energy functionals and canonical kahler metrics yau y2 has conjectured that a fano manifold admits a kahler einstein metric if and only if it is stable in the sense of geometric invariant theory there has been much progress on this conjecture by tian t1 t2 donaldson do1 do2 and others the mabuchi energy functional plays a central role in these ideas we study the ek functionals introduced by x x chen and g tian ct1 which generalize the mabuchi energy we show that if a fano manifold admits a kahler einstein metric then the functional e1 is bounded from below and modulo holomorphic vector fields is proper this answers affirmatively a question raised by chen c2 in fact we show that e1 is proper if and only if there exists a kahler einstein metric giving a new analytic criterion for the existence of this canonical metric with possible implications for the study of stability we also show that on a fano kahler einstein manifold all of the functionals ek are bounded below on the space of metrics with nonnegative ricci curvature

k polystability of q fano varieties admitting kahler einstein metrics it is shown that any possibly singular fano variety x admitting a kahler einstein metric is k polystable thus confirming one direction of the yau tian donaldson conjecture in the setting of mathbb q fano varieties equipped with their anti canonical polarization the proof is based on a new formula expressing the donaldson futaki invariants in terms of the slope of the ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti canonical line bundle of x one consequence is that a toric fano variety x is k polystable iff it is k polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope p associated to x the results also extend to the logarithmic setting and in particular to the setting of kahler einstein metrics with edge cone singularities applications to geodesic stability bounds on the ricci potential and perelman s lambda entropy functional on k unstable fano manifolds are also given

unstable products of smooth curves we give examples of smooth manifolds with negative first chern class which are slope unstable with respect to certain polarisations and so have kahler classes that do not admit any constant scalar curvature kahler metrics these manifolds also have unstable hilbert and chow points we compare this to the work of song weinkove on the j flow

weighted composition operators from the logarithmic weighted type space to the weighted bergman space in cn let w be a bounded circular strictly convex domain in c n with c 2 boundary and h w the space of all analytic functions on w let u h w and f be a holomorphic self map of w the weighted composition operator uc f on h w is defined by uc f f z u z f f z where f h w and z w let h l o g c b w b 0 c r be the logarithmic weighted type space on w and a a p w p 0 a 1 the weighted bergman space on w here we characterize the boundedness and compactness of the weighted composition operator uc f h l o g c b w a a p w

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

bessel functions heat kernel and the conical kahler ricci flow abstract following donaldson s openness theorem on deforming a conical kahler einstein metric we prove a parabolic schauder type estimate with respect to conical metrics as a corollary we show that the conical kahler ricci flow exists for short time the key is to establish the relevant heat kernel estimates where we use the weber formula on bessel function of the second kind and carslaw s heat kernel representation in 8

a remark on a standard and linear vector network equilibrium problem with capacity constraints in this paper weak vector equilibrium principle with capacity constraints is introduced a necessary condition that a vector minimum cost flow is a vector equilibrium flow with capacity constraints is obtained when the number of paths connecting with each pair of source and sink is less than or equal to 2 a sufficient condition for a vector minimum cost flow to be a vector equilibrium flow is also obtained a generalized weak vector equilibrium principle is also introduced without any additional assumption a necessary and sufficient condition for a weak vector minimum cost flow to be a generalized weak vector equilibrium flow is obtained

toric surfaces k stability and calabi flow let x be a toric surface and u be a normalized symplectic potential on the corresponding polygon p suppose that the riemannian curvature is bounded by a constant c 1 and int partial p u d sigma 0 depending only on c 1 c 2 and p such that donaldson s m condition holds for u as an application we show that if x p is analytic relative k stable then the modified calabi flow converges to an extremal metric exponentially fast by assuming that the calabi flow exists for all time and the riemannian curvature is uniformly bounded along the calabi flow

compact moduli spaces of del pezzo surfaces and kahler einstein metrics we prove that the gromov hausdorff compactification of the moduli space of kahler einstein del pezzo surfaces in each degree agrees with certain algebro geometric compactification in particular this recovers tian s theorem on the existence of kahler einstein metrics on smooth del pezzo surfaces and classifies all the degenerations of such metrics the proof is based on a combination of both algebraic and differential geometric techniques

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

k stability and parabolic stability parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces in this paper we show that the three notions of parabolic polystability k polystability and existence of constant scalar curvature kahler metrics on the iterated blowup are equivalent for certain polarizations close to the boundary of the kahler cone

geometric properties of boundary sections of solutions to the monge ampere equation and applications abstract in this paper we establish several geometric properties of boundary sections of convex solutions to the monge ampere equation the engulfing and separating properties and volume estimates as applications we prove a covering lemma of besicovitch type a covering theorem and a strong type p p estimate for the maximal function corresponding to boundary sections moreover we show that the monge ampere setting forms a space of homogeneous type

absolute optimal solution for a compact and convex game abstract this paper investigates the existence of absolute optimal solutions for a partition p in continuous and quasiconcave games we show that the p consistency property introduced in the paper together with the quasiconcavity and continuity of payoffs permits the existence of p absolute optimal solutions in games with compact and convex strategy spaces the p consistency property is a general condition that cannot be dispensed with for the existence of p absolute optimal solutions we also characterize the existence of p absolute optimal solutions by providing necessary and sufficient conditions moreover we suggest an algorithm for efficiently computing p absolute optimal solutions

toric partial density functions and stability of toric varieties let l h rightarrow x omega denote a polarized toric kahler manifold fix a toric submanifold y and denote by hat rho tk x rightarrow mathbb r the partial density function corresponding to the partial bergman kernel projecting smooth sections of l k onto holomorphic sections of l k that vanish to order at least tk along y for fixed t 0 such that tk in mathbb n we prove the existence of a distributional expansion of hat rho tk as k rightarrow infty including the identification of the coefficient of k n 1 as a distribution on x this expansion is used to give a direct proof that if omega has constant scalar curvature then x l must be slope semi stable with respect to y cf ross and thomas in j differ geom 72 3 429 466 2006 similar results are also obtained for more general partial density functions these results have analogous applications to the study of toric k stability of toric varieties

uniform k stability for extremal metrics on toric varieties abstract in this paper we prove that for toric varieties the uniform k stability is a necessary condition for the existence of extremal metrics

relative k stability for kahler manifolds we study the existence of extremal kahler metrics on kahler manifolds after introducing a notion of relative k stability for kahler manifolds we prove that kahler manifolds admitting extremal kahler metrics are relatively k stable along the way we prove a general l p lower bound on the calabi functional involving test configurations and their associated numerical invariants answering a question of donaldson when the kahler manifold is projective our definition of relative k stability is stronger than the usual definition given by szekelyhidi in particular our result strengthens the known results in the projective case even for constant scalar curvature kahler metrics and rules out a well known counterexample to the naive version of the yau tian donaldson conjecture in this setting

optimal lower bounds for donaldson s j functional abstract in this paper we provide an explicit formula for the optimal lower bound of donaldson s j functional in the sense of finding explicitly the optimal constant in the definition of coercivity which always exists and takes negative values in general this constant is positive precisely if the j equation admits a solution and the explicit formula has a number of applications first this leads to new existence criteria for constant scalar curvature kahler csck metrics in terms of tian s alpha invariant moreover we use the above formula to discuss calabi dream manifolds and an analogous notion for the j equation and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the kahler cone even when the underlying kahler classes admit csck metrics as a final application we show that if the lejmi szekelyhidi conjecture holds then the optimal bound coincides with its algebraic counterpart the set of j semistable classes equals the closure of the set of uniformly j stable classes in the kahler cone and there exists an optimal degeneration for uniform j stability

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 abreu equation with degenerated boundary conditions abstract the abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds we study the dirichlet problem of the abreu equation with degenerated boundary conditions the solutions provide the kahler metrics of constant scalar curvature on the complex torus

hyperdiscriminant polytopes chow polytopes and mabuchi energy asymptotics letx n p n be a smooth linearly normal algebraic variety it is shown that the mabuchi energy of x fsjx restricted to the bergman metrics is completely determined by the x hyperdiscriminant of format n 1 and the chow form of x as a corollary it is shown that the mabuchi energy is bounded from below for all degenerations in g if and only if the hyperdiscriminant polytope dominates the chow polytope for all maximal algebraic tori h of g

a multicriteria pareto optimal path algorithm abstract in this paper we present an algorithm for finding in a multicriteria network pareto optimal paths one each for each efficient objective vector validation of the algorithm and two illustrative examples are included computational experience is also reported this algorithm is a generalization of an earlier one for the bicriterion case

on the proper moduli spaces of smoothable kahler einstein fano varieties in this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth kahler einstein fano manifolds inside an appropriate hilbert scheme in particular we prove that being k semistable is a zariski open condition and we establish the uniqueness of the gromov hausdorff limit for a punctured flat family of kahler einstein fano manifolds based on these we construct a proper scheme parameterizing the s equivalent classes of q gorenstein smoothable k semistable q fano varieties and we verify various necessary properties to guarantee that it is a good moduli space

vector equilibrium problem and vector optimization this paper examines the vector equilibrium model based on a vector cost consideration this is a generalization of the well known wardrop traffic equilibrium principle where road users choose paths based on just a single cost the concept of parametric equilibria is introduced and used to establish relations with parametric complementarity and variational inequality problems relations with some vector optimization problems via scalarization techniques are given under appropriate conditions some solution methods for solving vector equilibrium problems are also discussed

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

modified futaki invariant and equivariant riemann roch formula abstract in this paper we give a new version of the modified futaki invariant for a test configuration associated to the soliton action on a fano manifold our version will naturally come from toric test configurations defined by donaldson for toric manifolds as an application we show that the modified k energy is proper for toric invariant kahler potentials on a toric fano manifold

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

deligne pairings and the knudsen mumford expansion let x b be a proper flat morphism between smooth quasiprojective varieties of relative dimension n and l x a line bundle which is ample on the fibers we establish formulas for the first two terms in the knudsen mumford expansion for det l k in terms of deligne pairings of l and the relative canonical bundle k this generalizes the theorem of deligne 1 which holds for families of relative dimension one as a corollary we show that when x is smooth the line bundle associated to x b which was introduced in phong sturm 12 coincides with the cm bundle defined by paul tian 10 11 in a second and third corollaries we establish asymptotics for the k energy along bergman rays generalizing the formulas obtained in 11

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

relative chow stability and extremal metrics abstract we prove that the existence of extremal metrics implies asymptotically relative chow stability an application of this is the uniqueness up to automorphisms of extremal metrics in any polarization

vanishing sequences and okounkov bodies we define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety building on previous work of the first author with huayi chen we prove an equidistribution result for vanishing sequences of large powers of a big line bundle and study the limit measure in particular the latter is described in terms of restricted volumes for divisorial valuations we also show on an example that the associated concave function on the okounkov body can be discontinuous at boundary points

birational superrigidity and slope stability of fano manifolds we prove a relation between birational superrigidity of fano manifold and its slope stability in the sense of ross and thomas j alg geom 16 201 205 2007

an obstacle problem for a class of monge ampere type functionals abstract in this paper we study an obstacle problem for monge ampere type functionals whose euler lagrange equations are a class of fourth order equations including the affine maximal surface equation and abreu s equation

interior estimates for the n dimensional abreuʼs equation abstract we study the abreu s equation in n dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform k stability

w4 p solution to the second boundary value problem of the prescribed affine mean curvature and abreu s equations abstract the second boundary value problem of the prescribed affine mean curvature equation is a nonlinear fourth order geometric partial differential equation it was introduced by trudinger and wang in 2005 in their investigation of the affine plateau problem in affine geometry the previous works of trudinger wang chau weinkove and the author solved this global problem in w 4 p under some restrictions on the sign or integrability of the affine mean curvature we remove these restrictions in this paper and obtain w 4 p solution to the second boundary value problem when the affine mean curvature belongs to l p with p greater than the dimension our self contained analysis also covers the case of abreu s equation

thresholds valuations and k stability abstract let x be a normal complex projective variety with at worst klt singularities and l a big line bundle on x we use valuations to study the log canonical threshold of l as well as another invariant the stability threshold the latter generalizes a notion by fujita and odaka and can be used to characterize when a q fano variety is k semistable or uniformly k stable it can also be used to generalize volume bounds due to fujita and liu the two thresholds can be written as infima of certain functionals on the space of valuations on x when l is ample we prove that these infima are attained in the toric case toric valuations achieve these infima and we obtain simple expressions for the two thresholds in terms of the moment polytope of l

