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

on energy functionals kahler einstein metrics and the moser trudinger onofri neighborhood abstract we prove that the existence of a kahler einstein metric on a fano manifold is equivalent to the properness of the energy functionals defined by bando chen ding mabuchi and tian on the set of kahler metrics with positive ricci curvature we also prove that these energy functionals are bounded from below on this set if and only if one of them is this answers two questions raised by x x chen as an application we obtain a new proof of the classical moser trudinger onofri inequality on the two sphere as well as describe a canonical enlargement of the space of kahler potentials on which this inequality holds on higher dimensional fano kahler einstein manifolds

blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

on the construction of nadel multiplier ideal sheaves and the limiting behavior of the ricci flow in this note we construct nadel multiplier ideal sheaves using the ricci flow on fano manifolds this extends a result of phong sesum and sturm these sheaves like their counterparts constructed by nadel for the continuity method can be used to obtain an existence criterion for kahler einstein metrics

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data a semidiscrete mixed finite element approximation to parabolic initial boundary value problems is introduced and analyzed superconvergence estimates for both pressure and velocity are obtained the estimates for the errors in pressure and velocity depend on the smoothness of the initial data including the limiting cases of data in l 2 and data in h r for r sufficiently large because of the smoothing properties of the parabolic operator these estimates for large time levels essentially coincide with the estimates obtained earlier for smooth solutions however for small time intervals we obtain the correct convergence orders for nonsmooth data

hamiltonian surface charges using external sources in this work we interpret part of the boundary conditions as external sources in order to partially solve the integrability problem present in the computation of surface charges associated to gauge symmetries in the hamiltonian formalism we start by describing the hamiltonian structure of external symmetries preserving the action up to a transformation of the external sources of the theory we then extend these results to the computation of surface charges for field theories with non trivial boundary conditions

extremal metrics on blowups in this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of kahler manifolds which already carry an extremal metric as a particular case we construct extremal metrics on mathbb p 2 blown up k points in general position with k lt m 2

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

analysis of two grid method for semi linear elliptic equations by new mixed finite element scheme this article considers two grid stable mixed finite element method based on the less regularity of flux velocity in practice for the semi linear elliptic equations approximated by the p 0 p 1 velocity pressure pair which satisfies the inf sup condition this method involves solving a small semi linear system on a coarse mesh with mesh size h and a linear system problem on a fine mesh with mesh size h o h which can still maintain the asymptotically optimal accuracy it provides an approximate solution with the convergence rate of the same order as the usual mixed finite element solution solving the semi linear elliptic equations on a fine mesh hence the two grid scheme can reduce the computational cost finally numerical tests confirm the theoretical results of the present method

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

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

geodesics in the space of kahler cone metrics i in this paper we study the dirichlet problem of the geodesic equation in the space of k ahler cone metrics cal h cal b that is equivalent to a homogeneous complex monge amp ere equation whose boundary values consist of k ahler metrics with cone singularities our approach concerns the generalization of the function space defined by donaldson to the case of k ahler manifolds with boundary moreover we introduce a subspace cal h c of cal h cal b which we define by prescribing appropriate geometric conditions our main result is the existence uniqueness and regularity of c 1 1 cal b geodesics whose boundary values lie in cal h c moreover we prove that such geodesic is the limit of a sequence of c 2 alpha cal b approximate geodesics under the c 1 1 cal b norm as a geometric application we prove the metric space structure of cal h c

the α invariant on certain surfaces with symmetry groups the global holomorphic invariant introduced by tian is closely related to the existence of kahler einstein metrics we apply the result of tian yau and zelditch on polarized kahler metrics to approximate plurisubharmonic functions and compute the invariant on cp 2 ncp 2 for n 1 2 3

on a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth on etudie le comportement asymptotique de varietes de riemann on construit des coordonnees a l infini sous certaines conditions intrinseques

the α invariant on toric fano manifolds the global holomorphic invariant plays an important role in the study of the existence of kahler einstein metrics on complex manifolds with positive first chern class in this paper we apply the tian yau zelditch expansion of the szego kernel on polarized kahler metrics to approximate almost plurisubharmonic functions and derive a formula to compute the invariant on toric fano manifolds

surface charge algebra in gauge theories and thermodynamic integrability surface charges and their algebra in interacting lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus in the case of exact solutions and symmetries the surface charges are interpreted as a pfaff system integrability is governed by frobenius theorem and the charges associated with the derived symmetry algebra are shown to vanish in the asymptotic context we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended comparison with hamiltonian and covariant phase space methods is made all approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context

