asymptotics of complete kahler metrics of finite volume on quasiprojective manifolds let x be a quasiprojective manifold given by the complement of a divisor d with normal crossings in a smooth projective manifold x using a natural compactification of x by a manifold with corners x we describe the full asymptotic behavior at infinity of certain complete kahler metrics of finite volume on x when these metrics evolve according to the ricci flow we prove that such asymptotic behaviors persist at later times by showing that the associated potential function is smooth up to the boundary on the compactification x however when the divisor d is smooth with kx d 0 so that the ricci flow converges to a kahler einstein metric we show that this kahler einstein metric has a rather different asymptotic behavior at infinity since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary

on the long time behavior of type iii ricci flow solutions we show that three dimensional homogeneous ricci flow solutions that admit finite volume quotients have long time limits given by expanding solitons we show that the same is true for a large class of four dimensional homogeneous solutions we give an extension of hamilton s compactness theorem that does not assume a lower injectivity radius bound in terms of riemannian groupoids using this we show that the long time behavior of type iii ricci flow solutions is governed by the dynamics of an mathbb r action on a compact space

spectral multipliers from hl1 x to lq x abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that the heat kernels p t x y corresponding to the semigroup e t l satisfy gaussian upper bounds but possess no regularity in variables x and y in this article we prove a spectral multiplier theorem for f l from h l 1 x to l q x for some 1 q 2 if the function f possesses the sobolev norm of order s with suitable bounds and s n 1 q 1 2 where n is a measure of the dimension of the space we also study the weighted l p l q estimates for spectral multiplier theorem

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

weighted norm inequalities gaussian bounds and sharp spectral multipliers abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l whose kernels p t x y have gaussian upper bounds but there is no assumption on the regularity in variables x and y in this article we study weighted l p norm inequalities for spectral multipliers of l we show that sharp weighted hormander type spectral multiplier theorems follow from gaussian heat kernel bounds and appropriate l 2 estimates of the kernels of the spectral multipliers these results are applicable to spectral multipliers for large classes of operators including laplace operators acting on lie groups of polynomial growth or irregular non doubling domains of euclidean spaces elliptic operators on compact manifolds and schrodinger operators with non negative potentials

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

global spectra polytopes and stacky invariants given a convex polytope we define its geometric spectrum a stacky version of batyrev s stringy e functions and we prove a stacky version of a formula of libgober and wood about the e polynomial of a smooth projective variety as an application we get a closed formula for the variance of the geometric spectrum and a noether s formula for two dimensional fano polytopes polytopes whose vertices are primitive lattice points a fano polytope is not necessarily smooth we also show that this geometric spectrum is equal to the algebraic spectrum the spectrum at infinity of a tame laurent polynomial whose newton polytope is the polytope alluded to this gives an explanation and some positive answers to hertling s conjecture about the variance of the spectrum of tame regular functions

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

asymptotics of kahler einstein metrics on quasi projective manifolds and an extension theorem on holomorphic maps such a framed manifold is also called canonically polarized let x x c be the complement of c in x according to results of r kobayashi h tsuji g tian and s t yau x possesses a uniquely determined complete kahlereinstein metric x of constant negative ricci curvature 1 in this paper the asymptotic behavior of this metric will be studied since kx c c kc 0 there exists also a kahler einstein metric c on c with ricci curvature 1 using a canonical section of c as a local coordinate function on x we restrict the complete kahler einstein metric x to the locally defined sets c 0 0 the notion of locally uniform convergence of x c 0 where 0 0 makes sense and we can state

