on the geometry of sasakian einstein 5 manifolds on simply connected five manifolds sasakian einstein metrics coincide with riemannian metrics admitting real killing spinors which are of great interest as models of near horizon geometry for three brane solutions in superstring theory 24 we expand on the recent work of demailly and kollar 14 and johnson and kollar 20 who give methods for constructing kahler einstein metrics on log del pezzo surfaces by a previous result of the first two authors 9 circle v bundles over log del pezzo surfaces with kahler einstein metrics have sasakian einstein metrics on the total space of the bundle here these simply connected 5 manifolds arise as links of isolated hypersurface singularities which by the well known work of smale 36 together with 11 must be diffeomorphic to s 5 l s 2 s 3 more precisely using methods from mori theory in algebraic geometry we prove the existence of 14 inequivalent sasakian einstein structures on s 2 s 3 and infinite families of such structures on l s 2 s 3 with 2 l 7 we also discuss the moduli problem for these sasakian einstein structures

bruhat poisson structure on cpn and integrable systems in this note we present a simple formula for the bruhat poisson structure on complex projective spaces in terms of the momentum coordinates we also give a simple description of a family of functions in involution on compact hermitian symmetric spaces obtained via the bi hamiltonian approach using the bruhat poisson structure and an invariant symplectic structure we compute these functions explicitly on cpn and relate them to the gelfand tsetlin coordinates we also show how the lenard scheme can be applied

six dimensional nearly kahler manifolds of cohomogeneity one ii let m be a six dimensional manifold endowed with a cohomogeneity one action of g su2 su2 and m rm reg subset m its subset of regular points we show that mreg admits a smooth 2 parameter family of g invariant non isometric strict nearly kahler structures and that a 1 parameter subfamily of such structures smoothly extends over a singular orbit of type s3 this determines a new class of examples of nearly kahler structures on t s3

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

eigenvalue boundary problems for the dirac operator on a compact riemannian spin manifold with mean convex boundary we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental dirac operator including the global aps condition and a riemannian version of the local mit bag condition we show that friedrich s inequality for the eigenvalues of the dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems more precisely we prove that for both the aps and the mit conditions the equality cannot be achieved and for the other two conditions the equality characterizes respectively half spheres and domains bounded by minimal hypersurfaces in manifolds carrying non trivial real killing spinors

a spinorial energy functional critical points and gradient flow let m be a compact spin manifold on the universal bundle of unit spinors we study a natural energy functional whose critical points if dim m ge 3 are precisely the pairs g varphi consisting of a ricci flat riemannian metric g together with a parallel g spinor varphi we investigate the basic properties of this functional and study its negative gradient flow the so called spinor flow in particular we prove short time existence and uniqueness for this flow

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

the cyclicity of period annuli of a class of quadratic reversible systems with two centers abstract quadratic perturbations of a class of quadratic reversible systems are studied the associated systems can have at most four limit cycles in any compact region of the period annuli and if four limit cycles exist then their distribution is 3 1 in other words three of them are nested and surrounding one focus the other limit cycle is surrounding the different focus

limit cycles bifurcated from a class of quadratic reversible center of genus one abstract we investigate the bifurcation of limit cycles in a class of planar quadratic reversible non hamiltonian systems this systems has a center of genus one the exact upper bound of the number of limit cycles is given

real ade equivariant co homotopy and super m branes a key open problem in m theory is the identification of the degrees of freedom that are expected to be hidden at ade singularities in spacetime comparison with the classification of d branes by k theory suggests that the answer must come from the right choice of generalized cohomology theory for m branes here we show that real equivariant cohomotopy on superspaces is a consistent such choice at least rationally after explaining this new approach we demonstrate how to use elmendorf s theorem in equivariant homotopy theory to reveal ade singularities as part of the data of equivariant s 4 valued super cocycles on 11d super spacetime we classify these super cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super cocycles on strata of intersecting black m brane species we find that on each singular stratum the black brane s instanton contribution namely its super nambu goto green schwarz action appears as the homotopy datum associated to the morphisms in the orbit category

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

extrinsic killing spinors under intrinsic and extrinsic curvature assumptions on a riemannian spin manifold and its boundary we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic killing spinors of non negative killing constant as a corollary we prove that a complete ricci flat spin manifold with mean convex boundary isometric to a round sphere is necessarily a flat disc

classification of the centers their cyclicity and isochronicity for the generalized quadratic polynomial differential systems in this paper we classify the centers the cyclicity of their hopf bifurcation and the isochronicity of the polynomial differential systems in r2 of degree d that in complex notation z x iy can be written as z i z zz d 22 az2 bzz cz 2 where d 2 is an arbitrary even positive integer r and a b c c note that if d 2 we obtain the well known class of quadratic polynomial differential systems which can have a center at the origin

a unified proof on the weak hilbert 16th problem for n 2 abstract the weak hilbert 16th problem for n 2 was solved by horozov and iliev proc london math soc 69 1994 198 244 zhang and li adv in math 26 1997 445 460 gavrilov invent math 143 2001 449 497 and li and zhang nonlinearity 15 2002 1775 1992 by using different methods for different cases the aim of this paper is to give a unified and easier proof for all cases the proof is restricted to the real domain combines geometric and analytical methods and uses deformation arguments

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

cones over pseudo riemannian manifolds and their holonomy by a classical theorem of gallot 1979 a riemannian cone over a complete riemannian manifold is either flat or has irreducible holonomy we consider metric cones with reducible holonomy over pseudo riemannian manifolds first we describe the local structure of the base of the cone when the holonomy of the cone is decomposable for instance we find that the holonomy algebra of the base is always the full pseudo orthogonal lie algebra one of the global results is that a cone over a compact and complete pseudo riemannian manifold is either flat or has indecomposable holonomy then we analyse the case when the cone has indecomposable but reducible holonomy which means that it admits a parallel isotropic distribution this analysis is carried out first in the case where the cone admits two complementary distributions and second for lorentzian cones we show that the first case occurs precisely when the local geometry of the base manifold is para sasakian and that of the cone

einstein matrices on s3 r3 and r4 bundles on construit a partir d un espace de base de kahler quaternionique 4n dimensionnel des metriques de cohomogeneite 1 en dimensions 4n 3 dont les surfaces de niveau sont l espace fibre s 2 des structures presque complexes sur la variete de base on obtient les conditions sur les fonctions metriques qui proviennent du fait qu on impose l equation d einstein et on obtient des solutions pour des espaces de dimension 4n 3 compactes et non compactes deux metriques 7 dimensionnelles ricci plates avec holonomie g 2 sont incluses dans les solutions non compactes

the cyclicity of the period annulus of the quadratic hamiltonian triangle abstract the paper studies quadratic hamiltonian centers surrounded by a separatrix contour having a form of a triangle it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to three

group theoretic preliminaries to the solution of the schrodinger equation for the manifold su 4 s u 2 u 2 irreducible representations irreps of a compact lie group g of class one w r t a lie subgroup h are those that contain the identity irrep of h once in their decompositions w r t h in the case of g su 4 and h s u 2 u 2 the class one irreps are identified and for them a general formula for their decomposition into irreps of h is given this admits a useful graphical presentation the relevance of these results to the solution of the schrodinger equation of su 4 s u 2 u 2 and the state labelling difficulties encountered in implementing this solution are discussed for g su n 1 and h s u n 1 u 2 n 4 the class one irreps of g and hence the spectrum of the corresponding schrodinger equation have also been determined

5d super yang mills on y p q sasaki einstein manifolds on any simply connected sasaki einstein five dimensional manifold one can construct a super yang mills theory which preserves at least two supersymmetries we study the special case of toric sasaki einstein manifolds known as y p q manifolds we use the localisation technique to compute the full perturbative part of the partition function the full equivariant result is expressed in terms of a certain special function which appears to be a curious generalisation of the triple sine function as an application of our general result we study the large n behaviour for the case of single hypermultiplet in adjoint representation and we derive the n 3 behaviour in this case

limit cycle bifurcations by perturbing a quadratic integrable system with a triangle abstract finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem which is related to the second part of hilbert s 16th problem this paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle it is proved that the cyclicity of the triangle in quadratic systems is at least three

