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

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

metrics on semistable and numerically effective higgs bundles we provide notions of numerical effectiveness and numerical flatness for higgs vector bundles on compact kahler manifolds in terms of fibre metrics we prove several properties of bundles satisfying such conditions and in particular we show that numerically flat higgs bundles have vanishing chern classes and that they admit filtrations whose quotients are stable flat higgs bundles we compare these definitions with those previously given in the case of projective varieties finally we study the relations between numerically effectiveness and semistability establishing semistability criteria for higgs bundles on projective manifolds of any dimension

multilevel algorithms considered as iterative methods on semidefinite systems for the representation of piecewise d linear functions instead of the usual finite element basis a generating system is introduced that contains the nodal basis functions of the finest level and of all coarser levels of discretization this approach enables the author to work directly with multilevel decompositions of a function for a partial differential equation the galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the one dimensional 1d case only about twice in the two dimensional 2d case about 4 3 times and in the three dimensional 3d case about 8 7 times as many unknowns as the usual system furthermore the semidefinite system possesses not just one but many solutions however the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem the author shows that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite sy

projective contact manifolds the present work is concerned with the study of complex projective manifolds x which carry a complex contact structure in the first part of the paper we show that if kx is not nef then either x is fano and b2 x 1 or x is of the form ty where y is a projective manifold in the second part of the paper we consider contact manifolds where kx is nef

rotational symmetry of conical kahler ricci solitons we show that expanding kahler ricci solitons which have positive holomorphic bisectional curvature and are c 2 asymptotic to a conical kahler manifold at infinity must be the u n rotationally symmetric expanding solitons constructed by cao

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

a new positivity condition for the curvature of hermitian manifolds in this note we introduce a new type of positivity condition for the curvature of a hermitian manifold which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non kahler case we derive a bochner formula for closed 1 1 forms from which this condition appears naturally and prove that if a hermitian manifold satisfies our positivity condition then any class alpha in h 1 1 bc x can be represented by a closed 1 1 form which is parallel with respect to the bismut connection lastly we show that such a curvature positivity condition holds on certain generalized hopf manifolds and on certain vaisman manifolds

black box multigrid for systems this paper extends to systems the black box multigrid method in which one provides a logically rectangular matrix problem and in which the coarser problems necessary to achieve a multigrid solution are automatically derived the extension is straightforward if the components of the vector solution of the system are defined at the same grid locations otherwise as in the case of staggered grids the situation is less satisfactory

murre s conjectures and explicit chow kunneth projectors for varieties with a nef tangent bundle in this paper we investigate murre s conjectures on the structure of rational chow groups and exhibit explicit chow kiinneth projectors for some examples more precisely the examples we study are the varieties which have a nef tangent bundle for surfaces and threefolds which have a nef tangent bundle explicit chow kunneth projectors are obtained which satisfy murre s conjectures and the motivic hard lefschetz theorem is verified

algebraic multigrid based on element interpolation amge we introduce amge an algebraic multigrid method for solving the discrete equations that arise in ritz type finite element methods for partial differential equations assuming access to the element stiffness matrices we have that amge is based on the use of two local measures which are derived from global measures that appear in existing multigrid theory these new measures are used to determine local representations of algebraically smooth error components that provide the basis for constructing effective interpolation and hence the coarsening process for amg here we focus on the interpolation process choice of the coarse grids based on these measures is the subject of current research we develop a theoretical foundation for amge and present numerical results that demonstrate the efficacy of the method

a new kind of integral transform and its application to periodic solutions in this paper we suggest a new kind of integral transform and establish the main theorems of that this approach allows us to show some general existence theorems for periodic solutions to the semilinear wave equation with the forced term having double frequency a novelty of this transform is to prove the existence of a combination oscillation for the continuous system without the so called difficulty of small divisors by j j stoker nonlinear vibrations interscience new york 1950 in addition our integral transform can be applied to rather general reaction diffusion systems however as a concrete example we show that maginu s model f rothe j math biol 7 1979 375 384 has multiple nonconstant t periodic plane wave solutions

