volume comparison without curvature assumption let x n m n be any pair of compact connected riemannian manifolds the only assumption on m n is that the gromov hausdorff distance varepsilon between m n and x n is smaller than the normalized injectivity radius of x n using a transport of measures and a new notion of barycentre we construct a c 1 gromov hausdorff c 0 cdot varepsilon 2 3 approximation h m n rightarrow x n whose energy and jacobian determinant are sharply bounded from above by 1 c 1 cdot varepsilon 1 3 the constants c i are explicit implying a sharp lower bound for the ratio between the volumes of m n and x n this result extends to the non compact case when x n and m n are only supposed to be quasi isometric

bounds on the kth eigenvalues of trees and forests abstract we prove the upper bound k t n k 1 for the kth largest eigenvalue of a tree t 1 k n 2 and show that this bound is best possible when n 0 mod k we further obtain the strict inequality t 1 k 1 t 1 k t t 1 where t n k for the case n 0 mod k 2 k n 2 our upper bound is also proved to be the best possible upper bound for the kth largest eigenvalues of the forests in all cases

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

euclidean spaces as weak tangents of infinitesimally hilbertian metric measure spaces with ricci curvature bounded below we show that in any infinitesimally hilbertian cd k n space at almost every point there exists a euclidean weak tangent i e there exists a sequence of dilations of the space that converges to a euclidean space in the pointed measured gromov hausdorff topology the proof follows by considering iterated tangents and the splitting theorem for infinitesimally hilbertian cd 0 n spaces

analysis and geometry on manifolds with integral ricci curvature bounds ii we extend several geometrical results for manifolds with lower ricci curvature bounds to situations where one has integral lower bounds in particular we generalize colding s volume convergence results and extend the cheeger colding splitting theorem

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

on the kth laplacian eigenvalues of trees with perfect matchings abstract let t n be the set of all trees of order n with perfect matchings in this paper we prove that for any tree t t n its k th largest laplacian eigenvalue k t satisfies k t 2 when n 2 k and k t n 2 k 2 n 2 k 2 4 2 when n 2 k moreover this upper bound is sharp when n 0 mod 2 k

a new notion of angle between three points in a metric space we give a new notion of angle in general metric spaces more precisely given a triple a points p x q in a metric space x d we introduce the notion of angle cone pxq as being an interval pxq pxq pxq where the quantities pxq are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally lipschitz functions on a general metric space our definition in the euclidean plane gives the standard angle between three points and in a riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q we show that in general the angle cone is not single valued even in case the metric space is a smooth riemannian manifold if x is in the cut locus of p or q but if we endow the metric space with a positive borel measure m obtaining the metric measure space x d m then under quite general assumptions which include many fundamental examples as riemannian manifolds finite dimensional alexandrov spaces with curvature bounded from below gromov hausdorff limits of riemannian manifolds with ricci curvature bounded from below and normed spaces with strictly convex norm fixed p q x the angle cone at x is single valued for m a e x x we prove some basic properties of the angle cone such as the invariance under homotheties of the space and we analyze in detail the case x d m is a measured gromov hausdorff limit of a sequence of riemannian manifolds with ricci curvature bounded from below showing the consistency of our definition with a recent construction of honda 24

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

poincare type inequalities and finding good parameterizations a very important question in geometric measure theory is how geometric features of a set translate into analytic information about it reifenberg bull am math soc 66 312 313 1960 proved that if a set is well approximated by planes at every point and at every scale then the set is a bi holder image of a plane it is known today that carleson type conditions on these approximating planes guarantee a bi lipschitz parameterization of the set in this paper we consider an n ahlfors regular rectifiable set m subset mathbb r n d that satisfies a poincare type inequality involving lipschitz functions and their tangential derivatives then we show that a carleson type condition on the oscillations of the tangent planes of m guarantees that m is contained in a bi lipschitz image of an n plane we also explore the poincare type inequality considered here and show that it is in fact equivalent to other poincare type inequalities considered on general metric measure spaces

