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

global existence and blow up of solutions to a nonlocal quasilinear degenerate parabolic system abstract this paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system u t div u p 2 u a v x t d x v t div v q 2 v b u x t d x with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where p q 2 1 and a b 0 are positive constants under appropriate hypotheses we first establish the local existence and uniqueness of solutions then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between and p 1 q 1 in the special case of q 1 and p 1 we conclude that the solution exists globally if p 1 d x q 1 d x 1 a b while if p 1 d x q 1 d x 1 a b then the solution blows up in finite time here x and x denote the unique solution of the following elliptic problem div p 2 1 in x 0 and div q 2 1 in x 0 respectively

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

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

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

extinction behavior of solutions for a quasilinear parabolic system with nonlocal sources this paper deals with the extinction properties of solutions for a quasilinear parabolic system u t div u p 2 u v x t dx v t div v q 2 v u x t dx with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where 1 p q 2 and 0 by using the comparison principle and energy estimate methods we obtain the sufficient conditions of extinction solutions

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

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

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

extinction decay and blow up for keller segel systems of fast diffusion type abstract we consider the quasi linear keller segel system of singular type where the principal part u m represents a fast diffusion like 0 m 1 we first construct a global weak solution with small initial data in the scaling invariant norm l n q m 2 for all dimensions n 2 and all exponents q 2 as for the large initial data we show that there exists a blow up solution in the case of n 2 in the second part the decay property in l r with 1 r for 1 2 n m 1 with the mass conservation is shown on the other hand in the case of 0 m 1 2 n the extinction phenomenon of solution is proved it is clarified that the case of m 1 2 n exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 2 n for the borderline case of m 1 2 n our solution decays in l r exponentially as t

extinction and positivity for the evolution p laplacian equation in rn abstract the aim of this paper is to discuss extinction and positivity for the solution of the initial boundary value problem of evolution p laplacian equation it is proved that the weak solution will be extincted for 1 p 2 and will be positive for p 2 for large t

blow up for a degenerate reaction diffusion system with nonlinear nonlocal sources this paper investigates the global existence and blow up of nonnegative solution of the systemu email protected m u p email protected wv q 1dx v email protected n v p email protected wu q 2dx x t email protected 0 t with homogeneous dirichlet boundary conditions where email protected r n is a bounded domain with smooth boundary email protected m n 1 p 1 p 2 q 1 q 2 0 the results depend crucially on the number p i q i m n the domain w and the initial data u 0 x v 0 x moreover we obtain the blow up rate of the blow up solution under some appropriate hypotheses

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

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

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

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

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 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

extinction and non extinction for a p laplacian equation with nonlinear source abstract in this paper we deal with the extinction of solutions of the initial boundary value problem of the p laplacian equation u t div u p 2 u u q in a bounded domain of r n with n 2 for 1 p 2 we show that q p 1 is the critical exponent of extinction for the weak solution furthermore for 1 p 2 and q p 1 we prove that the extinction and non extinction of the solution depends strongly on the first eigenvalue of the problem div p 2 p 2 in 0

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

long time extinction of solutions of some semilinear parabolic equations we study the long time behavior of solutions of semilinear parabolic equation of the following type tu u a0 x uq 0 where a0 x d0exp x x 2 d0 0 1 q 0 and is a positive continuous radial function we give a dini like condition on the function by two different methods which implies that any solution of the above equation vanishes in a finite time the first one is a variant of a local energy method and the second one is derived from semi classical limits of some schrodinger operators

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

vanishing solutions of anisotropic parabolic equations with variable nonlinearity abstract we study the property of finite time vanishing of solutions of the homogeneous dirichlet problem for the anisotropic parabolic equations u t i 1 n d i a i x t u d i u p i x t 2 d i u c x t u x t 2 u f x t with variable exponents of nonlinearity p i x t x t 1 we show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data if the solution does not identically vanish in a finite time we estimate the rate of vanishing of the solution as t we establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one

extinction and positivity for a system of semilinear parabolic variational inequalities abstract a simple model of chemical kinetics with two concentrations u and v can be formulated as a system of two parabolic variational inequalities with reaction rates v p and u q for te diffusion processes of u and v respectively it is shown that if pq u and v are comparable then at least one of the concentrations becomes extinct in finite time on the other hand for any p q 0 there are initial values for which both concentrations do not become extinct in any finite time