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

constructions in sasakian geometry we first generalize the join construction described previously by the first two authors 4 for quasi regular sasakian einstein orbifolds to the general quasi regular sasakian case this allows for the further construction of specific types of sasakian structures that are preserved under the join operation such as positive negative or null sasakian structures as well as sasakian einstein structures in particular we show that there are families of sasakian einstein structures on certain 7 manifolds homeomorphic to s 2 s 5 we next show how the join construction emerges as a special case of lerman s contact fibre bundle construction 32 in particular when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric sasakian manifold finally we study toric sasakian manifolds in dimension 5 and show that any simply connected compact oriented 5 manifold with vanishing torsion admits regular toric sasakian structures this is accomplished by explicitly constructing circle bundles over the equivariant blow ups of hirzebruch surfaces

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

energy properness and sasakian einstein metrics in this paper we show that the existence of sasakian einstein metrics is closely related to the properness of corresponding energy functionals under the condition of admitting no nontrivial hamiltonian holomorphic vector field we prove that the existence of sasakian einstein metric implies a moser trudinger type inequality at the end of this paper we also obtain a miyaoka yau type inequality in sasakian geometry

traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity this paper deals with the existence of traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity based on two different mixed quasi monotonicity reaction terms we propose new conditions on the reaction terms and new definitions of upper and lower solutions by using schauder s fixed point theorem and a new cross iteration scheme we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions the general results obtained have been applied to type k monotone and type k competitive diffusive lotka volterra systems

persistence in a periodic competitor competitor mutualist diffusion system in this paper we investigate the existence and the asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a competitor competitor mutualist model under dirichlet boundary conditions we shall prove that under certain conditions this system is persistent and under some other conditions it exhibits extinction or more exactly the species of mutualist extincts

on the metric structure of non kahler complex surfaces we give a characterization of a locally conformally kahler l c k metric with parallel lee form on a compact complex surface using the kodaira classification of surfaces we classify the compact complex surfaces admitting such structures this gives a classification of sasakian structures on compact three manifolds a weak version of the above mentioned characterization leads to an explicit construction of l c k metrics on all hopf surfaces we characterize the locally homogeneous l c k metrics on geometric complex surfaces and we prove that some inoue surfaces do not admit any l c k metric

homogeneous contact manifolds and resolutions of calabi yau cones in the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds the main feature of our approach is to describe the cartan ehresmann connection gauge field for principal u 1 bundles over complex flag manifolds by using elements of representation theory of simple lie algebras this description allows us to compute explicitly the expression of the contact form for any boothby wang fibration over complex flag manifolds boothby and wang in ann math 68 721 734 1958 as well as their underlying sasaki structures by following conlon and hein duke math j 162 2855 2902 2013 van coevering math ann 2009 https doi org 10 1007 s00208 009 0446 1 and goto j math soc jpn 64 1005 1052 2012 as an application of our results we use the cartan remmert reduction grauert in math ann 146 331 368 1962 and the calabi ansatz technique calabi in ann sci ecole norm sup 4 12 269 294 1979 to provide many explicit examples of crepant resolutions of calabi yau cones with certain homogeneous sasaki einstein manifolds realized as links of isolated singularities these concrete examples illustrate the existence part of the conjecture introduced in martelli and sparks phys rev d 79 6 065009 2009

toric geometry sasaki einstein manifolds and a new infinite class of ads cft duals recently an infinite family of explicit sasaki einstein metrics yp q on s2 s3 has been discovered where p and q are two coprime positive integers with q p these give rise to a corresponding family of calabi yau cones which moreover are toric aided by several recent results in toric geometry we show that these are kahler quotients open image in new window namely the vacua of gauged linear sigma models with charges p p p q p q thereby generalising the conifold which is p 1 q 0 we present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold open image in new window for all q p with fixed p we hence find that the yp q manifolds are ads cft dual to an infinite class of open image in new window superconformal field theories arising as ir fixed points of toric quiver gauge theories with gauge group su n 2p as a non trivial example we show that y2 1 is an explicit irregular sasaki einstein metric on the horizon of the complex cone over the first del pezzo surface the dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its ir fixed point using the ads cft correspondence the value we obtain is a quadratic irrational number and remarkably agrees with a recent purely field theoretic calculation using a maximisation

a momentum construction for circle invariant kahler metrics examples of kahler metrics of constant scalar curvature are relatively scarce over the past two decades several workers in geometry and physics have used symmetry reduction to construct complete kahler metrics of constant scalar curvature by ode methods one fruitful idea the calabi ansatz is to begin with an hermitian line bundle p l h m g m over a kahler manifold and to search for kahler forms p m dd c f t in some disk subbundle where t is the logarithm of the norm function and f is a function of one variable our main technical result theorem a is the calculation of the scalar curvature for an arbitrary kahler metric g arising from the calabi ansatz this suggests geometric hypotheses which we call a constancy to impose upon the base metric g m and hermitian structure h in order that the scalar curvature of g be specified by solving an ode we show that constancy is necessary and sufficient for the calabi ansatz to work in the following sense under the assumption of constancy the disk bundle admits a one parameter family of complete kahler metrics of constant scalar curvature that restrict to g m on the zero section theorems b and d an analogous result holds for the punctured disk bundle theorem c a simple criterion determines when such a metric is einstein conversely in the absence of constancy the calabi ansatz yields at most one metric of constant scalar curvature in either the disk bundle or the punctured disk bundle theorem e many of the metrics constructed here seem to be new including a complete negative einstein kahler metric on the disk subbundle of a stable vector bundle over a riemann surface of genus at least two and a complete scalar flat kahler metric on c 2

asymptotic behavior of solutions of a periodic diffusion system this paper deals with a periodic reaction diffusion system of plankton allelopathy under homogeneous neumann boundary conditions based on the result of ahmad and lazer we show some estimates and nonexistence results for the positive solutions of the system furthermore we investigate the asymptotic behavior of the solutions of the system that is one species dies out and the other exists as time t tends to infinity

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

kahler einstein metrics and volume minimization abstract we prove that if a polarized q fano variety admits a kahler einstein metric then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing up the vertex this is also generalized to the logarithmic and the orbifold settings a refinement of this result together with the result in 41 gives an equivalent characterization of k semistability for any smooth fano manifold we also prove that the valuation associated to the reeb vector field of a smooth sasaki einstein metric minimizes the normalized volume over the corresponding kahler cone these results strengthen and generalize the minimization result of martelli sparks yau

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

persistence in reacting diffusing systems interaction of two predators and one prey many phenomena in ecology biology biochemistry chemistry and physics can be mathematically formulated as reaction diffusion systems an important and interesting problem is the prediction of time evolution of the various density distributions and their relations to the corresponding stationary distributions the interaction of two predators competing for a single prey has occupied an important place in the ecological literature 2 in 5 we discussed the asymptotic behaviour of solutions for a reaction diffusion system modelling the interaction of two predators and one prey but the emphasis was on the coexistence of the three populations rather than on persistence in other words we analyzed the behaviour of the solutions of the reaction diffusion system in order to answer the following questions under which conditions would neither or one population of predators survive if one predator survives how to determine the limiting behaviour of the surviving predator and of the prey in the present paper we want to answer a different question namely as to when all of the components of the ecosystem survive and how to determine the limiting behaviour of both the surviving predators and of their prey as to persistence we shall use a definition mutuated from that currently used in ordinary differential systems 2 namely a population with density p t x is said to persist if 0 x 0 and lim inf p t x 0 for any x in the habitat a system is said to persist if each component

a reaction diffusion system of a competitor competitor mutualist model we investigate the homogeneous dirichlet problem and neumann problem to a reactiondiffusion system of a competitor competitor mutualist model the existence uniqueness and boundedness of the solutions are established by means of the comparison principle and the monotonicity method for the dirichlet problem we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities for the neumann problem we analyze the contant equilibrium solutions and their stabilities the main method used in studying of the stabilities is the spectral analysis to the linearized operators the o d e problem to the same model was proposed and studied by b rai h i freedman and j f addicott

hard lefschetz theorem for sasakian manifolds we prove that on a compact sasakian manifold m g of dimension 2n 1 for any 0 p n the wedge product with d defines an isomorphism between the spaces of harmonic forms n p m and n p 1 m therefore it induces an isomorphism between the de rham cohomology spaces h m and h m such isomorphism is proven to be independent of the choice of a compatible sasakian metric on a given contact manifold as a consequence an obstruction for a contact manifold to admit sasakian structures is found

kahler einstein metrics and the generalized futaki invariant in 1983 futaki introduced his famous invariant this invariant generalizes the obstruction of kazdan warner to prescribing gauss curvature on s 2 cf ful l the futaki invariant is defined for any compact k ihler manifold with positive first chern class that has nontrivial holomorphic vector fields it is a lie algebraic character from the lie algebra of holomorphic vector fields into and its vanishing is a necessary condition for the existence of a k ihler einstein metric on the underlying manifold therefore it can be used to test the existence of k ihlereinstein metrics on a given compact k ihler manifold with positive first chern class an excellent reference on the futaki invariant is futaki s book fu2 until now all known nontrivial obstructions to k ihler einstein metrics come from holomorphic vector fields this suggests the following conjecture

normal cr structures on compact 3 manifolds we study normal cr compact manifolds in dimension 3 for a choice of a cr reeb vector field we associate a sasakian metric on them and we classify those metrics as a consequence the underlying manifolds are topologically finite quotients of s 3 or of a non flat circle bundle over a riemann surface of positive genus in the latter case we prove that their cr automorphisms group is a finite extension of s 1 and we classify the normal cr structures on these manifolds

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

frankel conjecture and sasaki geometry abstract we classify simply connected compact sasaki manifolds of dimension 2 n 1 with positive transverse bisectional curvature in particular the kahler cone corresponding to such manifolds must be bi holomorphic to c n 1 0 as an application we recover the theorem of mori and siu yau on the frankel conjecture and extend it to certain orbifold version the main idea is to deform such sasaki manifolds to the standard round sphere in two steps both fixing the complex structure on the kahler cone first we deform the metric along the sasaki ricci flow and obtain a limit sasaki ricci soliton with positive transverse bisectional curvature then by varying the reeb vector field which essentially decreases the volume functional we deform the sasaki ricci soliton to a sasaki einstein metric with positive transverse bisectional curvature i e a round sphere the second deformation is only possible when one treats simultaneously regular and irregular sasaki manifolds even if the manifold one starts with is regular quasi regular i e kahler manifolds orbifolds

periodic solutions of parabolic systems with nonlinear boundary conditions this paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions the approach to the problem is by the method of upper and lower solutions and its associated monotone iterations this method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial boundary value problems a sufficient condition for the stability of a periodic solution is also given these results are applied to three model problems arising from chemical kinetics ecology and population biology

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

the existence of kahler einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group on utilise la methode de continuite pour demontrer l existence de metriques de kahler einstein pour des varietes de kahler de fibre linge positif anticanonique sous l hypothese additionnelle de l existence d un bon groupe de symetries fini ou compact

the geometric dual of a maximisation for toric sasaki einstein manifolds we show that the reeb vector and hence in particular the volume of a sasaki einstein metric on the base of a toric calabi yau cone of complex dimension n may be computed by minimising a function z on mathbb r n which depends only on the toric data that defines the singularity in this way one can extract certain geometric information for a toric sasaki einstein manifold without finding the metric explicitly for complex dimension n 3 the reeb vector and the volume correspond to the r symmetry and the a central charge of the ads cft dual superconformal field theory respectively we therefore interpret this extremal problem as the geometric dual of a maximisation we illustrate our results with some examples including the yp q singularities and the complex cone over the second del pezzo surface

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

a note of sasakian metrics with constant scalar curvature in this paper we consider compact sasakian manifolds with constant scalar curvature under some positive curvature assumption we prove that such sasakian metrics must be einstein metrics

sasakian manifolds with purely transversal bach tensor we show that a 2n 1 dimensional sasakian manifold m g with a purely transversal bach tensor has constant scalar curvature 2n 2n 1 equality holding if and only if m g is einstein for dimension 3 m is locally isometric to the unit sphere s3 for dimension 5 if in addition m g is complete then it has positive ricci curvature and is compact with finite fundamental group 1 m

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

asymptotically log fano varieties abstract motivated by the study of fano type varieties we define a new class of log pairs that we call asymptotically log fano varieties and strongly asymptotically log fano varieties we study their properties in dimension two under an additional assumption of log smoothness and give a complete classification of two dimensional strongly asymptotically log smooth log fano varieties based on this classification we formulate an asymptotic logarithmic version of calabi s conjecture for del pezzo surfaces for the existence of kahler einstein edge metrics in this regime we make some initial progress towards its proof by demonstrating some existence and non existence results among them a generalization of matsushima s result on the reductivity of the automorphism group of the pair and results on log canonical thresholds of pairs one by product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime of which a folklore conjecture concerning the case of a fano manifold with an anticanonical divisor is a special case

uniqueness and examples of compact toric sasaki einstein metrics in 11 it was proved that given a compact toric sasaki manifold with positive basic first chern class and trivial first chern class of the contact bundle one can find a deformed sasaki structure on which a sasaki einstein metric exists in the present paper we first prove the uniqueness of such einstein metrics on compact toric sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure and secondly remark that the result of 11 implies the existence of compatible einstein metrics on all compact sasaki manifolds obtained from the toric diagrams with any height or equivalently on all compact toric sasaki manifolds whose cones have flat canonical bundle we further show that there exists an infinite family of inequivalent toric sasaki einstein metrics on s 5 sharp k s 2 times s 3 for each positive integer k

the global attractor of a competitor competitor mutualist reaction diffusion system with time delays abstract the aim of this paper is to investigate the asymptotic behavior of time dependent solutions of a three species reaction diffusion system in a bounded domain under a neumann boundary condition the system governs the population densities of a competitor a competitor mutualist and a mutualist and time delays may appear in the reaction mechanism it is shown under a very simple condition on the reaction rates that the reaction diffusion system has a unique constant positive steady state solution and for any nontrivial nonnegative initial function the corresponding time dependent solution converges to the positive steady state solution an immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable moreover the state state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion all the conclusions for the time delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays

canonical sasakian metrics let m be a closed manifold of sasaki type a polarization of m is defined by a reeb vector field and for any such polarization we consider the set of all sasakian metrics compatible with it on this space we study the functional given by the square of the l2 norm of the scalar curvature we prove that its critical points or canonical representatives of the polarization are sasakian metrics that are transversally extremal we define a sasaki futaki invariant of the polarization and show that it obstructs the existence of constant scalar curvature representatives for a fixed cr structure of sasaki type we define the sasaki cone of structures compatible with this underlying cr structure and prove that the set of polarizations in it that admit a canonical representative is open we use our results to describe fully the case of the sphere with its standard cr structure showing that each element of its sasaki cone can be represented by a canonical metric we compute their sasaki futaki invariant and use it to describe the canonical metrics that have constant scalar curvature and to prove that only the standard polarization can be represented by a sasaki einstein metric

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

asymptotic behaviour of solutions of periodic competition diffusion system on considere le systeme d equations de reaction diffusion u t k 1 u a bu cv v t k 2 v v d eu fv ou k 1 k 2 a b f sont des fonctions suffisamment lisses definies sur un cylindre ou est un domaine borne lisse de r n

dynamics of lotka volterra competition reaction diffusion systems with degenerate diffusion abstract this paper is dealt with a class of lotka volterra competition reaction diffusion system and a competitor competitor mutualist system with density dependent diffusion in a bounded domain under the three basic types of dirichlet neumann and robin boundary conditions the governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor competitor mutualist system involves three degenerate equations the goal of the paper is to show the existence of positive steady state solutions or quasi solutions the existence and uniqueness of a classical global time dependent solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions or quasi solutions the above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species in the case of neumann boundary condition the system has a unique constant positive steady state solution which is a global attractor of all the competing species the above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction diffusion systems and the corresponding semilinear reaction diffusion systems in which some or all of the competing species may be in extinction

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

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

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

on eta einstein sasakian geometry a compact quasi regular sasakian manifold m is foliated by one dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact kahler orbifold in the case when the transverse space is also einstein the corresponding sasakian manifold m is said to be sasakian einstein in this article we study einstein geometry as a class of distinguished riemannian metrics on contact metric manifolds in particular we use a previous solution of the calabi problem in the context of sasakian geometry to prove the existence of einstein structures on many different compact manifolds including exotic spheres we also relate these results to the existence of einstein weyl and lorenzian sasakian einstein structures

examples of asymptotically conical ricci flat kahler manifolds previously the author has proved that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r these manifolds can be considered to be generalizations of the ricci flat ale kahler spaces known by the work of p kronheimer d joyce and others this article considers further the problem of constructing examples we show that every 3 dimensional gorenstein toric kahler cone admits a crepant resolution for which the above theorem applies this gives infinitely many examples of asymptotically conical ricci flat manifolds then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit ricci flat kahler cone metrics by the work of c boyer k galicki j kollar and others we concentrate on 3 dimensional examples two families of hypersurface examples are given which are distinguished by the condition b3 y 0 or b3 y 0

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

quasisolutions and dynamics of time periodic nonquasimonotone reaction diffusion systems abstract in this paper using the upper lower solution method we investigate the properties of periodic quasisolutions for time periodic nonquasimonotone reaction diffusion systems and present some existence asymptotic behavior results for two ecological models

hamiltonian constructions of kahler einstein metrics and kahler metrics of constant scalar curvature assuming the existence of a real torus acting through holomorphic isometries on a kahler manifold we construct an ansatz for kahler einstein metrics and an ansatz for kahler metrics with constant scalar curvature using this hamiltonian approach we solve the differential equations in special cases and find in particular a family of constant scalar curvature kahler metrics describing a non linear superposition of the bergman metric the calabi metric and a higher dimensional generalization of the lebrun kahler metric the superposition contains kahler einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spacepn we also build such kahler geometries on kahler quotients of higher cohomogeneity