convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

numerical solutions of neutral stochastic functional differential equations this paper examines the numerical solutions of neutral stochastic functional differential equations nsfdes d x t u x t f x t dt g x t dw t t geq 0 the key contribution is to establish the strong mean square convergence theory of the euler maruyama approximate solution under the local lipschitz condition the linear growth condition and contractive mapping these conditions are generally imposed to guarantee the existence and uniqueness of the true solution so the numerical results given here are obtained under quite general conditions although the way of analysis borrows from x mao lms j comput math 6 2003 pp 141 161 to cope with u x t several new techniques have been developed

crosshole seismic inversion with normalized full waveform amplitude data we investigate a simple scheme for full waveform amplitude spectrum inversion of crosshole seismic data with an unknown source wavelet the method is based on our 2d 2 5d finite element method of acoustic wave modeling the normalized amplitude data defined as the spectral ratio of the original trace amplitude to the average amplitude for the entire common shot gather are used for full waveform inversion in the frequency domain in essence the normalization of amplitudes removes the source wavelet and is easily carried out in the time domain or frequency domain from crosshole seismic surveying two synthetic models simulating respectively a dipping fracture model and a complicated sedimentary structure are examined with the inversion scheme the numerical results show that clear images of the targets can be obtained by using the single and multiple frequency data for comparison two other amplitude inversion schemes in which the source signatures are known and estimated with an iterative procedure were carried out for the same models the results show that the iteratively source estimated procedure also produces satisfactory images of the velocity structure and yields an approximate amplitude of the source wavelet with the multifrequency data three such schemes yield very competitive results

on stability and the convergence of the kahler ricci flow assuming uniform bounds for the curvature the exponential convergence of the kahler ricci flow is established under two con ditions which are a form of stability the mabuchi energy is bound ed from below and the dimension of the space of holomorphic vector fields in an orbit of the diffeomorphism group cannot jump up in the limit

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

the kahler ricci flow on fano bundles we study the behavior of the kahler ricci flow on some fano bundles which is a trivial bundle on one zariski open set we show that if the fiber is mathbb p m blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kahler class then the fibers collapse in finite time and the metrics converge sub sequentially in gromov hausdorff sense to a metric on the base

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

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

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

calabi flow geodesic rays and uniqueness of constant scalar curvature kahler metrics we prove that constant scalar curvature kahler metric adjacent to a fixed kahler class is unique up to isomorphism the proof is based on the study of a fourth order evolution equation namely the calabi flow from a new geometric perspective and on the geometry of the space of kahler metrics

the kahler ricci flow with positive bisectional curvature we show that the kahler ricci flow on a manifold with positive first chern class converges to a kahler einstein metric assuming positive bisectional curvature and certain stability conditions

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

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

k stability of constant scalar curvature kahler manifolds abstract we show that a polarised manifold with a constant scalar curvature kahler metric and discrete automorphisms is k stable this refines the k semistability proved by s k donaldson

the kahler ricci flow and k polystability we consider the k ahler ricci flow on a fano manifold we show that if the curvature remains uniformly bounded along the flow the mabuchi energy is bounded below and the manifold is k polystable then the manifold admits a k ahler einstein metric the main ingredient is a result that says that a sufficiently small perturbation of a csck manifold admits a csck metric if it is k polystable

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

deformation of extremal metrics complex manifolds and the relative futaki invariant let mathcal x omega be a closed polarized complex manifold g be an extremal metric on mathcal x that represents the kahler class and g be a compact connected subgroup of the isometry group isom mathcal x g assume that the futaki invariant relative to g is nondegenerate at g consider a smooth family mathcal m to b of polarized complex deformations of mathcal x omega simeq mathcal m 0 theta 0 provided with a holomorphic action of g which is trivial on b then for every t in b sufficiently small there exists an h 1 1 mathcal x dimensional family of extremal kahler metrics on mathcal m t whose kahler classes are arbitrarily close to t we apply this deformation theory to show that certain complex deformations of the mukai umemura 3 fold admit kahler einstein metrics