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

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

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

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

generalized coset codes for distributed binning in many multiterminal communication problems constructions of good source codes involve finding distributed partitions into bins of a collection of quantizers associated with a group of source encoders further computationally efficient procedures to index these bins are also required in this work we consider a constructive approach for distributed binning in an algebraic framework several application scenarios fall under the scope of this paper including the ceo problem distributed source coding and n channel symmetric multiple description source coding with n 2 specifically in this exposition we consider the case of two codebooks while focusing on the gaussian ceo problem with mean squared error reconstruction and with two symmetric observations this problem deals with distributed encoding of correlated noisy observations of a source into descriptions such that the joint decoder having access to them can reconstruct the source with a fidelity criterion we employ generalized coset codes constructed in a group theoretic setting for this approach and analyze the performance in terms of distance properties and decoding algorithms

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

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

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

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 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 the existence of hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the existence of hermitian positive definite solutions of the general nonlinear matrix equation x s a x t a q is studied systematically and deeply a new estimate of hermitian positive definite solutions is derived based on a fixed point theorem some new sufficient conditions and new necessary conditions for the existence of hermitian positive definite solutions are obtained in the end a necessary and sufficient condition for the existence of a hermitian positive definite solution is proved

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