isomorphisms between the batalin vilkovisky antibracket and the poisson bracket one may introduce at least three different lie algebras in any lagrangian field theory i the lie algebra of local brst cohomology classes equipped with the odd batalin vilkovisky antibracket which has attracted considerable interest recently ii the lie algebra of local conserved currents equipped with the dickey bracket and iii the lie algebra of conserved integrated charges equipped with the poisson bracket we show in this paper that the subalgebra of i in ghost number 1 and the other two algebras are isomorphic for a field theory without gauge invariance we also prove that in the presence of a gauge freedom ii is still isomorphic to the subalgebra of i in ghost number 1 while iii is isomorphic to the quotient of ii by the ideal of currents without charge in ghost number different from 1 a more detailed analysis of the local brst cohomology classes in the hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended poisson bracke

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

mixed finite element methods for nonlinear second order elliptic problems mixed finite element methods are developed to approximate the solution of the dirichlet problem for the most general quasi linear second order elliptic operator in divergence form existence and uniqueness of the approximation are proved and optimal error estimates in l 2 are demonstrated for both the scalar and vector functions approximated by the method error estimates are also derived in l q 2 leq q leq infty newton s method is presented and analyzed to solve the nonlinear algebraic equations

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

investigations on two kinds of two grid mixed finite element methods for the elliptic eigenvalue problem two kinds of two grid new mixed finite element schemes for the elliptic eigenvalue problem based on less regularity of flux are considered in the new mixed variational formulation the flux belongs to the square integrable space instead of the classical h div and the choices of finite element pairs become simple and easy the two grid methods consist of solving a small elliptic eigenvalue problem on the coarse mesh and then solving a linear algebraic system on the fine mesh which can still maintain asymptotically optimal accuracy finally the performance of two kinds of two grid schemes are compared in efficiency and precision aspects by a series of numerical experiments numerical results show that the accelerated two grid method is a viable choice for the lowest order approximations of the elliptic eigenvalue problem

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

on einstein hermitian 4 manifolds let m h be a compact 4 dimensional einstein manifold and suppose that h is hermitian with respect to some complex structure j on m then either m j h is kahler einstein or else up to rescaling and isometry it is one of the following two exceptions the page metric on mathbb cp 2 overline mathbb cp 2 or the einstein metric on mathbb cp 2 2 overline mathbb cp 2 discovered in on conformally kahler einstein manifolds j amer math soc 21 2008 pp 1137 1168

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

mixed finite element methods for quasilinear second order elliptic problems a mixed finite element method is developed to approximate the solution of a quasilinear second order elliptic partial differential equation the existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived

lp error estimates of two grid schemes of expanded mixed finite element methods abstract we present a scheme for solving two dimensional semi linear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid scheme based on the newton iteration method we give the l p error estimate to the variable p in the two grid solution it is shown that the algorithm achieve asymptotically optimal approximation applying the two grid methods as long as the mesh sizes satisfy h o h 1 3 for semi linear problems

existence of kahler einstein metrics and multiplier ideal sheaves on del pezzo surfaces we apply nadel s method of multiplier ideal sheaves to show that every complex del pezzo surface of degree at most six whose automorphism group acts without fixed points has a kahler einstein metric in particular all del pezzo surfaces of degree 4 5 or 6 and certain special del pezzo surfaces of lower degree are shown to have a kahler einstein metric these existence statements are not new but the proofs given in the present paper are less involved than earlier ones by siu tian and tian yau

central kahler metrics the determinant of the ricci endomorphism of a kahler metric is called its central curvature a notion well defined even in the riemannian context this work investigates two types of kahler metrics in which this curvature potential gives rise to a potential for a gradient holornorphic vector field these metric types generalize the kahler einstein notion as well as that of bando and mabuchi 1986 whenever possible the central curvature is treated in analogy with the scalar curvature and the metrics are compared with the extremal kahler metrics of calabi an analog of the futaki invariant is employed both invariants belonging to a family described in the language of holomorphic equivariant cohomology it is shown that one of the metric types realizes the minimum of an l 2 functional defined on the space of kahler metrics in a given kahler class for metrics of constant central curvature results are obtained regarding existence uniqueness and a partial classification in complex dimension two consequently on a manifold of fano type such metrics and kahler einstein metrics can only exist concurrently an existence result for the case of non constant central curvature is slated and proved in a sequel to this work

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

relative k stability and modified k energy on toric manifolds in this paper we give a sufficient condition for both the relative k stability and the properness of modified k energy associated to calabi s extremal metrics on toric manifolds in addition several examples of toric manifolds which satisfy the sufficient condition are presented

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

numerical kahler einstein metric on the third del pezzo the third del pezzo surface admits a unique kahler einstein metric which is not known in closed form the manifold s toric structure reduces the einstein equation to a single monge ampere equation in two real dimensions we numerically solve this nonlinear pde using three different algorithms and describe the resulting metric the first two algorithms involve simulation of ricci flow in complex and symplectic coordinates respectively the third algorithm involves turning the pde into an optimization problem on a certain space of metrics which are symplectic analogues of the algebraic metrics used in numerical work on calabi yau manifolds our algorithms should be applicable to general toric manifolds using our metric we compute various geometric quantities of interest including laplacian eigenvalues and a harmonic 1 1 form the metric and 1 1 form can be used to construct a klebanov tseytlin like supergravity solution

on the poisson brackets of differentiable generators in classical field theory the canonical formulation of field theory on open spaces is considered it is proved under appropriate assumptions that the poisson bracket of two differentiable generators is also a differentiable generator

local symmetries and constraints the general relationship between local symmetries occurring in a lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied first a prescription applicable to an arbitrary lagrangian field theory for the construction of phase space from the manifold of field configurations on space time is given next a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses as special cases the usual gauge transformations of yang mills theory and general relativity local symmetries on phase space are then defined via projection from field configuration space it is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space moreover the constraints thereby obtained are always first class and the poisson bracket algebra of the constraint functions is isomorphic to the lie bracket algebra of the local symmetries on the constraint submanifold of phase space the differences that occur in the structure of constraints in yang mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space in yang mills theory all the field independent local symmetries project to all of phase space whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are field dependent as by products of the present work definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary lagrangian field theory and the noether current density associated with an arbitrary local symmetry a number of properties of these currents are established and some relationships between them are obtained

the generalized kahler ricci flow abstract in this paper we study a generalized kahler ricci flow in which the ricci term ric is replaced by ric where is a closed real semi positive 1 1 form we first introduce the generalized functionals w and and study their monotonicity along the generalized kahler ricci flow then extend some of perelman s estimates to the generalized kahler ricci flow we also study the asymptotic behavior of the generalized ricci potential

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

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

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

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

on the existence and nonexistence of extremal metrics on toric kahler surfaces abstract in this paper we study the existence of extremal metrics on toric kahler surfaces we show that on every toric kahler surface there exists a kahler class in which the surface admits an extremal metric of calabi we found a toric kahler surface of 9 t c 2 fixed points which admits an unstable kahler class and there is no extremal metric of calabi in it moreover we prove a characterization of the k stability of toric surfaces by simple piecewise linear functions as an application we show that among all toric kahler surfaces with 5 or 6 t c 2 fixed points cp 2 3 cp 2 is the only one which allows vanishing futaki invariant and admits extremal metrics of constant scalar curvature

covariant hamiltonian boundary conditions in general relativity for spatially bounded space time regions we investigate the covariant hamiltonian symplectic structure of general relativity for spatially bounded regions of space time with a fixed time flow vector for existence of a well defined hamiltonian variational principle taking into account a spatial boundary it is necessary to modify the standard arnowitt deser misner hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field the most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space time region a main result is that we obtain a covariant derivation of dirichlet neumann and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface together with the associated hamiltonian boundary terms as well we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current our analysis uses a noether charge method which extends and unifies several results developed in recent literature for general relativity as an illustration of the method we apply it to the maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a well defined hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of minkowski space time

two level picard and modified picard methods for the navier stokes equations abstract iterative methods of picard type for the navier stokes equations are known to converge only for quite small reynolds numbers however we study methods involving just one such iteration at general reynolds numbers for the initial approximation a coarse mesh of width h0 is used the corrected approximation is computed by just one picard or modified picard step on a fine mesh of width h1 for example h1 may be of order o h 0 2 when linear velocity elements are used the resulting method requires the solution of a small system of nonlinear equations on the coarse mesh and only one larger linear system on the fine mesh this two level picard method is proven to converge for fixed reynolds number as h 0 further the fine mesh solution satisfies a quasi optimal error bound the error constants grow as re as for the usual finite element method one very heuristic explanation why one step of the divergent picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower order derivatives thus they are approximated with higher accuracy on the coarse mesh this is linked with a smoothing property of the fine mesh step

central charges in the canonical realization of asymptotic symmetries an example from three dimensional gravity it is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level this is done by studying three dimensional gravity with a negative cosmological constant the asymptotic symmetry group in that case is eitherr so 2 or the pseudo conformal group in two dimensions depending on the boundary conditions adopted at spatial infinity in the latter situation a nontrivial central charge appears in the algebra of the canonical generators which turns out to be just the virasoro central charge