half flat structures and special holonomy it was proven by hitchin that any solution of his evolution equations for a half flat su 3 structure on a compact six manifold m defines an extension of m to a seven manifold with holonomy in g2 we give a new proof which does not require the compactness of m more generally we prove that the evolution of any half flat g structure on a six manifold m defines an extension of m to a ricci flat seven manifold n for any real form g of sl 3 c if g is noncompact then the holonomy group of n is a subgroup of the noncompact form gof g c similar results are obtained for the extension of nearly half flat structures by nearly parallel g2 or g structures as well as for the extension of cocalibrated g2 and g structures by parallel spin 7 and spin0 3 4 structures respectively as an application we obtain that any six dimensional homogeneous manifold with an invariant half flat structure admits a canonical extension to a seven manifold with a parallel g2 or g structure for the group h3 h3 where h3 is the three dimensional heisenberg group we describe all left invariant half flat structures and develop a method to explicitly determine the resulting parallel g2 or g structure without integrating in particular we construct three eight parameter families of metrics with holonomy equal to g2 and g moreover we obtain a strong rigidity result for the metrics induced by a half flat structure on h3 h3 satisfying z z 0 where z denotes the centre finally we describe the special geometry of the space of stable three forms satisfying a reality condition considering all possible reality conditions we find four different special kahler manifolds and one special para kahler manifold

asymptotically euclidean manifolds and twistor spinors based on the berger simons holonomy classification we characterize all riemannian spin manifolds carrying a twistor spinor with at least one zero in particular the dimension n of the manifold is either even or n 7 outside the set of zeros of the twistor spinor the metric is conformal to either a flat metric or a ricci flat and locally irreducible metric

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

imaginary killing spinors in lorentzian geometry we study the geometric structure of lorentzian spin manifolds which admit imaginary killing spinors the discussion is based on the cone construction and a normal form classification of skew adjoint operators in signature 2 n 2 derived geometries include brinkmann spaces lorentzian einstein sasaki spaces and certain warped product structures exceptional cases with decomposable holonomy of the cone are possible

solution of the schrodinger equation of the complex manifold cpn passing from the cpn su n 1 u n lagrangian in kahler form to the hamiltonian in terms of polar coordinates this paper describes the solutions of the cpn schrodinger equation whose energy eigenspaces carry the irreducible representations 0n 2 0 1 2 of su n 1 in highest weight notation a full account is given of the u n structure of these eigenspaces and of how this emerges within our solution of the schrodinger equation by separation of variables explicit solutions of the radial equation which give rise to the derivation of spectrum and energy eigenspace details are presented for 1 2 and related to jacobi polynomials

conformal killing forms on riemannian manifolds conformal killing forms are a natural generalization of conformal vector fields on riemannian manifolds they are defined as sections in the kernel of a conformally invariant first order differential operator we show the existence of conformal killing forms on nearly kahler and weak g2 manifolds moreover we give a complete description of special conformal killing forms a further result is a sharp upper bound on the dimension of the space of conformal killing forms

spin c manifolds and complex contact structures in this paper we extend our notion of projectable spinors 9 ch 1 to the framework of spin c manifolds and deduce the basic formulas relating spinors on the base and the total space of riemannian submersions with totally geodesic one dimensional fibres some geometric applications concerning positive kahler einstein complex contact manifolds e g their characterisation as twistor spaces over positive quaternionic kahler manifolds are also given

new g _2 holonomy cones and exotic nearly kahler structures on s 6 and s 3 times s 3 there is a rich theory of so called strict nearly kaehler manifolds almost hermitian manifolds generalising the famous almost complex structure on the 6 sphere induced by octonionic multiplication nearly kaehler 6 manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional lie group g 2 the metric cone over a riemannian 6 manifold m has holonomy contained in g 2 if and only if m is a nearly kaehler 6 manifold a central problem in the field has been the absence of any complete inhomogeneous examples we prove the existence of the first complete inhomogeneous nearly kaehler 6 manifolds by proving the existence of at least one cohomogeneity one nearly kaehler structure on the 6 sphere and on the product of a pair of 3 spheres we conjecture that these are the only simply connected inhomogeneous cohomogeneity one nearly kaehler structures in six dimensions

on the abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree abstract in this study we determine the number of zeros for abelian integrals in four cases of quadratic reversible centers of genus one based on the results of li et al we prove that the upper bound of the associated number of zeros for abelian integrals with orbits formed by quartics quintics sextics and eight degree curves under polynomial perturbations of arbitrary degree n depends linearly on n

complete solution of the schrodinger equation of the complex manifold cp2 passing from the cp2 lagrangian in kahler form to the hamiltonian in terms of polar coordinates this paper gives the complete set of solutions of the corresponding schrodinger equation in a manner that makes fully explicit the su 3 description of the energy eigenspaces the solutions of the self adjoint equation for the radial coordinates are derived most easily directly but also related to jacobi polynomials

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

on the classification of elliptic foliations induced by real quadratic fields with center abstract related to the study of hilbert s infinitesimal problem is the problem of determining the existence and estimating the number of limit cycles of the linear perturbation of hamiltonian fields a classification of the elliptic foliations in the projective plane induced by the fields obtained by quadratic fields with center was already studied by several authors in this work we devise a unified proof of the classification of elliptic foliations induced by quadratic fields with center this technique involves using a formula due to cerveau lins neto to calculate the genus of the generic fiber of a first integral of foliations of these kinds furthermore we show that these foliations induce several examples of linear families of foliations which are not bimeromorphically equivalent to certain remarkable examples given by lins neto

kahlerian twistor spinors on a kahler spin manifold kahlerian twistor spinors are a natural analogue of twistor spinors on riemannian spin manifolds they are defined as sections in the kernel of a first order differential operator adapted to the kahler structure called kahlerian twistor penrose operator we study kahlerian twistor spinors and give a complete description of compact kahler manifolds of constant scalar curvature admitting such spinors as in the riemannian case the existence of kahlerian twistor spinors is related to the lower bound of the spectrum of the dirac operator

inhomogeneous fuchs equations and the limit cycles in a class of near integrable quadratic systems we study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension five components in the intersection of two strata in the centre manifold and q 4 the study of limit cycles for this degenerate case requires us to investigate not the first but the second variation m 2 of the displacement function we prove that up to three limit cycles can emerge from the period annulus surrounding the centre this implies that the cyclicity of period annuli of nearby systems in and q 4 is at most three as well our approach relies upon the possibility of deriving appropriate picard fuchs equations satisfied by the four independent integrals included in m 2

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

the cyclicity and period function of a class of quadratic reversible lotka volterra system of genus one abstract in this paper we investigate a class of quadratic reversible lotka volterra system of genus one z i z z 2 b z 2 with b 3 5 it is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two moreover we prove that the period function of its period trajectories is monotone increasing

quadratic perturbations of quadratic codimension four centers abstract we study the stratum in the set of all quadratic differential systems x p 2 x y y q 2 x y with a center known as the codimension four case q 4 it has a center and a node and a rational first integral the limit cycles under small quadratic perturbations in the system are determined by the zeros of the first poincare pontryagin melnikov integral i we show that the orbits of the unperturbed system are elliptic curves and i is a complete elliptic integral then using picard fuchs equations and the petrov s method based on the argument principle we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center

embedding into manifolds with torsion we introduce a class of special geometries associated to the choice of a differential graded algebra contained in lambda mathbb r n we generalize some known embedding results that effectively characterize the real analytic riemannian manifolds that can be realized as submanifolds of a riemannian manifold with special holonomy to this more general context in particular we consider the case of hypersurfaces inside nearly kahler and einstein sasaki manifolds proving that the corresponding evolution equations always admit a solution in the real analytic case

the twistor equation in lorentzian spin geometry in this paper we discuss the twistor equation in lorentzian spin geometry in particular we explain the local conformal structure of lorentzian manifolds which admit twistor spinors inducing lightlike dirac currents furthermore we derive all local geometries with singularity free twistor spinors that occur up to dimension 7

turbulence models for wall boundary layers boundary layer equations with boundary conditions corresponding to a wide range of plane two dimensional flows including the 14 mandatory test cases of the 1968 afosr ifd standford conference and those over the sub and supercritical airfoils of nakayama have been solved results were obtained from turbulence models based on an algebraic eddy viscosity formula the equations for turbulent kinetic energy and dissipation rate and equations for the reynolds stresses and dissipation rate they are shown to depend upon the model in an almost random manner the cost penalty associated with the higher order model is quantified and it is concluded that for the flows considered a well founded algebraic eddy viscosity model is preferred

limit cycles near a homoclinic loop by perturbing a class of integrable systems abstract in this paper the problem of limit cycle bifurcation is investigated by perturbing a class of integrable systems with a homoclinic loop under the assumption that the homoclinic loop passes through a degenerate singular point at the origin the asymptotic expansion of the melnikov function along the level curves of the first integral inside the homoclinic loop is studied near the loop meanwhile the formulas for the first coefficients in the expansion are given which can be used to study the number of limit cycles near the homoclinic loop finally an example is provided to demonstrate the obtained results