volume comparison in the presence of a gromov hausdorff varepsilon approximation i let m g be any compact connected riemannian manifold of dimension n we use a transport of measures and the barycentre to construct a map from m g onto a flat torus mathbb r n lambda g 0 in such a way that its jacobian is sharply bounded from above we make no assumptions on the topology of m g and on its curvature and geometry but we only assume the existence of a measurable gromov hausdorff varepsilon approximation between mathbb r n lambda g 0 and m g when the hausdorff approximation is continuous with non vanishing degree this leads to a sharp volume comparison rm vol m g geq left 1 37 n 2 2 frac varepsilon rm inj mathbb r n lambda g 0 right frac n 2 vert rm deg h vert rm vol mathbb r n lambda g 0 where rm inj mathbb r n lambda g 0 is the injectivity radius of the flat torus notice that the above inequality is valid for any varepsilon frac rm inj mathbb r n lambda g 0 37 n 2 2

on the convergence rate in the central limit theorem for associated processes on donne des taux de convergence uniforme dans le theoreme central limite pour des processus associes au troisieme moment fini la stationarite n etant pas requise

volume comparison and its applications in riemann finsler geometry abstract we introduce a new geometric quantity the mean covariation for finsler metrics and establish a volume comparison theorem as an application we obtain some precompactness and finiteness theorems for finsler manifolds

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

three circles theorems for harmonic functions we proved two three circles theorems for harmonic functions on manifolds in integral sense as one application on manifold with nonnegative ricci curvature whose tangent cone at infinity is the unique metric cone with unique conic measure we showed the existence of nonconstant harmonic functions with polynomial growth this existence result recovered and generalized the former result of ding and led to a complete answer of ni s conjecture furthermore in similar context combining the techniques of estimating the frequency of harmonic functions with polynomial growth which were developed by colding and minicozzi we confirmed their conjecture about the uniform bound of frequency

negative association of random variables with applications random variables x 1 cdots x k are said to be negatively associated na if for every pair of disjoint subsets a 1 a 2 of 1 2 cdots k operatorname cov lbrack f x 1 i in a 1 g x j j in a 2 rbrack leq 0 for all nondecreasing functions f g the basic properties of negative association are derived especially useful is the property that nondecreasing functions of mutually exclusive subsets of na random variables are na this property is shown not to hold for several other types of negative dependence recently proposed one consequence is the inequality p x i leq x i i 1 cdots k leq prod k 1p x i leq x i for na random variables x 1 cdots x k and the dual inequality resulting from reversing the inequalities inside the square brackets in another application it is shown that negatively correlated normal random variables are na other na distributions are the a multinomial b convolution of unlike multinomials c multivariate hypergeometric d dirichlet and e dirichlet compound multinomial negative association is shown to arise in situations where the probability measure is permutation invariant applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures in a somewhat striking example na and positive association representing quite strong opposing types of dependence are shown to exist side by side in models of categorical data analysis

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

note on the kth laplacian eigenvalues of trees with perfect matchings abstract denote by k t the k th laplacian eigenvalue of tree t let t n be the set of all trees of order n with perfect matchings in this paper we characterize the tree t in t 4 t with 2 t t 2 t 2 4 2 and the tree in t 6 t with 3 t t 2 t 2 4 2 these results affirm problem of j x li w c shiu a chang 2010 5 for the cases k 2 and k 3 respectively

embeddings of riemannian manifolds with heat kernels and eigenfunctions we show that any closed n dimensional riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points both this time and this number can be bounded in terms of the dimension a lower bound on the ricci curvature the injectivity radius and the volume it follows that the manifold can be embedded by a finite number of eigenfunctions of the laplace operator again this number only depends on the geometric bounds and the dimension in addition both maps can be made arbitrarily close to an isometry in the appendix we derive quantitative estimates of the harmonic radius so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well 2016 wiley periodicals inc

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

weak convergence for weighted empirical processes of dependent sequences in this paper we establish weak convergence theorems for weighted empirical processes of strong mixing p mixing and associated sequences we apply these results to obtain weak convergence of integral functionals of empirical processes and of mean residual life processes in reliability theory to carry out the proofs we develop two rosenthal type inequalities for strong mixing and associated sequences

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

a spectra comparison theorem and its applications we give a sharp comparison between the spectra of two riemannian manifolds y g and x g 0 under the following assumptions x g 0 has bounded geometry y g admits a continuous gromov hausdorff varepsilon approximation onto x g 0 of non zero absolute degree and the volume of y g is almost smaller than the volume of x g 0 these assumptions imply no restrictions on the local topology or geometry of y g in particular no curvature assumption is supposed or inferred

precise asymptotics in the baum katz and davis laws of large numbers let x x1 x2 be a sequence of i i d random variables such that ex 0 let z be a random variable possessing a stable distribution g with exponent 1 2 assume the distribution of x is attracted to g and set sn x1 xn we prove that n 1nr p 2p sn en1 p e p p r p 1 pr pe z p p r p 1 as e 0 for 1 p r under the additional assumption that there is normal attraction to g and that n 11np sn en1 p p p logeas e 0 for 1 p we close with a related result for the limiting case r p 2

local spectral convergence in rcd k n spaces abstract in this paper we give necessary and sufficient conditions for the validity of the local spectral convergence in balls in the setting of rcd spaces

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

on the strong convergence rate for positively associated random variables abstract for sequences of the positively associated random variables which are strictly weaker than the classical associated random ones introduced by esary et al 1967 7 strong convergence rate n 1 2 log n 1 2 is obtained which reaches the available one for independent random variables in terms of berstein type inequality further we give the corresponding precise asymptotics with respect to the rate mentioned above which extend and improve the relevant results in fu 2009 8

central limit theorems for associated random variables and the percolation model we prove a central limit theorem for families x n n mathbf n in lambda n of associated random variables indexed by subsets lambda n of mathbb z d as n rightarrow infty this is an extension of the newman wright invariance principle for associated stationary sequences x n n geq 1 satisfying sum n operatorname cov x 1 x n infty but with the stationarity property replaced by conditions on the moments of the x s the theorem has applications to the voter model and the percolation model in the latter case it provides an extension of a central limit theorem of the authors 4 by reducing the severity of the moment conditions also we prove a central limit theorem for certain non stationary non associated families of random variables which arise in percolation theory this includes for example a central limit theorem for the number of open clusters contained within the circuit gamma n of mathbb z 2 where gamma n is a sequence of circuits which satisfy a regularity condition and whose interiors overset circ gamma n satisfy overset circ gamma n rightarrow infty as n rightarrow infty

exact rates in log law for positively associated random variables abstract let x n n 1 be a strictly stationary sequence of positively associated random variables with mean zero and finite variance set s n k 1 n x k m n max k n s k n 1 suppose 2 e x 1 2 2 k 2 e x 1 x k in this paper we study the exact convergence rates of a kind of weighted infinite series of p m n e n log n p s n e n log n and i s n e n log n as e 0 respectively

proof of a problem on laplacian eigenvalues of trees abstract denote by k l t the k th laplacian eigenvalue of a tree t let t k 2 t be the set of all trees of order 2 t k with perfect matchings in this note the trees t in t k 2 t with k l t t 2 t 2 4 2 are characterized which solves problem of j x li w c shiu and a chang in 3 completely

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

associated random variables and martingale inequalities many of the classical submartingale inequalities including doob s maximal inequality and upcrossing inequality are valid for sequences sj such that the sj 1 sj s are associated positive mean random variables and for more general demisubmartingales the demisubmartingale maximal inequality is used to prove weak convergence to the two parameter wiener process of the partial sum processes constructed from a stationary two parameter sequence of associated random variables xijwith mathop sum limits i mathop sum limits j text cov left x text oo x ij right infty

on the exponential inequalities for negatively dependent random variables a number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors the proofs use the truncation technique together with the control of the bounded terms and unbounded terms in this paper we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables our results improve on the corresponding ones in the literature

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

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

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry