 that surface groups are locally extended residually finite before giving a precise statement of our main result we must review a few elementary notions from 3 manifold theory

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

quasiconformal homeomorphisms and the convex hull boundary we investigate the relationship between an open simply connected region s 2 and the boundary y of the hyperbolic convex hull in h 3 of s 2 a counterexample is given to thurston s conjecture that these spaces are related by a 2 quasiconformal homeomorphism which extends to the identity map on their common boundary in the case when the homeomorphism is required to respect any group of mobius transformations which preserves we show that the best possible universal lipschitz constant for the nearest point retraction r y is 2 we find explicit universal constants 0 c 2 c 1 such that no pleating map which bends more than c 1 in some interval of unit length is an embedding and such that any pleating map which bends less than c 2 in each interval of unit length is embedded we show that every k quasiconformal homeomorphism d 2 d 2 is a k k quasi isometry where k is an explicitly computed function the multiplicative constant is best possible and the additive constant k is best possible for some values of k

description of all conformally invariant differential operators acting on scalar functions we present an algorithm to obtain all conformally invariant differential operators acting on scalar functions and taking values on scalar functions as well there are two cases under consideration operators acting on functions defined on the euclidean space r4 and on the minkowski space in both cases each conformally invariant operator of order k is nonlinear and is a function of a finite set of functionally independent invariant operators of order up to k in particular the second order independent operators are only three and we give their explicit realization the applied technique is based on the jet bundle formalism analysis of the group action and a dimensional reduction as a simple illustration of this method we consider the case of differential operators between analytic functions on c which are invariant under the action of the modular group a power series which generates all functionally independent invariant operators is derived

duality and sufficiency in control problems with invexity recently mond chandra and husain 3 established duality results for a variational problem under more general convexity conditions called invex than mond and hanson 4 in a similar fashion the results of mond and hanson s for duality in control problems are extended to more general convex or invex functions it is also shown that for invex functions the necessary conditions for optimality in the control problem are also sufficient

hausdorff dimension and dendritic limit sets let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3 space h3 such that h3 has positive injectivity radius it is known that the limit set is a dendrite of hausdorff dimension 2 we show that the cut point set of the limit set has hausdorff dimension strictly less than 2

cusps are dense we show cusps are dense in bers boundary for teichmuller space the proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic riemann surface

approximation by maximal cusps in boundaries of deformation spaces of kleinian groups let m be a compact oriented irreducible atoroidal 3 manifold with nonempty boundary let cc0 m denote the space of convex cocompact kleinian groups uniformizing m we show that any kleinian group in the boundary of cc0 m whose limit set is the whole sphere can be approximated by maximal cusps density of maximal cusps on the boundary of schottky space is derived as a corollary we further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3 manifold with connected conformal boundary

ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3 manifolds we show that geometrically finite kleinian groups are dense in the boundary of the quasiconformal deformation space of any geometrically finite kleinian group

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

optimality and duality for multiple objective optimization under generalized type i univexity abstract in this paper we extend the classes of generalized type i vector valued functions introduced by aghezzaf and hachimi in j global optim 18 2000 91 101 to generalized univex type i vector valued functions and consider a multiple objective optimization problem involving generalized type i univex functions a number of kuhn tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution the mond weir and general mond weir type duality results are also presented

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

connectivity of the space of ending laminations we prove that for any closed surface of genus at least four and any punctured surface of genus at least two the space of ending laminations is connected a theorem of e klarreich 28 theorem 1 3 implies that this space is homeomorphic to the gromov boundary of the complex of curves it follows that the boundary of the complex of curves is connected in these cases answering the conjecture of p storm other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate kleinian groups

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

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

on ahlfors finiteness theorem abstract ahlfors finiteness theorem and its two complements namely the area inequalities of bers and the finiteness of the cusps due to sullivan are some of the central results in the modern theory of kleinian groups their proofs are analytic whereas their conclusions have a geometric flavor in this paper we have attempted to explain the topological and group theoretic genesis of these theorems our approach is based on a relative version of the theorem a finitely generated 3 manifold groups is finitely presented due to scott and shalen