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

parallel spinors and holonomy groups in this paper we complete the classification of spin manifolds admitting parallel spinors in terms of the riemannian holonomy groups more precisely we show that on a given n dimensional riemannian manifold spin structures with parallel spinors are in one to one correspondence with lifts to spinn of the riemannian holonomy group with fixed points on the spin representation space in particular we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors

symmetry operators of killing spinors and superalgebras in ads5 we construct the first order symmetry operators of killing spinor equation in terms of odd killing yano forms by modifying the schouten nijenhuis bracket of killing yano forms we show that the symmetry operators of killing spinors close into an algebra in ads5 spacetime since the symmetry operator algebra of killing spinors corresponds to a jacobi identity in extended killing superalgebras we investigate the possible extensions of killing superalgebras to include higher degree killing yano forms we found that there is a superalgebra extension but no lie superalgebra extension of the killing superalgebra constructed out of killing spinors and odd killing yano forms in ads5 background

unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops abstract in this paper we present a complete study of quadratic 3 parameter unfoldings of some integrable system belonging to the classqr3 and having two centers and two unbounded heteroclinic loops we restrict to unfoldings that are transverse toqr3 obtain a versal bifurcation diagram and all global phase portraits including the precise number and configuration of the limit cycles it is proved that 3 is the maximal number of limit cycles surrounding a single focus and only the 1 1 configuration can occur in case of simultaneous nests of limit cycles essentially the proof relies on a careful analysis of a related non conservative abelian integral

integrability conditions for killing spinors the conditions for the existence of solutions ofd c are discussed in general it is not sufficient to consider only the first integrability condition d dv 2c2 v in particular the second integrability condition is needed to explain why in certain cases only for one choice of sign does a solution exist the killing spinor tensors as defined by walker and penrose are shown to be the spinorial equivalent of conformal killing tensors their relationship to the killing spinors and spinor vectors used in supergravity is given

centers with equal period functions abstract we give a sufficient condition for systems with symmetries to have periodic solutions with equal periods we show that the main result can be applied both to hamiltonian and to non hamiltonian systems we apply the main results to produce planar centers with equal period functions

spin c geometry of kahler manifolds and the hodge laplacian on minimal lagrangian submanifolds from the existence of parallel spinor fields on calabi yau hyper kahler or complex flat manifolds we deduce the existence of harmonic differential forms of different degrees on their minimal lagrangian submanifolds in particular when the submanifolds are compact we obtain sharp estimates on their betti numbers which generalize those obtained by smoczyk in 49 when the ambient manifold is kahler einstein with positive scalar curvature and especially if it is a complex contact manifold or the complex projective space we prove the existence of kahlerian killing spinor fields for some particular spin c structures using these fields we construct eigenforms for the hodge laplacian on certain minimal lagrangian submanifolds and give some estimates for their spectra these results also generalize some theorems by smoczyk in 50 finally applications on the morse index of minimal lagrangian submanifolds are obtained

the first eigenvalue of the dirac operator on quaternionic kahler manifolds in 15 we proved a lower bound for the spectrum of the dirac operator on quaternionic kahler manifolds in the present article we show that the only manifolds in the limit case i e the only manifolds where the lower bound is attained as an eigenvalue are the quaternionic projective spaces we use the equivalent formulation in terms of the quaternionic killing equation introduced in 16 and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

einstein metrics on s 3 r 3 and r 4 bundles starting from a 4n dimensional quaternionic kahler base space we construct metrics of cohomogeneity one in 4n 3 dimensions whose level surfaces are thes 2 bundle space of almost complex structures on the base manifold we derive the conditions on the metric functions that follow from imposing the einstein equation and obtain solutions both for compact and non compact 4n 3 dimensional spaces included in the non compact solutions are two ricci flat 7 dimensional metrics withg 2 holonomy we also discuss two other ricci flat solutions one on ther 4 bundle overs 3 and the other on anr 4 bundle overs 4 these haveg 2 and spin 7 holonomy respectively

einstein manifolds of dimension five with small first eigenvalue of the dirac operator soit m n une variete de spin compacte d einstein a courbure scalaire positive r 0 et on note par d t s t s l operateur de dirac agissant sur des sections du fibre spinoriel on etudie le probleme de la classification des varietes pour le cas a 5 dimensions

generalized cylinders in semi riemannian and spin geometry we use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory it has the advantage of being very simple and the result directly extends to semi riemannian manifolds and to embeddings into spaces of constant curvature we also give a new way to identify spinors for different metrics and to derive the variation formula for the dirac operator moreover we show that generalized killing spinors for codazzi tensors are restrictions of parallel spinors finally we study the space of lorentzian metrics and give a criterion when two lorentzian metrics on a manifold can be joined in a natural manner by a 1 parameter family of such metrics

