stochastic modeling of stock price process induced from the conjugate heat equation abstract currency can be considered as a ruler for values of commodities then the price is the measured value by the ruler we can suppose that inflation and variation of exchange rate are caused by variation of the scale of the ruler in geometry variation of the scale means that the metric is time dependent the conjugate heat equation is the modified heat equation which satisfies the heat conservation law for the time dependent metric space we propose a new model of stock prices by using the stochastic process whose transition probability is determined by the kernel of the conjugate heat equation our model of stock prices shows how the volatility term is affected by inflation and exchange rate this model modifies the black scholes equation in light of inflation and exchange rate

positive scalar curvature with skeleton singularities we study positive scalar curvature on the regular part of riemannian manifolds with singular uniformly euclidean l infty metrics that consolidate gromov s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of einstein manifolds we show that in all dimensions edge singularities with cone angles le 2 pi along codimension 2 submanifolds do not affect the yamabe type in three dimensions we prove the same for more general singular sets which are allowed to stratify along 1 skeletons exhibiting edge singularities angles le 2 pi and arbitrary l infty isolated point singularities we derive as an application of our techniques positive mass theorems for asymptotically flat manifolds with analogous singularities

ricci measure for some singular riemannian metrics we define the ricci curvature as a measure for certain singular torsion free connections on the tangent bundle of a manifold the definition uses an integral formula and vector valued half densities we give relevant examples in which the ricci measure can be computed in the time dependent setting we give a weak notion of a ricci flow solution on a manifold

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

a formula for the core of an ideal the core of an ideal is the intersection of all its reductions for large classes of ideals i we explicitly describe the core as a colon ideal of a power of a single reduction and a power of i

a flow tangent to the ricci flow via heat kernels and mass transport abstract we present a new relation between the short time behavior of the heat flow the geometry of optimal transport and the ricci flow we also show how this relation can be used to define an evolution of metrics on non smooth metric measure spaces with ricci curvature bounded from below

lower bounds on ricci curvature and quantitative behavior of singular sets let yn denote the gromov hausdorff limit m n i stackrel d mathrm gh longrightarrow y n of v noncollapsed riemannian manifolds with mathrm ric m n i geq n 1 the singular set mathcal s subset y has a stratification mathcal s 0 subset mathcal s 1 subset cdots subset mathcal s where y in mathcal s k if no tangent cone at y splits off a factor k 1 isometrically here we define for all 0 0 r 1 the k th effective singular stratum mathcal s k eta r satisfying bigcup eta bigcap r mathcal s k eta r mathcal s k sharpening the known hausdorff dimension bound dim mathcal s k leq k we prove that for all y the volume of the r tubular neighborhood of mathcal s k eta r satisfies mathrm vol t r mathcal s k eta r cap b frac 1 2 y leq c n mathrm v eta r n k eta the proof involves a quantitative differentiation argument this result has applications to einstein manifolds let mathcal b r denote the set of points at which the c2 harmonic radius is r if also the m n i are kahler einstein with l2 curvature bound rm l 2 leq c then mathrm vol mathcal b r cap b frac 1 2 y leq c n mathrm v c r 4 for all y in the kahler einstein case without assuming any integral curvature bound on the m n i we obtain a slightly weaker volume bound on mathcal b r which yields an a priori lp curvature bound for all p 2 the methodology developed in this paper is new and is applicable in many other contexts these include harmonic maps minimal hypersurfaces mean curvature flow and critical sets of solutions to elliptic equations

improved estimate of the singular set of dir minimizing q valued functions via an abstract regularity result abstract in this note we prove an abstract version of a recent quantitative stratification principle introduced by cheeger and naber 2013 6 7 using this general regularity result paired with an e regularity theorem we provide a new estimate of the minkowski dimension of the set of higher multiplicity points of a dir minimizing q valued function the abstract principle is applicable to several other problems we recover recent results in the literature and we obtain also some improvements in more classical contexts

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

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

the largest non vanishing degree of graded local cohomology modules abstract this paper gives characterizations for the largest non vanishing degree of the local cohomology modules of a graded ring s in terms of a reduction of s and of the associated graded ring of an ideal i in terms of any reduction of i as a consequence this invariant can be computed explicitly for the associated graded ring of any ideal generated by a d sequence and of ideals having small analytic deviation

convex functionals and the stratification of the singular set of their stationary points abstract we prove partial regularity of stationary solutions and minimizers u from a set r n to a riemannian manifold n for the functional f x u u 2 d x the integrand f is convex and satisfies some ellipticity and boundedness assumptions we also develop a new monotonicity formula and an regularity theorem for such stationary solutions with no restriction on their images we then use the idea of quantitative stratification to show that the k th strata of the singular set of such solutions are k rectifiable

super ricci flows and improved gradient and transport estimates we introduce brownian motions on time dependent metric measure spaces proving their existence and uniqueness we prove contraction estimates for their trajectories assuming that the time dependent heat flow satisfies transport estimates with respect to every l p kantorovich distance p in 1 infty these transport estimates turn out to characterize super ricci flows introduced by sturm j funct anal 275 12 3504 3569 2015

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

stratification for the singular set of approximate harmonic maps the aim of this note is to extend the results in naber and valtorta ann math 2 185 131 227 https doi org 10 4007 annals 2017 185 1 3 2017 to the case of approximate harmonic maps more precisely we will proved that the singular strata mathcal s k u of an approximate harmonic map are k rectifiable and we will show effect bounds on the quantitative strata in the process we will simplify many of the arguments from naber and valtorta 2017 and in particular we produce a new main covering lemmas which vastly simplifies the older argument

regularity of einstein manifolds and the codimension 4 conjecture in this paper we are concerned with the regularity of noncollapsed riemannian manifolds m n g with bounded ricci curvature as well as their gromov hausdor limit spaces m n dj dgh x d where dj denotes the riemannian distance our main result is a solution to the codimension 4 conjecture namely thatx is smooth away from a closed subset of codimension 4 we combine this result with the ideas of quantitative stratication to prove a priori l q estimates on the full curvaturejrmj for all q v 0 and diam m d contains at most a nite number of dieomorphism classes a local version is used to show that noncollapsed 4 manifolds with bounded ricci curvature have a priori l 2 riemannian curvature estimates

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

displacement interpolations from a hamiltonian point of view abstract one of the most well known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the riemannian wasserstein metric and the ricci curvature lower bound of the underlying riemannian manifold there are also generalizations of this result to the finsler manifolds and manifolds with a ricci flow background in this paper we study displacement interpolations from the point of view of hamiltonian systems and give a unifying approach to the above mentioned results

ricci flow entropy and optimal transportation let a smooth family of riemannian metrics g tau satisfy the backwards ricer flow equation on a compact oriented n dimensional manifold m suppose two families of normalized n forms omega tau 0 and omega over bar tau 0 satisfy the forwaids in tau heat equation on m generated by the connection laplacian delta g t if these n forms represent two evolving distributions of particles over m the minimum root mean square distance w 2 omega t omega over tilde t t to transport the particles of omega t onto those of omega over tilde t is shown to be non increasing as a function of t without sign conditions on the curvature of mg t moreover this contractivity property is shown to characterize supersolutions to the ricci flow

the resolution of the generic residual intersection of a complete intersection the concept of residual intersection introduced by artin and nagata l in 1972 is a fruitful generalization of linkage as the following two examples attest let i be a strongly cohen macaulay ideal in a cohenmacaulay local ring r if z satisfies the condition g then huneke 6 proposition 4 31 has proved that the extended rees algebra r it t 1 is defined by an ideal which is obtained from i by way of residual intersection since the extended rees algebra is a deformation of both the rees algebra r it and the associated graded algebra gz i it contains considerable information about the analytic properties of i the definitions of residual intersection strongly cohen macaulay and g may be found in section 4 huneke 6 theorem 4 11 has also shown that the ideal i generated by the maximal order minors of a generic n x m matrix is a residual intersection of a generic codimension two cohen macaulay ideal since j is rather poorly behaved with respect to linkage s it is promising that it is close to a well understood ideal once we weaken the tie from linkage to residual intersection

ℒ optimal transportation for ricci flow we introduce the notion of l optimal transportation and use it to construct a natural monotonic quantity for ricci flow which includes a selection of other monotonicity results including some key discoveries of perelman 13 both related to entropy and to l length and a recent result of mccann and the author 11

on period maps that are open embeddings for certain complex projective manifolds such as k3 surfaces and their higher dimensional analogues the complex symplectic projective manifolds the period map takes values in a locally symmetric variety of type iv it is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor we explain that phenomenon and get our hands on the complementary divisor in terms of geometric data

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

residual intersections in cohen macaulay rings abstract residual intersection generalizes the notion of linkage the central questions in residual intersection theory are for a residual intersection j of i when is r j cohen macaulay and what is the canonical module for r j our main result gives answers to the two central questions in the case where r is merely cohen macaulay

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

a complex ball uniformization of the moduli space of cubic surfaces via periods of k3 surfaces in this paper we show that the moduli space of nodal cubic surfaces is iso morphic to a quotient of a 4 dimensional complex ball by an arithmetic subgroup of the unitary group this complex ball uniformization uses the periods of certain k3 surfaces which are naturally associated to cubic surfaces a similar uniformization is given for dif ferent covers of the moduli space corresponding to geometric markings of the picard group or a choice of a line on the surface we also give a detailed description of the boundary com ponents corresponding to singular surfaces

regularity and compactness of harmonic einstein equations let m x g be pointed riemannian manifold with mathrm vol b 1 x ge mathrm v 0 and satisfy harmonic einstein equation mathrm ric g nabla u otimes nabla u lambda g with lambda le n 1 where u m g rightarrow n h is a harmonic map to a fixed compact riemannian manifold n h then for any p 1 we prove the l p curvature estimate open image in new window as a consequence if n h has nonpositive sectional curvature we have mathrm ric le c n mathrm v n that means harmonic einstein equation automatically implies bounded ricci curvature provided nonpositive sectional curvature of n h let x x infty d u infty be the limit of a sequence of harmonic einstein manifolds m i x i g i u i we show that the singular set of x is closed and the convergence is smooth on the regular part we also prove an orbifold type compactness theorem of harmonic einstein equations with bounded int m mathrm rm n 2 without assuming nonpositive sectional curvature of n h which generalizes xu s compactness result in yiyan adv math 231 2 680 708 2012

rees algebras of ideals having small analytic deviation in this article we identify two large families of ideals of a cohen macaulay sometimes gorenstein local ring whose rees algebras are cohen macaulay our main results imply for example that if r m is a regular local ring and p is a prime ideal of r such that p n is unmixed for all n 1 then the rees algebra r pt is cohen macaulay if either dim r p 2 o dim r p 3 r p is cohen macaulay and r p is integrally closed

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

a polyhedron comparison theorem for 3 manifolds with positive scalar curvature the study of comparison theorems in geometry has a rich history in this paper we establish a comparison theorem for polyhedra in 3 manifolds with nonnegative scalar curvature answering affirmatively a dihedral rigidity conjecture by gromov for a large collections of polyhedra with interior non negative scalar curvature and mean convex faces we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding euclidean model unless it is isometric to a flat polyhedron

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

the cauchy riemann equations on product domains we establish the l 2 theory for the cauchy riemann equations on product domains provided that the cauchy riemann operator has closed range on each factor we deduce regularity of the canonical solution on p 1 forms in special sobolev spaces represented as tensor products of sobolev spaces on the factors of the product this leads to regularity results for smooth data

on the structure of almost einstein manifolds in this paper we study the structure of the limit space of a sequence of almost einstein manifolds which are generalizations of einstein manifolds roughly speaking such manifolds are the initial manifolds of some normalized ricci flows whose scalar curvatures are almost constants over space time in the l1 sense ricci curvatures are bounded from below at the initial time under the non collapsed condition we show that the limit space of a sequence of almost einstein manifolds has most properties which is known for the limit space of einstein manifolds as applications we can apply our structure results to study the properties of kahler manifolds

non explosion of diffusion processes on manifolds with time dependent metric we study the problem of non explosion of diffusion processes on a manifold with time dependent riemannian metric in particular we obtain that brownian motion cannot explode in finite time if the metric evolves under backwards ricci flow our result makes it possible to remove the assumption of non explosion in the pathwise contraction result established by arnaudon et al arxiv 0904 2762 to appear in sem prob as an important tool which is of independent interest we derive an ito formula for the distance from a fixed reference point generalising a result of kendall ann prob 15 1491 1500 1987

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

arithmetic torelli maps for cubic surfaces and threefolds it has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety we give a construction which works for cubics over an arithmetic base and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties this answers away from the prime 2 an old question of deligne and a recent question of kudla and rapoport

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

the compactified picard scheme of the compactified jacobian abstract let c be an integral projective curve in any characteristic given an invertible sheaf l on c of degree 1 form the corresponding abel map a l c j which maps c into its compactified jacobian and form its pullback map a l pic j 0 j which carries the connected component of 0 in the picard scheme back to the jacobian if c has at worst double points then a l is known to be an isomorphism we prove that a l always extends to a map between the natural compactifications pic j j and that the extended map is an isomorphism if c has at worst ordinary nodes and cusps

critical sets of elliptic equations given a solution u to a linear homogeneous second order elliptic equation with lipschitz coefficients we introduce techniques for giving improved estimates of the critical set u u x u x 0 as well as the first estimates on the effective critical set r u which roughly consists of points x such that the gradient of u is small somewhere on br x compared to the nonconstancy of u the results are new even for harmonic functions on n given such a u the standard first order stratification lk of u separates points x based on the degrees of symmetry of the leading order polynomial of u u x in this paper we give a quantitative stratification of u which separates points based on the number of almost symmetries of approximate leading order polynomials of u at various scales we prove effective estimates on the volume of the tubular neighborhood of each which lead directly to n 2 minkowski type estimates for the critical set of u with some additional regularity assumptions on the coefficients of the equation we refine the estimate to give new proofs of the uniform n 2 hausdorff measure estimate on the critical set and singular sets of u 2014 wiley periodicals inc

structure theory of singular spaces abstract in this paper we develop a structure theory of einstein manifolds or manifolds with lower ricci curvature bounds for certain singular spaces that arise as geometric limits of sequences of riemannian manifolds this theory generalizes the results that were obtained by cheeger colding and naber in the smooth setting in the course of the paper we will carefully characterize the assumptions that we have to impose on this sequence of riemannian manifolds in order to guarantee that the individual results hold an important aspect of our approach is that we don t need impose any ricci curvature bounds on the sequence of riemannian manifolds leading to the singular limit the ricci curvature bounds will only be required to hold on the regular part of the limit and we will not impose any synthetic curvature condition on its singular part the theory developed in this paper will have applications in the blowup analysis of certain geometric equations in which we study scales that are much larger than the local curvature scale in particular this theory will have applications in the study of ricci flows of bounded scalar curvature which we will describe in a subsequent paper

log sobolev inequality for the continuum sine gordon model we prove a multiscale generalisation of the bakry emery criterion for a measure to satisfy a log sobolev inequality our criterion relies on the control of an associated pde well known in renormalisation theory the polchinski equation it implies the usual bakry emery criterion but we show that it remains effective for measures which are far from log concave indeed using our criterion we prove that the massive continuum sine gordon model with beta 6 pi satisfies asymptotically optimal log sobolev inequalities for glauber and kawasaki dynamics these dynamics can be seen as singular spdes recently constructed via regularity structures but our results are independent of this theory

super ricci flows for weighted graphs abstract we present a notion of super ricci flow for time dependent finite weighted graphs a challenging feature is that these flows typically encounter singularities where the underlying graph structure changes our notion is robust enough to allow the flow to continue past these singularities as a crucial tool for this purpose we study the heat flow on such singular time dependent weighted graphs with changing graph structure we then give several equivalent characterizations of super ricci flows in terms of a discrete dynamic bochner inequality gradient and transport estimates for the heat flow and dynamic convexity of the entropy along discrete optimal transport paths the latter property can be used to show that our notion of super ricci flow is consistent with classical super ricci flows for manifolds or metric measure spaces in a discrete to continuum limit

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete