a new curvature condition preserved by the ricci flow abstract in this paper we establish a new curvature condition preserved by the ricci flow which is named as 2 parameters nonnegative curvature condition it relies on the first second and third eigenvalues of the riemannian curvature operator based on this we prove the strong maximum principle for the 2 parameters nonnegativity along ricci flow

the ricci flow on manifolds with boundary we study the short time existence and regularity of solutions to a boundary value problem for the ricci deturck equation on a manifold with boundary using this we prove the short time existence and uniqueness of the ricci flow prescribing the mean curvature and conformal class of the boundary with arbitrary initial data finally we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded

flag manifolds with strongly positive curvature we obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the wallach flag manifolds w 6 w 12 and w 24 which are respectively the manifolds of complete flags in mathbb c 3 mathbb h 3 and mathbb c mathrm a 3 together with our earlier work this concludes the classification of simply connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature

surgery stable curvature conditions we give a simple criterion for a pointwise curvature condition to be stable under surgery namely a curvature condition c which is understood to be an open convex mathrm o n invariant cone in the space of algebraic curvature operators is stable under surgeries of codimension at least c provided it contains the curvature operator corresponding to s c 1 times mathbb r n c 1 c ge 3 this is used to generalize the well known classification result of positive scalar curvature in the simply connected case in the following way any simply connected manifold m n n ge 5 which is either spin with vanishing alpha invariant or else is non spin admits for any epsilon 0 a metric such that the curvature operator satisfies r epsilon left r right

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

einstein four manifolds of three nonnegative curvature operator in this paper we prove that einstein four manifolds of 3 positive curvature operator are isometric to s 4 g 0 or mathbb c p 2 g fs and einstein four manifolds of 3 nonnegative curvature operator are isometric to s 4 g 0 mathbb c p 2 g fs or s 2 times s 2 g 0 oplus g 0 up to rescaling we also prove that the first eigenvalue of the laplace operator for einstein four manifolds with mathrm ric g and nonnegative sectional curvature is bounded above by frac 4 3 4 frac 1 3 the basic idea of the proofs is to construct an integrated subharmonic function and the main ingredients of the proofs are curvature decompositions in particular berger decomposition the weitzenbock formula and the refined kato inequality along with the proofs we also discover an alternative proof for the weitzenbock formula using berger decomposition and an alternative proof for the refined kato inequality using derdzinski s argument

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

on the blow up of four dimensional ricci flow singularities in this paper we prove a conjecture by feldman ilmanen knopf in 14 that the gradient shrinking soliton metric they constructed on the tautological line bundle over cp 1 is the uniform limit of blow ups of a type i ricci ow singularity on a closed manifold we use this result to show that limits of blow ups of ricci ow singularities on closed four dimensional manifolds do not necessarily have non negative ricci curvature

manifolds with positive curvature operators are space forms the ricci flow was introduced by hamilton in 1982 h1 in order to prove that a compact three manifold admitting a riemannian metric of positive ricci curvature is a spherical space form in dimension four hamilton showed that compact four manifolds with positive curvature operators are spherical space forms as well h2 more generally the same conclusion holds for compact four manifolds with 2 positive curvature operators che recall that a curvature operator is called 2 positive if the sum of its two smallest eigenvalues is positive in arbitrary dimensions huisken hu described an explicit open cone in the space of curvature operators such that the normalized ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature hamilton conjectured that in all dimensions compact riemannian manifolds with positive curvature operators must be space forms in this paper we confirm this conjecture more generally we show the following

collapsed 5 manifolds with pinched positive sectional curvature abstract let m be a closed 5 manifold of pinched curvature 0 sec m 1 we prove that m is homeomorphic to a spherical space form if one of the following conditions holds i the center of the fundamental group has index w a constant depending on ii 1 4 1 10 6 2 and the fundamental group is a non cyclic group of order c a constant iii the volume is less than and the fundamental group is either isomorphic to a spherical 5 space group or has an odd order and it has a center of index w a constant

on wilking s criterion for the ricci flow wilking has recently shown that one can associate a ricci flow invariant cone of curvature operators c s which are nonnegative in a suitable sense to every ad so n mathbb c invariant subset s subset mathbf so n mathbb c in this article we show that if s is an ad so n mathbb c invariant subset of mathbf so n mathbb c such that s cup 0 is closed and c s subset c s denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds a the connected sum of any two riemannian manifolds with curvature operators in c s also admits a metric with curvature operator in c s b the normalized ricci flow on any compact riemannian manifold m with curvature operator in c s converges to a metric of constant positive sectional curvature we also point out that if s is an arbitrary ad so n mathbb c subset then c s is contained in the cone of curvature operators with nonnegative isotropic curvature

fractal measures with uniform marginals abstract we provide several constructions of self affine probability measures on the unit square with uniform marginals these constructions include and extend constructions of previous authors and are parameterized in a natural way in addition for each different construction we determine the dimension of the parameter space and thus the level of flexibility for instance for approximation purposes each construction allows finally we give some simple approximation results showing how to approximate any measure with uniform marginals on the unit square with a fractal measure resulting from one of our constructions

integral pinched shrinking ricci solitons abstract we prove that a n dimensional 4 n 6 compact gradient shrinking ricci soliton satisfying a l n 2 pinching condition is isometric to a quotient of the round s n the proof relies mainly on sharp algebraic curvature estimates the yamabe sobolev inequality and an improved rigidity result for integral pinched einstein metrics

upper bound on the products of particle interactions in cellular automata abstract particle like objects are observed to propagate and interact in many spatially extended dynamical systems for one of the simplest classes of such systems one dimensional cellular automata we establish a rigorous upper bound on the number of distinct products that these interactions can generate the upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatio temporal information that a particle stores along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata thereby elucidating why that approach is of general utility the upper bound is tested against several relatively complex domain particle cellular automata and found to be tight

generalization of the hamilton ivey estimate to the higher dimensional ricci flow with a vanishing weyl tensor abstract in this study we consider curvature estimates of the ricci flow and we generalize the hamilton ivey pinching estimate to a higher dimensional ricci flow with a vanishing weyl tensor we also prove a pinching estimate for the ricci curvature under the ricci flow

on closed manifolds with harmonic weyl curvature abstract we derive point wise and integral rigidity gap results for a closed manifold with harmonic weyl curvature in any dimension in particular there is a generalization of tachibana s theorem for non negative curvature operator the key ingredients are new bochner weitzenbock lichnerowicz type formulas for the weyl tensor which are generalizations of identities in dimension four

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

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

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

the ricci flow under almost non negative curvature conditions we generalize most of the known ricci flow invariant non negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time as an illustration of the contents of the paper we prove that metrics whose curvature operator has eigenvalues greater than 1 can be evolved by the ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than c here the time of existence and the constant c only depend on the dimension and the degree of non collapsedness we obtain similar generalizations for other invariant curvature conditions including positive biholomorphic curvature in the kahler case we also get a local version of the main theorem as an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest including a classification for non collapsed manifolds with almost non negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds we also obtain a short time existence result for the ricci flow on open manifolds with almost non negative curvature without requiring upper curvature bounds

rigidity of einstein manifolds with positive scalar curvature we prove that if m n n ge 4 is a compact einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition r 0 sigma n k max where sigma n in frac 1 4 1 is an explicit positive constant depending only on n then m must be isometric to a spherical space form moreover we prove that if an n ge 4 dimensional compact einstein manifold satisfies k min ge eta n r 0 where eta n in frac 1 4 1 is an explicit positive constant then m is locally symmetric it should be emphasized that the pinching constant eta n is optimal when n is even we then obtain some rigidity theorems for einstein manifolds under n 2 th ricci curvature and normalized scalar curvature pinching conditions finally we extend the theorems above to einstein submanifolds in a riemannian manifold and prove that if m is an n ge 4 dimensional compact einstein submanifold in the simply connected space form f n c with constant curvature c ge 0 and the normalized scalar curvature r 0 of m satisfies r 0 frac a n a n 4n 8 c h 2 where a n n 3 5n 2 8n and h is the mean curvature of m then m is isometric to a standard n sphere

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

ancient solutions to the ricci flow with pinched curvature we show that any ancient solution to the ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature

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

strongly nonnegative curvature we prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition their curvature operator can be modified with a 4 form to become positive semidefinite