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

the limits on boundary of orbifold kahler einstein metrics and kahler ricci flows over quasi projective manifolds in this paper we consider a sequence of orbifold kahler einstein metrics omega x m m and normalized kahler ricci flows omega x m t m on a quasi projective manifold x overline x setminus overline d for a projective manifold overline x and a divisor overline d of overline x with simple normal crossings such that k overline x overline d is ample for sufficiently large m omega x m or omega x m t is the orbifold kahler einstein metric or orbifold normalized kahler ricci flow on x equipped with the orbifold structure with respect to the divisor 1 m overline d respectively the main theorem of this paper is that the limit on the boundary overline d or the inside x of the sequence omega x m t m as m rightarrow infty is exactly the complete normalized kahler ricci flow on overline d or x respectively moreover our method of the proof also leads to a result that on the boundary overline d the sequence omega x m m converges to the complete kahler einstein metric on overline d as m rightarrow infty

numerical characterization of the kahler cone of a compact kahler manifold the goal of this work is to give a precise numerical description of the kahler cone of a compact kahler manifold our main result states that the kahler cone depends only on the intersection form of the cohomology ring the hodge structure and the homology classes of analytic cycles if x is a compact kahler manifold the kahler cone k of x is one of the connected components of the set p of real 1 1 cohomology classes which are numerically positive on analytic cycles i e y p 0 for every irreducible analytic set y in x p dim y this result is new even in the case of projective manifolds where it can be seen as a generalization of the well known nakai moishezon criterion and it also extends previous results by campana peternell and eyssidieux the principal technical step is to show that every nef class which has positive highest self intersection numberx n 0 contains a kahler current this is done by using the calabi yau theorem and a mass concentration technique for monge ampere equations the main result admits a number of variants and corollaries including a description of the cone of numerically effective 1 1 classes and their dual cone another important consequence is the fact that for an arbitrary deformation x s of compact kahler manifolds the kahler cone of a very general fibre xt is independent of t i e invariant by parallel transport under the 1 1 component of the gauss manin connection

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

spectral multiplier theorems via h infty calculus and r bounds we prove spectral multiplier theorems for hormander classes mathcal h alpha p for 0 sectorial operators a on banach spaces assuming a bounded h infty sigma sigma calculus for some sigma in 0 pi and norm and certain r bounds on one of the following families of operators the semigroup e za on mathbb c the wave operators e isa for s in mathbb r the resolvent lambda a 1 on mathbb c backslash mathbb r the imaginary powers a it for t in mathbb r or the bochner riesz means 1 a u alpha for u 0 in contrast to the existing literature we neither assume that a operates on an l p scale nor that a is self adjoint on a hilbert space furthermore we replace generalized gaussian or poisson bounds and maximal estimates by the weaker notion of r bounds which allow for a unified approach to spectral multiplier theorems in a more general setting in this setting our results are close to being optimal moreover we can give a characterization of the r bounded mathcal h alpha 1 calculus in terms of r boundedness of bochner riesz means

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

chern numbers for singular varieties and elliptic homology a fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones intersection homology for example directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so here we use ideas from minimal model theory to define some characteristic numbers for singular varieties generalizing the chern numbers of a smooth variety this was suggested by goresky and macpherson as a next natural problem after the definition of intersection homology 11 we find that only a subspace of the chern numbers can be defined for singular varieties a convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism whereas a singular variety can at most have a fundamental class in a weaker homology theory elliptic homology we use this idea to give an algebro geometric definition of elliptic homology complex bordism modulo flops equals elliptic homology this paper was inspired by some questions asked by jack morava the descriptions of elliptic homology given by gerald hohn 13 were also an important influence thanks to dave bayer mike stillman john stembridge sheldon katz and stein stromme for their computer algebra programs macaulay sf and schubert which helped in guessing the right answer

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

kahler ricci flow with degenerate initial class in the earlier joint work 3 we introduced the weak kahler ricci flow for various geometric motivations in this current work we take further consideration on setting up the weak flow by allowing the initial class to be not necessary kahler it s shown that the construction is compatible with the earlier construction in kahler case we also discuss the convergence as t 0 which is of great interest in this topic and provide related motivation 1 motivation and set up of weak flow the kahler ricci flow the complex version of ricci flow has been under intensive study over the last twenty some years in 18 and more recently 17 g tian proposed the intriguing program of constructing globally existing weak kahlerricci flow with canonical singular limit at time infinity and applying it to the study of algebraic manifolds and even kahler manifolds in general it can be viewed as the analytic version of the famous minimal model program in algebraic geometry in general people should expect the classic smooth kahler ricci flow to encounter singularity at some finite time which is completely decided by cohomology information according to the optimal existence result in 20 just as what people have been doing and had successes in cases for ricci flow surgeries for the underlying manifold should be expected for the kahler ricci flow we naturally expect the surgery to have flavours from algebraic geometry a simple example would be surfaces of general type one only needs the blowingdown of 1 curves when applying the general construction at the end of 3 to push the flow through the finite time of singularities where the measure restriction is actually not so involved as explained later in example 4 4 the degenerate class at the singularity time would become kahler for the new manifold because those 1 curves causing the cohomology degeneration would eventually been crushed to points things can get significantly more complicated for higher dimension for example a standard procedure called flip is introduced in the algebraic geometry context which is of great importance for the business about the algebraic minimal model program simply speaking one needs to blow up the manifold and then perform a different blowing down process naturally we should expect the transformation of the degenerate class is still not kahler in this note we confirm that this is not a problem under the assumption that formally the kahler ricci flow is instantly date received on january 6 2012 and accepted on august 22 2012 2010 mathematics subject classification primary 53c44 secondary 14e30 58j35

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

joint functional calculi and a sharp multiplier theorem for the kohn laplacian on spheres let box b be the kohn laplacian acting on 0 j forms on the unit sphere in mathbb c n in a recent paper of casarino cowling sikora and the author a spectral multiplier theorem of mihlin hormander type for box b is proved in the case 0 j n 1 here we prove an analogous theorem in the exceptional cases j 0 and j n 1 including a weak type 1 1 endpoint estimate we also show that both theorems are sharp the proof hinges on an abstract multivariate multiplier theorem for systems of commuting operators

kahler structures on spin 6 manifolds we show that many spin 6 manifolds have the homotopy type but not the homeomorphism type of a kahler manifold moreover for given betti numbers there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type finally on a fixed spin 6 manifold the chern numbers take on only finitely many values on all possible kahler structures

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

topologically invariant chern numbers of projective varieties abstract we prove that a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least three we prove that only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving these results solve a long standing problem of hirzebruch s we also determine the linear combinations of chern numbers that can be bounded in terms of betti numbers

characteristic numbers of algebraic varieties a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least 3 only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving in the space of chern numbers there are 2 distinguished subspaces one spanned by the euler and pontryagin numbers and the other spanned by the hirzebruch todd numbers their intersection is the span of the euler number and the signature

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

semilinear elliptic equations with the pseudo relativistic operator on a bounded domain abstract we study the dirichlet problem for the semilinear equations involving the pseudo relativistic operator on a bounded domain m 2 m u u p 1 u in with the dirichlet boundary condition u 0 on here p 1 and the operator m 2 m is defined in terms of spectral decomposition in this paper we investigate existence and nonexistence of a nontrivial solution depending on the choice of p m and precisely we show that i if p is not h 1 subcritical p n 2 n 2 and is star shaped the equation has no nontrivial solution for all m 0 i i if p is not h 1 2 supercritical 1 p n 1 n 1 then there exists a least energy solution for all m 0 and any bounded domain i i i finally in the intermediate range n 1 n 1 p n 2 n 2 the problem has a nontrivial solution provided that m is sufficiently large and the problem 0 1 u u p 1 u in u 0 on admits a non degenerate nontrivial solution for example when is a ball or an annulus

ricci flow on surfaces with cusps we consider the normalized ricci flow tg r g with initial condition a complete metric g0 on an open surface m where m is conformal to a punctured compact riemann surface and g0 has ends which are asymptotic to hyperbolic cusps we prove that when m 0 and 0 the flow g t converges exponentially to the unique complete metric of constant gauss curvature 2 in the conformal class