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

differential and sensitivity properties of gap functions for minty vector variational inequalities the purpose of this paper is to investigate differential properties of a class of set valued maps and gap functions involving minty vector variational inequalities relationships between their contingent derivatives are discussed an explicit expression of the contingent derivative for the class of set valued maps is established optimality conditions of solutions for minty vector variational inequalities are obtained

alpha invariant and k stability of q fano varieties abstract we give a purely algebro geometric proof that if the invariant of a q fano variety x is greater than dim x dim x 1 then x o x k x is k stable the key of our proof is a relation among the seshadri constants the invariant and k stability it also gives applications concerning the automorphism group

analytic efficient solution set for multi criteria quadratic programs abstract we present active set methods to evaluate the exact analytic efficient solution set for multi criteria convex quadratic programming problems mcqp subject to linear constraints the idea is based on the observations that a strictly convex programming problem admits a unique solution and that the efficient solution set for a multi criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized the case of bi criteria quadratic programming bcqp is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives in particular we note that the efficient solution set of a bcqp problem is a curve on the surface of a polytope the extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation two numerical examples are given to illustrate the proposed methods

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

affine techniques on extremal metrics on toric surfaces abstract this paper consists of a package of real and complex affine techniques for the abreu equation on toric surfaces manifolds in particular as an application of this package we give an interior estimate for the ricci tensor of toric surfaces

global w 2 p estimates for solutions to the linearized monge ampere equations in this paper we establish global w 2 p estimates for solutions to the linearized monge ampere equations under natural assumptions on the domain monge ampere measures and boundary data our estimates are affine invariant analogues of the global w 2 p estimates of winter for fully nonlinear uniformly elliptic equations and also linearized counterparts of savin s global w 2 p estimates for the monge ampere equations

geodesic einstein metrics and nonlinear stabilities in this paper we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic einstein metric on this line bundle which generalize the classical stabilities and hermitian einstein metrics of holomorphic vector bundles we introduce a donaldson type functional and show that this functional attains its absolute minimum at geodesic einstein metrics and we also discuss the relations between the existence of geodesic einstein metrics and the nonlinear stabilities of the line bundle as an application we will prove that a holomorphic vector bundle admits a finsler einstein metric if and only if it admits a hermitian einstein metric which answers a problem posed by s kobayashi

kahler geometry of horosymmetric varieties and application to mabuchi s k energy functional we introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties we develop kahler geometry on these varieties with applications to canonical metrics in mind as a generalization of the guillemin abreu donaldson geometry of toric varieties namely we associate convex functions with hermitian metrics on line bundles and express the curvature form in terms of this function as well as the corresponding monge ampere volume form and scalar curvature we then provide an expression for the mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by li zhou and zhu on group compactifications

the global existence and convergence of the calabi flow on cn zn izn in this note we study the long time existence of the calabi flow on x cn zn izn assuming the uniform bound of the total energy we establish the non collapsing property of the calabi flow by using donaldson s estimates and streets regularity theorem next we show that the curvature is uniformly bounded along the calabi flow on x when the dimension is 2 partially confirming chen s conjecture moreover we show that the calabi flow exponentially converges to the flat kahler metric for arbitrary dimension if the curvature is uniformly bounded partially confirming donaldson s conjecture

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

kahler non collapsing eigenvalues and the calabi flow abstract we first prove a new compactness theorem of kahler metrics which confirms a prediction in 17 then we establish several eigenvalue estimates along the calabi flow combining the compactness theorem and these eigenvalue estimates we generalize the method developed for the kahler ricci flow in 22 to obtain several new small energy theorems of the calabi flow

minimizing normalized volumes of valuations for any mathbb q gorenstein klt singularity x o we introduce a normalized volume function widehat mathrm vol that is defined on the space of real valuations centered at o and consider the problem of minimizing widehat mathrm vol we prove that the normalized volume has a uniform positive lower bound by proving an izumi type estimate for any mathbb q gorenstein klt singularity furthermore by proving a properness estimate we show that the set of real valuations with uniformly bounded normalized volumes is compact and hence reduce the existence of minimizers for widehat mathrm vol to a conjectural lower semicontinuity property for a smooth point we use an inequality of de fernex ein musta to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of widehat mathrm vol finally the relation to fujita s work on divisorial stability is pointed out

on the calabi flow in this paper we study the calabi flow on a polarized kahler manifold and some related problems we first give a precise statement on the short time existence of the calabi flow for any c 3 a m initial kahler potential as an application we prove a stability result any metric near a constant scalar curvature kahler csck metric will flow to a nearby csck metric exponentially fast secondly we prove that a compactness theorem in the space of the kahler metrics given uniform ricci bound and potential bound as an application we prove the calabi flow can be extended once ricci curvature stays uniformly bounded lastly we prove a removing singularity result about a weak constant scalar curvature metric in a punctured disc

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

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

conical kahler einstein metrics revisited in this paper we introduce the interpolation degeneration strategy to study kahler einstein metrics on a smooth fano manifold with cone singularities along a smooth divisor that is proportional to the anti canonical divisor by interpolation we show the angles in 0 2 that admit a conical kahler einstein metric form a connected interval and by degeneration we determine the boundary of the interval in some important cases as a first application we show that there exists a kahler einstein metric on mathbb p 2 with cone singularity along a smooth conic degree 2 curve if and only if the angle is in 2 2 when the angle is 2 3 this proves the existence of a sasaki einstein metric on the link of a three dimensional a2 singularity and thus answers a question posed by gauntlett martelli sparks yau as a second application we prove a version of donaldson s conjecture about conical kahler einstein metrics in the toric case using song wang s recent existence result of toric invariant conical kahler einstein metrics

on some isometries on the bergman privalov class on the unit ball abstract bergman privalov class a n b consists of all holomorphic functions on the unit ball b c n such that f a n b ln 1 f z d v z where 1 d v z c n 1 z 2 d v z d v z is the normalized lebesgue volume measure on b and c n is the normalization constant that is v b 1 under a mild condition we characterize surjective isometries not necessarily linear on a n b and prove that t is a surjective multiplicative isometry not necessarily linear on a n b if and only if it has the form t f f or t f f for every f a n b where is a unitary transformation of the unit ball the corresponding results for the case of the bergman privalov class on the unit polydisk d n are also given our results extend and complement recent results by o hatori and y iida

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

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

on existence of vector equilibrium flows with capacity constraints of arcs in this paper we propose a weak vector equilibrium principle with capacity constraints of arcs by proving the existence of solutions for the weighted variational inequality we establish the existence results of weak vector traffic equilibrium flows with capacity constraints of arcs

weighted composition operators from bergman privalov type spaces to weighted type spaces on the unit ball abstract motivated by recent paper x zhu appl math comput 215 12 2010 4340 4346 we characterize the boundedness and compactness of weighted composition operators from the bergman privalov type space to weighted type and the little weighted type space on the unit ball in c n

filtrations and test configurations we introduce a strengthening of k stability based on filtrations of the homogeneous coordinate ring this allows for considering certain limits of families of test configurations which arise naturally in several settings we prove that if a manifold with no automorphisms admits a csck metric then it satisfies this stronger stability notion we also discuss the relation with the birational transformations in the definition of b stability

note on k stability of pairs we prove that a pair x d with x fano and d an anti canonical divisor is k unstable for negative angles and is k semistable for zero angle

variational solutions to extremal metrics on toric surfaces in this paper we prove the interior regularity of minimizers of donaldson s version of k energy associated to calabi s extremal metrics on toric surfaces a new idea is to use solutions to the first boundary value problem of abreu s equation to approximate a minimizer of k energy by the regularity we also prove the uniqueness of minimizers in dimension 2

interior second derivative estimates for solutions to the linearized monge ampere equation let rn be a bounded convex domain and c be a convex function such that is sufficiently smooth on and the monge ampere measure det d2 is bounded away from zero and infinity in the corresponding linearized monge ampere equation is trace d2u f where det d2 d2 1 is the matrix of cofactors of d2 we prove a conjecture in gt about the relationship between lp estimates for d2u and the closeness between det d2 and one as a consequence we obtain interior w2 p estimates for solutions to such equation whenever the measure det d2 is given by a continuous density and the function f belongs to lq for some q max p n