on the accuracy of impulse methods for fluid flow a lagrangian numerical method based on impulse variables is analyzed the relation between discrete impulse vectors and vortex dipoles with a prescribed dipole moment is presented and used to adapt the high accuracy cutoff functions of vortex methods for use in impulse based methods the long time implementation of the impulse method is shown to contain a source of error whose growth in time leads to a severe loss of accuracy in interior flows applications for which impulse methods are well suited are identified

la premiere valeur propre de l operateur de dirac sur les varietes kahleriennes compactes k d kirchberg ki1 gave a lower bound for the first eigenvalue of the dirac operator on a spin compact kahler manifoldm of odd complex dimension with positive scalar curvature we prove that manifolds of real dimension 8l 6 satisfying the limiting case are twistor space cf sa of quaternionic kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l 2 satisfying the limiting case is the complex projective spacecp4l 1

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

contact spheres and hyperkahler geometry a taut contact sphere on a 3 manifold is a linear 2 sphere of contact forms all defining the same volume form in the present paper we completely determine the moduli of taut contact spheres on compact left quotients of su 2 the only closed manifolds admitting such structures we also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles

quadratic centers defining elliptic surfaces abstract let x be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one to each x we associate an elliptic surface a smooth complex compact surface which is a genus one fibration we give the list of all such vector fields and determine the corresponding elliptic surfaces

cohomogeneity one einstein sasaki 5 manifolds we consider hypersurfaces in einstein sasaki 5 manifolds which are tangent to the characteristic vector field we introduce evolution equations that can be used to reconstruct the 5 dimensional metric from such a hypersurface analogous to the nearly hypo and half flat evolution equations in higher dimensions we use these equations to classify einstein sasaki 5 manifolds of cohomogeneity one

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

on certain geometric aspects of cpn harmonic maps a weierstrass type system of equations corresponding to cpn harmonic maps is presented it constitutes a generalization of the previously constructed systems for cp1 and cp2 fields from the linear spectral problem for the cpn model a set of conserved quantities is derived and used for a construction of a generalized weierstrass representation for conformally parametrized surfaces immersed in multidimensional euclidean spaces based on this representation a possible geometrical interpretation of cpn harmonic maps is discussed

the cyclicity of quadratic reversible systems with a center of genus one and non morsean point in this paper we investigate a quadratic reversible system with non morsean point it is proved that the cyclicity of the period annulus under quadratic perturbations is less or equal to four

generalized killing spinors in dimension 5 we study the intrinsic geometry of hypersurfaces in calabi yau manifolds of real dimension 6 and more generally structures on 5 manifolds defined by a generalized killing spinor we prove that in the real analytic case such a 5 manifold can be isometrically embedded as a hypersurface in a calabi yau manifold in a natural way we classify nilmanifolds carrying invariant structures of this type and present examples of the associated metrics with holonomy

generic metrics and connections on spin and spin c manifolds we study the dependence of the dimension h0 g a of the kernel of the atyiah singer dirac operator cal d g a on a spinc manifold m on the metric g and the connection a the main result is that in the case of spin structures the value of h0 g for the generic metric is given by the absolute value of the index provided dim m in 3 4 in dimension 2 the mod 2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of riemann surfaces in the spinc case we also discuss upper bounds on h0 g a for generic metrics and we obtain a complete result in dimension 2 the much simpler dependence on the connection a and applications to seiberg witten theory are also discussed

ricci flat metrics harmonic forms and brane resolutions we discuss the geometry and topology of the complete non compact ricci flat stenzel metric on the tangent bundle of s n 1 we obtain explicit results for all the metrics and show how they can be obtained from first order equations derivable from a superpotential we then provide an explicit construction for the harmonic self dual p q forms in the middle dimension p q n 1 for the stenzel metrics in 2 n 1 dimensions only the p p forms are l 2 normalisable while for p q forms the degree of divergence grows with we also construct a set of ricci flat metrics whose level surfaces are u 1 bundles over a product of n einstein kahler manifolds and we construct examples of harmonic forms there as an application we construct new examples of deformed supersymmetric non singular m2 branes with such 8 dimensional transverse ricci flat spaces we show explicitly that the fractional d3 branes on the 6 dimensional stenzel metric found by klebanov and strassler is supported by a pure 2 1 form and thus it is supersymmetric while the example of pando zayas tseytlin is supported by a mixture of 1 2 and 2 1 forms we comment on the implications for the corresponding dual field theories of our resolved brane solutions