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

the conformal boundary and the boundary of the convex core in this note we investigate the relationship between the conformal boundary at infinity of a hyperbolic 3 manifold and the boundary of its convex core in particular we prove that the length of a curve in the conformal boundary gives an upper bound on the length of the corresponding curve in the boundary of the convex core a hyperbolic 3 manifold n is the quotient of hyperbolic 3 space h by a group of isometries the boundary at infinity of h may be identified with the riemann sphere and extends to act on as a group of conformal automorphisms the domain of discontinuity of is the largest invariant open subset of which acts on properly discontinuously if is not abelian then inherits a hyperbolic metric called the poincare metric which acts on as a group of isometries one may then consider cn to be the conformal boundary at infinity of the hyperbolic 3 manifold n the conformal boundary is the topological boundary of n h the first important result concerning the relationship between the geometry of the conformal boundary and the geometry of n is due to bers 4 who proved that if each component of is simply connected and is a closed curve in its conformal boundary cn then the geodesic in n in the homotopy class of has length ln at most twice the length l cn of in the conformal boundary if is homotopic to arbitrarily short closed curves in n then we say ln 0 canary 8 generalized this to the setting of arbitrary kleinian groups proving that given any 0 there exists k 0 such that if has injectivity radius bounded below by at each point then if is a closed curve in c n then ln kl cn

mixed type duality in multiobjective programming problems two mixed type duals are introduced for multiobjective programming and for multiobjective fractional programming respectively results about weak duality strong duality and strictly reverse duality are presented under more suitable conditions

duality for a class of multiobjective control problems with generalized invexity abstract b mond and i smart j math anal appl 136 1988 325 333 defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity d bhatia and p kumar j math anal appl 189 1995 676 692 defined another kind of invexity corresponding generalized invexity and discussed the duality for multiobjective control problems with such generalized invexity in this paper the duality results for multiobjective control problems with mond and smart s generalized invexity are discussed

a 4 dimensional kleinian group we give an example of a 4 dimensional kleinian group which is finitely generated but not finitely presented and is a subgroup of a cocompact kleinian group

the gauss bonnet theorem for cone manifolds and volumes of moduli spaces this paper generalizes the gauss bonnet formula to a class of stratied spaces called riemannian cone manifolds as an application we compute the volumes of the moduli spacesm0 n with respect to the complex hyperbolic metrics introduced by picard deligne mostow and thurston

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

weakly efficient solutions and pseudoinvexity in multiobjective control problems abstract in this paper we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem called w kt pseudoinvexity and w fj pseudoinvexity we prove that all kuhn tucker or fritz john points are weakly efficient solutions if and only if these conditions are fulfilled we relate weakly efficient solutions to optimal solutions of weighting control problems we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

efficient solutions in v kt pseudoinvex multiobjective control problems a characterization in this paper we introduce a new condition on functionals involved in a multiobjective control problem for which we define the v kt pseudoinvex control problem we prove that a v kt pseudoinvex control problem is characterized so that a kuhn tucker point is an efficient solution we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

average bending of convex pleated planes in hyperbolic three space if p is a pleated plane in 3 dimensional hyperbolic space h 3 and a geodesic in its intrinsic metric we define b p the average bending of p in the direction we show that if p is a convex pleated plane embedded in h 3 then b p k for some universal k furthermore if p is a boundary component of the convex hull of a quasi fuchsian group then b p b almost everywhere where b is a constant times the length of the bending lamination of the pleated surface x p we use these to prove a number of results about quasi fuchsian groups including a universal bound on the lipschitz constant for the map to infinity and a bound on the length of by a constant times the euler characteristic of x

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

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

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

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

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

an explicit bound for uniform perfectness of the julia sets of rational maps a compact set c in the riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate c julia sets of rational maps of degree at least two are uniformly perfect proofs have been given independently by ma ne and da rocha and by hinkkanen but no explicit bounds are given in this note we shall provide such an explicit bound and as a result give another proof of uniform perfectness of julia sets of rational maps of degree at least two as an application we provide a lower estimate of the hausdorff dimension of the julia sets we also give a concrete bound for the family of quadratic polynomials f c z z 2 c in terms of the parameter c

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

generalized invexity and duality in multiple objective variational problems multiple objective programming problems with the concept of weak minima are extended to multiple objective variational problems a number of weak strong and converse duality theorems are given under a variety of generalized invexity conditions

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

duality for multiobjective variational problems with invexity abstract in this paper wolfe and mond weir type duals for a class of nondifferentiable multiobjective variational problems are formulated under invexity assumptions on the objective and the constraint functions involved weak and strong duality theorems are proved to related properly efficient solutions for primal and dual problems later related problems with these variational problems are introduced

duality for multiobjective fractional variational problems abstract a class of multiobjective fractional variational problems is considered and duals are formulated under concavity assumptions on the functions involved duality theorems are proved through a parametric approach to relate efficient solutions of the primal and dual problems we generalize those results for control problems also