rational curves dynkin diagrams and fano manifolds with nef tangent bundle a fano manifold x with nef tangent bundle is of flag type if it has the same kind of elementary contractions as a complete flag manifold in this paper we present a method to associate a dynkin diagram mathcal d x with any such x based on the numerical properties of its contractions we then show that mathcal d x is the dynkin diagram of a semisimple lie group as an application we prove that campana peternell conjecture holds when x is a flag type manifold whose dynkin diagram is a n i e we show that x is the variety of complete flags of linear subspaces in mathbb p n

positivity of the diagonal abstract we study how the geometry of a projective variety x is reflected in the positivity properties of the diagonal x considered as a cycle on x x we analyze when the diagonal is big when it is nef and when it is rigid in each case we give several implications for the geometric properties of x for example when the cohomology class of x is big we prove that the hodge groups h k 0 x vanish for k 0 we also classify varieties of low dimension where the diagonal is nef and big

big vector bundles and complex manifolds with semi positive tangent bundles we classify compact kahler manifolds with semi positive holomorphic bisectional and big tangent bundles we also classify compact complex surfaces with semi positive tangent bundles and compact complex 3 folds of the form mathbb p t x whose tangent bundles are nef moreover we show that if x is a fano manifold such that mathbb p t x has nef tangent bundle then x cong mathbb p n

p1 bundles admitting another smooth morphism of relative dimension one abstract we give the complete classification of p 1 bundles over projective manifolds of picard number one each of which admit another smooth morphism of relative dimension one

stability conditions on threefolds with nef tangent bundles abstract in this paper we prove the bogomolov gieseker type inequality conjecture for threefolds with nef tangent bundles as a corollary there exist bridgeland stability conditions on these threefolds

on fano manifolds with nef tangent bundles admitting 1 dimensional varieties of minimal rational tangents let x be a fano manifold of picard number 1 with numerically effective tangent bundle according to the principal case of a conjecture of campana peternell s x should be biholomorphic to a rational homogeneous manifold g p where g is a simple lie group and p g is a maximal parabolic subgroup in our opinion there is no overriding evidence for the campana peternell conjecture for the case of picard number 1 to be valid in its full generality as part of a general programme that the author has undertaken with jun muk hwang to study uniruled projective manifolds via their varieties of minimal rational tangents a new geometric approach is adopted in the current article in a special case consisting of a recovering the generic variety of minimal rational tangents c x and b recovering the structure of a rational homogeneous manifold from c x the author proves that when b 4 x 1 and the generic variety of minimal rational tangents is 1 dimensional x is biholomorphic to the projective plane p 2 the 3 dimensional hyperquadric q 3 or the 5 dimensional fano homogeneous contact manifold of type g 2 to be denoted by k g 2 the principal difficulty is part a of the scheme we prove that c x c pt x x is a rational curve of degrees 3 and show that d 1 resp 2 resp 3 corresponds precisely to the cases of x p 2 resp q 3 resp k g 2 let be the normalization of a choice of a chow component of minimal rational curves on x nefness of the tangent bundle implies that is smooth furthermore it implies that at any point x x the normalization x of the corresponding chow space of minimal rational curves marked at is smooth after proving that x is a rational curve our principal object of study is the universal family u of giving a double fibration p u u x which gives p 1 bundles there is a rank 2 holomorphic vector bundle v on whose projectivization is isomorphic to p u we prove that v is stable and deduce the inequality d 4 from the inequality c 2 1 v 4c 2 v resulting from stability and the existence theorem on hermitian einstein metrics the case of d 4 is ruled out by studying the structure of the curvature tensor of the hermitian einstein metric on v in the special case where c 2 1 v 4c 2 v

fano 5 folds with nef tangent bundles and picard numbers greater than one we prove that smooth fano 5 folds with nef tangent bundles and picard numbers greater than one are rational homogeneous manifolds

higher direct images of dualizing sheaves of lagrangian fibrations let f x s be a lagrangian fibration between projective varieties we prove that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i i s if s is smooth suppose that x is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a k 3 surface the hodge numbers satisfy h p q s h p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where n dim s if s inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and x is an irreducible symplectic manifold there exists a hypersurface m f of the kuranishi space of x such that every member of the kuranishi family over m f admits a lagrangian fibration over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i

projective manifolds whose tangent bundles are numerically effective in 1979 mori mo proved the so called hartshorne frankel conjecture every projective n dimensional manifold with ample tangent bundle is isomorphic to the complex projective space p a differential geometric analogon assuming the existence of a k ihler metric on x with positive holomorphic bisectional curvature is independently due to siu yau sy thus it seems natural to classify projective manifolds x whose tangent bundle tx satisfy a degenerate condition of ampleness numerical effectivity abbreviated by nef this means that the tautological quotient line bundle d 1 on f tx is numerically effective i e c 0

ricci flow under local almost non negative curvature conditions abstract we find a local solution to the ricci flow equation under a negative lower bound for many known curvature conditions under a non collapsing assumption the flow exists for a uniform amount of time during which the curvature stays bounded below by a controllable negative number the curvature conditions we consider include 2 non negative and weakly pic 1 cases of which the results are new we complete the discussion of the almost preservation problem by bamler cabezas rivas wilking and the 2 non negative case generalizes a result in 3d by simon topping to higher dimensions as an application we use the local ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non negative curvature conditions and show that this limit space is bi holder homeomorphic to a smooth manifold

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

plurisubharmonic functions and the structure of complete kahler manifolds with nonnegative curvature in this paper we study global properties of continuous plurisubharmonic functions on complete noncompact kahler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds we prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation optimal liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established the results are then applied to prove several structure theorems on complete noncompact kahler manifolds with nonnegative bisectional or sectional curvature

fano n folds with nef tangent bundle and picard number greater than n 5 we prove that fano n folds with nef tangent bundle and picard number greater than n 5 are rational homogeneous manifolds

parallel subspace correction methods for nearly singular systems abstract in this paper we consider the parallel subspace correction psc methods for the nearly singular systems we apply the psc methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods our focus is to estimate the condition number of the preconditioned systems we deduce the parameter independent estimates on the psc preconditioners for the nearly singular systems under appropriate assumptions on subspace decomposition the main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces our estimates can be applied into actual problems and two examples are analyzed in this paper one is the elliptic problem with large jumps in the coefficients the other is the planar nearly incompressible elasticity problem with the scott vogelius finite element discretization we prove that the related parallel multilevel methods for both examples are convergent uniformly with respect to the coefficients and the mesh size

on manifolds whose tangent bundle is big and 1 ample a line bundle over a complex projective variety is called big and 1 ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1 dimensional fibers a vector bundle is called big and 1 ample if the relative hyperplane line bundle over its projectivisation is big and 1 ample the main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more whose tangent bundle is big and 1 ample is equal either to a projective space or to a smooth quadric since big and 1 ample bundles are almost ample the present result is yet another extension of the celebrated mori paper projective manifolds with ample tangent bundles ann of math 110 1979 593 606 the proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective in the appendix we re prove these results

lie algebraic curvature conditions preserved by the hermitian curvature flow the purpose of this paper is to prove that the hermitian curvature flow hcf on an hermitian manifold m g j preserves many natural curvature positivity conditions following ideas of wilking j reine angew math 679 223 247 2013 for an text ad gl t 1 0 m invariant subset s subset text end t 1 0 m and a nice function f text end t 1 0 m rightarrow mathbb r we construct a convex set of curvature operators c s f which is invariant under the hcf varying s and f we prove that the hcf preserves griffiths positivity dual nakano positivity positivity of holomorphic orthogonal bisectional curvature lower bounds on the second scalar curvature as an application we prove that stationary periodic solutions to the hcf can exist only on manifolds m with the trivial canonical bundle on the universal cover widetilde m

semistable and numerically effective principal higgs bundles we study miyaoka type semistability criteria for principal higgs g bundles e on complex projective manifolds of any dimension we prove that e has the property of being semistable after pullback to any projective curve if and only if certain line bundles obtained from some characters of the parabolic subgroups of g are numerically effective one also proves that these conditions are met for semistable principal higgs bundles whose adjoint bundle has vanishing second chern class in a second part of the paper we introduce notions of numerical effectiveness and numerical flatness for principal higgs bundles discussing their main properties for non higgs principal bundles we show that a numerically flat principal bundle admits a reduction to a levi factor which has a flat hermitian yang mills connection and as a consequence that the cohomology ring of a numerically flat principal bundle with coefficients in r is trivial to our knowledge this notion of numerical effectiveness is new even in the case of non higgs principal bundles

a lie algebraic approach to ricci flow invariant curvature conditions and harnack inequalities we consider a subset s of the complex lie algebra so n c and the cone c s of curvature operators which are nonnegative on s we show that c s defines a ricci flow invariant curvature condition if s is invariant under adso n c the analogue for kahler curvature operators holds as well although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions as an application we reprove that a compact kahler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to cp moreover the methods can also be applied to prove harnack inequalities we consider a lie algebra g endowed with a scalar product which is invariant under the adjoint representation of the lie algebra the reader should think of g either as the space so n of skew adjoint endomorphism of r with the scalar product a b 1 2 tr ab or of the lie subalgebra u n so 2n corresponding to the unitary group u n so 2n endowed with the induced scalar product we consider the space of selfadjoint endomorphisms of s g every selfadjoint endomorphism r s g is determined by the corresponding bilinear form x y 7 rx y the extension of this form to a complex bilinear form r g r c g r c c will be denoted with the same letter r notice that for any x g r c the number r x x is real where x 7 x is complex conjugation also recall that the space of algebraic curvature operators s b so n is a linear subspace of s so n similarly the space of algebraic kahler curvature operators s k u n is a linear subspace of s u n the subspaces are also invariant under the ricci flow ode on s g r r r where rx y 12 tr adx r ady r for x y g we have the following basic result theorem 1 let s be a subset of the complex lie algebra g r c and let gc denote a lie group with lie algebra g r c if s is invariant under the adjoint representation of gc then for h r the set c s h r s g r v v h for all v s is invariant under the ode r r r in many cases s is scaling invariant and then h 0 is the only meaningful choice for h 0 the set c s c s 0 is a cone and the curvature condition c s can be thought of as a nonnegativity condition we recall that for a o n invariant subset c s so n we say that a manifold satisfies c if the curvature operator

constrained matrix li yau hamilton estimates on kahler manifolds we derive a family of constrained matrix li yau hamilton estimates on kahler manifolds as a result we first get a constrained matrix li yau hamilton estimate for heat equation on a kahler manifold with fixed kahler metric secondly we get such estimate for forward conjugate heat equation on kahler manifolds with time dependent kahler metrics evolving under the kahler ricci flow

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

on the curvature of vortex moduli spaces we use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed riemann surface after computing the homotopy type of the universal cover of the moduli spaces which are symmetric products of the surface we prove that for genus g 1 the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case and this property extends to all kahler metrics on certain symmetric products our result rules out an established and natural conjecture on the geometry of the moduli spaces

an improved algebraic multigrid method for solving maxwell s equations we propose two improvements to the reitzinger and schoberl algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations the main focus in the reitzinger schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence of the overall multigrid method we illustrate how the reitzinger schoberl amg method loses h independence due to the somewhat limited approximation property of the grid transfer operators we present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates the first improvement is based on smoothing the reitzinger schoberl grid transfer operators the second improvement is obtained by using higher order nodal interpolation to derive the corresponding amg interpolation operators while not completely h independent the resulting amg cg method demonstrates improved convergence behavior while maintaining low operator complexity

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

a multigrid method for variable coefficient maxwell s equations this paper presents a multigrid method for solving variable coefficient maxwell s equations the novelty in this method is the use of interpolation operators that do not produce multilevel commutativity complexes that lead to multilevel exactness rather the effects of multilevel exactness are built into the level equations themselves on the finest level using a discrete t v formulation and on the coarser grids through the galerkin coarsening procedure of a t v formulation these built in structures permit the levelwise use of an effective hybrid smoother on the curl free near nullspace components and permit the development of interpolation operators for handling the curl free and divergence free error components separately the resulting block diagonal interpolation operator does not satisfy multilevel commutativity but has good approximation properties for both of these error components applying operator dependent interpolation for each of these error components leads to an effective multigrid scheme for variable coefficient maxwell s equations where multilevel commutativity based methods can degrade numerical results are presented to verify the effectiveness of this new scheme

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative