the stretch of a foliation and geometric superrigidity we consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type such objects can be considered as compact models for the geometry of the symmetric space based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions to achieve this we consider the heat flow equation along the leaves of a foliation a bochner formula on foliations and a geometric invariant for foliations with leafwise riemannian metrics called the stretch we obtain as applications a metric rigidity theorem for foliations and a rigidity type result for riemannian manifolds whose geometry is only partially symmetric

rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

rigidity of nonpositively curved graphmanifolds there are strong relations between the metric structure of a complete riemannian manifold m of nonpositive sectional curvature and the algebraic structure of its fundamental group in particular we have the following rigidity results for compact m the fundamental group i m determines whether the universal covering space x of m is a nontrivial riemannian product or a symmetric space of higher rank the product case follows from the work of lawson yau ly and gromollwolf gw combined with the results in e 1 the statement is also true i fm is complete and has finite volume is the case of symmetric spaces is stated in be it is based on the rigidity results of mostow m gromov bgs and eberlein e2 and the investigation of higher rank manifolds by ballmann brin burns eberlein and spatzier bbe bbs b this result also holds in the finite volume case these examples products and higher rank symmetric spaces have rank 2 in the sense of bbe on the other hand manifolds of higher rank are products or symmetric spaces b thus the above rigidity results reflect the rigidity properties of higher rank spaces in this paper we study the rigidity of a special class of rank 1 manifolds namely the 3 dimensional graphmanifolds of non positive curvature our main result says that the whole metric structure of nonpositively curved graphmanifolds is determined by the fundamental group see theorem 2 in sect 4 for a precise statement we should say a few words about the relevance of nonpositively curved graphmanifolds waldhausen w studied graphmanifolds from a topological viewpoint gromov g mentioned that a special class of graphmanifolds allow complete riemannian metrics of nonpositive curvature in this way he obtained the first examples of complete riemannian manifolds with bounded nonpositive curvature a 2 k 0 and finite volume with infinite topological type

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

stabilized finite element schemes with lbb stable elements for incompressible flows we study stabilized fe approximations of supg type to the incompressible navier stokes problem revisiting the analysis for the linearized model we show that for conforming lbb stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal order approximation then we analyze a reduced supg scheme often used in practice for lbb stable elements to provide the reduced scheme with appropriate stability estimates we introduce a modified lbb condition which is proved for a family of fe approximations the analysis is given for the linearized equations numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

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

remarks on non compact gradient ricci solitons in this paper we show how techniques coming from stochastic analysis such as stochastic completeness in the form of the weak maximum principle at infinity parabolicity and lp liouville type results for the weighted laplacian associated to the potential may be used to obtain triviality rigidity results and scalar curvature estimates for gradient ricci solitons under lp conditions on the relevant quantities

eigenvalues of the drifting laplacian on complete noncompact riemannian manifolds abstract in this paper we investigate eigenvalues of the eigenvalue problem with dirichlet boundary condition of the drifting laplacian on an n dimensional complete noncompact riemannian manifold some estimates for eigenvalues are obtained by utilizing cheng and yang recursion formula we give a sharp upper bound of the k th eigenvalue as we know product riemannian manifolds ricci solitons and self shrinkers are some important riemannian manifolds therefore we investigate the eigenvalues of the drifting laplacian on those riemannian manifolds in particular by some theorems of classification for ricci solitons we can obtain some eigenvalue inequalities of drifting laplacian on the ricci solitons with certain conditions

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

some applications of the hermite matrix polynomials series expansions this paper deals with hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems properties of hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well known methods results are applied to compute accurate approximations of certain differential systems in terms of hermite matrix polynomials

incomplete partial fractions for parallel evaluation of rational matrix functions abstract frequently one needs to evaluate expressions of the form p a 1q a b where a r n n b r n and p and q are polynomials with degree q degree p and such that no zero of p is an eigenvalue of a algorithms based on the partial fraction representation of q p when evaluating p a 1 q a b lend themselves well to implementation on a parallel computer but might yield poor accuracy we discuss how to determine an incomplete partial fraction representation of q p which allows parallel computation while retaining high accuracy

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

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

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

eigenvalue estimates for a class of elliptic differential operators in divergence form abstract we compute estimates for eigenvalues of a class of linear second order elliptic differential operators in divergence form with dirichlet boundary condition on a bounded domain in a complete riemannian manifold our estimates are based upon the weyl s asymptotic formula as an application we find a lower bound for the mean of the first k eigenvalues of the drifting laplacian in particular we have extended for this operator a partial solution given by cheng and yang for the generalized conjecture of polya we also derive a second yang type inequality due to chen and cheng and other two inequalities which generalize results by cheng and yang obtained for a domain in the unit sphere and for a domain in the projective space

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

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

navier stokes airfoil computations with e transition prediction including transitional flow regions the e ow around laminar airfoils is computed using a navier stokes method coupled to a transition prediction method based on thee n approach applying point transition at the predicted transition location produces a strong viscous inviscid interaction region that prevents the coupled system from converging whereas the introduction of transitional e ow regions resolves that problem the emphasis is not placed on the development of new transitional e ow models but primarily on producing convergence applying modie ed available models a comprehensive computational study is performed in a strong adverse and zero pressure gradient airfoil e ow region as the transitional lengths differ considerably for the different models a conventional model which is applicable in e ow regions wheretransition is predicted well upstream of laminar separation is proposed together with a special transitional length model fore owswheretheboundary layerstays laminarup to separation thee owsoverthedoal3 and the nlf 1 0416 laminar airfoils are investigated the coupled navier stokes and e n methods are shown to produce converged results furthermore the values for lift and drag are in excellent agreement with the free transition measurements

fundamental groups of manifolds of nonpositive curvature introduction the universal covering space h of a complete riemannian manifold m of nonpositive sectional curvature is diffeomorphic to r n dim f hence the homotopy type of m is completely determined by the isomorphism class of the fundamental group of m it is therefore only natural to expect strong relations between the geometric structure of m and the algebraic structure of in this paper we obtain several such relations a general assumption in the results we state below is that 1 the sectional curvature is nonpositive and bounded from below by some constant a and 2 the volume of m is finite we define the rank of a unit tangent vector of m rank u to be the dimension of the space of all parallel jacobi fields along the geodesic yv which has initial velocity v the minimum of rank over all v e sm is called the rank of m this agrees with the usual rank if m is a locally symmetric space manifolds of rank one resemble manifolds of strictly negative curvature see 2 3 and 2 below manifolds of higher rank are studied in 5 6 7 and 3 and the conclusive result is that h is a space of rank one or a symmetric space or a riemannian product of such spaces this is the basic ingredient in the proofs of our results it allows us more or less to consider only the cases that h is of rank one or a symmetric space as a first example of this principle we indicate in our preliminary section the proof of the following theorem theorem a either m is flat or contains a nonabelian free subgroup this is an improvement of the result of avez 1 that has exponential growth if m is compact and not flat

estimates for the eigenvalues of the bi drifting laplacian on complete metric measure spaces as we know it is very difficult to prove an eigenvalue inequality of ashbaugh cheng ichikawa mametsuka type for the bi drifting laplacian on the bounded domain of the complete metric measure spaces even assumed that the differential operator is bi laplacian on the n dimensional unit euclidean sphere this problem still remains open however a general inequality with respect to the bi drifting laplacian is successfully established under certain conditions in this paper applying the general inequality we prove some eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type on the gaussian shrinking ricci soliton and the n dimensional cigar metric measure spaces cmms for short in particular we obtain two interesting eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type for the cmms with lower dimensional topology

efficient orthogonal matrix polynomial based method for computing matrix exponential the matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields this paper presents an efficient method for computing matrix exponentials based on hermite matrix polynomial expansions hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods a backward error analysis of the approximation in exact arithmetic is given this analysis is used to provide a theoretical estimate for the optimal scaling of matrices two algorithms based on this method have been implemented as matlab functions they have been compared with matlab functions funm and expm obtaining greater accuracy in the majority of tests a careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs reaching in numerical tests relative higher average costs than expm of only 4 43 for the final hermite selected order and obtaining better accuracy results in the 77 36 of the test matrices the matlab implementation of the best hermite matrix polynomial based algorithm has been made available online

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

on mathematical modeling of fluid structure interactions with nonlinear effects in this paper the numerical simulation of aeroelastic interactions of flexibly supported two degrees of freedom 2 dof airfoil in two dimensional 2d incompressible viscous turbulent flow subjected to a gust sudden change of flow conditions is considered the flow is modeled by reynolds averaged navier stokes equations rans and by k turbulence model the considered flow problem is discretized in space using the fully stabilized finite element fe method implemented in the developed in house program which allows to solve interaction problems in order to treat the time dependent inlet boundary condition the standard stabilization procedure was modified further the under relaxation procedure was introduced in order to overcome the artificial instability of the coupling algorithm the aeroelastic response to a sudden gust is numerically analyzed with the aid of the developed fe code

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

axial isometries of manifolds of non positive curvature let m be a complete c riemannian manifold of non positive sectional curvature we say that a geodesic 9 ir m bounds a fiat strip of width c 0 a fiat half plane if there is a totally geodesic isometric immersion i 0 c x ir m i 0 oo x ir m such that i 0 t 9 0 a 9eodesic without fiat strip without fiat half plane is a geodesic which does not bound a flat strip a flat half plane we will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of m in fact many of the properties of a manifold of strictly negative curvature resp of a visibility manifold still remain true if one assumes only the existence of a closed geodesic without flat half plane we will discuss the existence of free non abelian subgroups of gl m the existence of infinitely many closed geodesics the density of closed geodesics and a transitivity property of the geodesic flow it is therefore interesting to give conditions which ensure the existence of a closed geodesic without flat half plane we will prove that m has a closed geodesic without flat half plane if vol m oo and if m contains a geodesic without flat half plane note that a geodesic is not boundary of a flat strip and a fortiori not boundary of a flat half plane if it passes through a point p e m such that the sectional curvature of all tangent planes at p is negative in the proofs of our results we investigate the action of rtl m as group of isometries on the universal covering space h of m in the proofs of many of our results we do not use the fact that this action is properly discontinuous and free we therefore formulate these results for arbitrary groups d of isometries of h the paper is organized as follows in sect 1 we fix some definitions and notations and quote some standard results of non positive curvature section 2 is the central section of this paper we investigate the properties of those isometries of h which correspond to closed geodesics in m we also prove

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

efficient evaluation of matrix polynomials abstract this paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state of the art paterson stockmeyer method examples of the application of the methods to the taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations and also with a recent method based on mixed rational and polynomial approximants for many years the paterson stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials in this paper we show that this statement is no longer true moreover for many years rational approximations have been considered more efficient than polynomial approximations although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine in this paper we show that in fact polynomial approximations provide a higher order of approximation than the state of the art computational methods for rational approximations for the same cost in terms of matrix products

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

upper bounds on the first eigenvalue for a diffusion operator via bakry emery ricci curvature let l delta nabla varphi cdot nabla be a symmetric diffusion operator with an invariant measure d mu e varphi dx on a complete riemannian manifold in this paper we prove li yau gradient estimates for weighted elliptic equations on the complete manifold with nabla varphi leq theta and dimensional bakry emery ricci curvature bounded below by some negative constant based on this we give an upper bound on the first eigenvalue of the diffusion operator l on this kind manifold and thereby generalize a cheng s result on the laplacian case math z 143 289 297 1975

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

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

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

on the geometric rank of homogeneous spaces of nonpositive curvature in this paper we investigate differential geometric properties of homogeneous hadamard manifolds and prove a rigidity result in this class of spaces a hadamard manifold h is a simply connected complete riemannian manifold of nonpositive curvature if in addition h is homogeneous then by the work of all aw1 aw2 and hei we know that h is isometric to a solvable lie group s a n with lie algebra s of type nc see aw1 and a left invariant metric here a exp s s denotes an abelian complement to the nilpotent commutator subgroup n exp s s assuming that h has a trivial flat de rham factor the number k dim a only depends on h whereas s need not be uniquely determined and is called the aloebraic rank of h k is an upper bound for the oeometric rank r rk h dr3 i e the maximal integer such that every geodesic is contained in a r fiat a totally geodesic copy of in h the geometric meaning of rk m is well understood for a huge class of hadamard manifolds m thanks to the following remarkable result by ballmann and burns spatzier we quote a generalisation from e3 and eh

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

a discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation abstract to avoid the local oscillations that still remain using the streamline upwind petrov galerkin formulation for the scalar convection diffusion equation the introduction of a nonlinear crosswind dissipation is proposed it is shown that the method is less overdiffusive than other discontinuity capturing techniques and has better numerical behavior the design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases

on complete gradient shrinking ricci solitons in this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons based on this we prove that a complete noncompact gradient shrinking ricci soliton has at most euclidean volume growth this latter result can be viewed as an analog of the well known volume comparison theorem of bishop that a complete noncompact riemannian manifold with nonnegative ricci curvature has at most euclidean volume growth

efficient mixed rational and polynomial approximation of matrix functions abstract this paper presents an efficient method for computing approximations for general matrix functions based on mixed rational and polynomial approximations a method to obtain this kind of approximation from rational approximations is given reaching the highest efficiency when transforming nondiagonal rational approximations with a higher numerator degree than the denominator degree then the proposed mixed rational and polynomial approximation can be successfully applied for matrix functions which have any type of rational approximation such as pade chebyshev etc with maximum efficiency for higher numerator degrees than the denominator degrees the efficiency of the mixed rational and polynomial approximation is compared with the best existing evaluating schemes for general polynomial and rational approximations providing greater theoretical accuracy with the same cost in terms of matrix multiplications it is well known that diagonal rational approximants are generally more accurate than the corresponding nondiagonal rational approximants which have the same computational cost using the proposed mixed approximation we show that the above statement is no longer true and nondiagonal rational approximants are in fact generally more accurate than the corresponding diagonal rational approximants with the same cost

numerical simulation of aeroelastic response of an airfoil in flow with laminar turbulence transition this paper is interested in numerical simulations of the interaction of the fluid flow with an airfoil particularly the problem of the aeroelastic response of the airfoil to a sudden gust is considered the main attention is paid to the finite element approximations of the incompressible viscous flow over a flexibly supported airfoil the gust is modelled using the time dependent boundary condition the structure vibration is governed by the nonlinear system of ordinary differential equations the flow is described using the reynolds averaged navier stokes equations the system is enclosed by the two equation k turbulence model together with the transition model based on the intermittency equation modelling of this laminar turbulence transition of the flow on the airfoil surface is the main novelty of the paper the motion of the computational domain is treated with the aid of the arbitrary lagrangian eulerian method the solution of the nonlinear coupled problem is discussed and numerically tested using a stabilized finite element method

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

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

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

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface