collapsing limits of the kahler ricci flow and the continuity method we consider the kahler ricci flow on certain calabi yau fibration which is a calabi yau fibration with one dimensional base or a product of two calabi yau fibrations with one dimensional bases assume the kahler ricci flow on total space admits a uniform lower bound for ricci curvature then the flow converges in gromov hausdorff topology to the metric completion of the regular part of generalized kahler einstein current on the base which is a compact length metric space homeomorphic to the base the analogue results for the continuity method on such calabi yau fibrations are also obtained moreover we show the continuity method starting from a suitable kahler metric on the total space of a fano fibration with one dimensional base converges in gromov hausdorff topology to a compact metric on the base during the proof we show the metric completion of the regular part of a generalized kahler einstein current on a riemann surface is compact

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

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

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

design of a chatter free terminal sliding mode controller for nonlinear fractional order dynamical systems this paper investigates the problem of robust control of nonlinear fractional order dynamical systems in the presence of uncertainties first a novel switching surface is proposed and its finite time stability to the origin is proved subsequently using the sliding mode theory a robust fractional control law is proposed to ensure the existence of the sliding motion in finite time we use a fractional lyapunov stability theory to prove the stability of the system in a given finite time in order to avoid the chattering which is inherent in conventional sliding mode controllers we transfer the sign function of the control input into the fractional derivative of the control signal the proposed chattering free sliding mode technique is then applied for stabilisation of a broad class of three dimensional fractional order chaotic systems via a single variable driving control input simulation results reveal that the proposed fractional sliding mode controller works well for chaos control of fractional ord

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

analysis of a new chaotic system abstract this paper reports a new three dimensional continuous quadratic autonomous chaotic system modified from the lorenz system in which each equation contains a single quadratic cross product term which is different from the lorenz rossler chen lu systems basic properties of the new system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics

a generalized tetrahedral property we present examples of metric spaces that are not riemannian manifolds nor dimensionally homogeneous that satisfy sormani s tetrahedral property we then note that euclidean cones over metric spaces with small diameter do not satisfy this property therefore we extend the tetrahedral property to a less restrictive one and prove that this generalized definition retains all the results of the original tetrahedral property proven by portegies sormani it provides a lower bound on the sliced filling volume and a lower bound on the volumes of balls thus sequences with uniform bounds on this generalized tetrahedral property also have subsequences which converge in both the gromov hausdorff and sormani wenger intrinsic flat sense to the same noncollapsed and countably rectifiable limit space

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

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

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

interpolated measures with bounded density in metric spaces satisfying the curvature dimension conditions of sturm abstract we construct geodesics in the wasserstein space of probability measures along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic using these geodesics we show that a local poincare inequality and the measure contraction property follow from the ricci curvature bounds defined by sturm we also show for a large class of convex functionals that a local poincare inequality is implied by the weak displacement convexity of the functional

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

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

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

isoperimetry for semilinear torsion problems in riemannian two manifolds abstract given m 2 g a smooth complete riemannian 2 manifold this note discusses three types of isoperimetric properties arising from u z o g o g z u d a g the non negative solution to the semilinear 0 1 torsion problem g u u in o subject to u o 0 for a relatively compact domain o m 2 with smooth boundary o

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

approximations of lipschitz maps via immersions and differentiable exotic sphere theorems abstract as our main theorem we prove that a lipschitz map from a compact riemannian manifold m into a riemannian manifold n admits a smooth approximation via immersions if the map has no singular points on m in the sense of f h clarke where dim m dim n as its corollary we have that if a bi lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense then they are diffeomorphic we have three applications of the main theorem the first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones the third one is that a compact n manifold m is a twisted sphere and there exists a bi lipschitz homeomorphism between m and the unit n sphere s n 1 which is a diffeomorphism except for a single point if m satisfies certain two conditions with respect to critical points of its distance function in the clarke sense moreover we have three corollaries from the third theorem the first one is that for any twisted sphere n of general dimension n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for a single point in particular there exists such a map between an exotic n sphere n of dimension n 4 and s n 1 the second one is that if an exotic 4 sphere 4 exists then 4 does not satisfy one of the two conditions above the third one is that for any grove shiohama type n sphere n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for one of points that attain their diameters

expanding solitons with non negative curvature operator coming out of cones we consider ricci flow of complete riemannian manifolds which have bounded non negative curvature operator non zero asymptotic volume ratio and no boundary we prove scale invariant estimates for these solutions using these estimates we show that there is a limit solution obtained by scaling down this solution at a fixed point in space this limit solution is an expanding soliton coming out of the asymptotic cone at infinity

large time behavior of the heat kernel in this paper we study the large time behavior of the heat kernel on complete riemannian manifolds with nonnegative ricci curvature which was studied by p li with additional maximum volume growth assumption following y ding s original strategy by blowing down the metric using cheeger and colding s theory about limit spaces of gromov hausdorff convergence combining with the gaussian upper bound of heat kernel on limit spaces we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure as one application we get the consistent large time limit of heat kernel in more general context which generalizes the former result of p li furthermore by choosing different sequences to blow down the suitable metric we show the first example manifold whose heat kernel has inconsistent limit behavior which answers an open question posed by p li negatively

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

rigidity of amalgamated product in negative curvature let gamma be the fundamental group of a compact n dimensional riemannian manifold x of sectional curvature bounded above by 1 we suppose that gamma is a free product of its subgroup a and b over the amalgamated subgroup c we prove that the critical exponent delta c of c satisfies delta c geq n 2 the equality happens if and only if there exist an embedded compact hypersurface y in x totally geodesic of constant sectional curvature 1 with fundamental group c and which separates x in two connected components whose fundamental groups are a and b similar results hold if gamma is an hnn extension or more generally if gamma acts on a simplicial tree without fixed point

chaos synchronization for a class of nonlinear oscillators with fractional order abstract in this paper a special kind of nonlinear chaotic oscillator the qi oscillator is studied in detail since such systems are shown to possess a relatively wide spectral bandwidth it is considerably beneficial to practical engineering in the secure communication field the chaos synchronization problem of the fractional order qi oscillators coupled in a master slave pattern is examined by applying three different kinds of methods the nonlinear feedback method the one way coupling method and the method based on the state observer suitable synchronization conditions are derived by using the lyapunov stability theory and most importantly a sufficient and necessary synchronization condition for the case with fractional order between 1 and 2 is presented results of numerical simulations validate the effectiveness and applicability of the proposed schemes

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

free vs locally free kleinian groups we prove that kleinian groups whose limit sets are cantor sets of hausdorff dimension 0 examples of non free purely hyperbolic kleinian groups whose limit set is a cantor set of hausdorff dimension 1 epsilon

on the tangent cone of kahler manifolds with ricci curvature lower bound let x be the gromov hausdorff limit of a sequence of pointed complete kahler manifolds m n i p i satisfying ric m i ge n 1 and the volume is noncollapsed we prove that there exists a lie group isomorphic to mathbb r acting isometrically on the tangent cone at each point of x moreover the action is locally free on the cross section this generalizes the metric cone theorem of cheeger colding to the kahler case we also discuss some applications to complete kahler manifolds with nonnegative bisectional curvature

riemannian ricci curvature lower bounds in metric measure spaces with finite measure in prior work 4 of the first two authors with savare a new riemannian notion of lower bound for ricci curvature in the class of metric measure spaces x d m was introduced and the corresponding class of spaces denoted by rcd k this notion relates the cd k n theory of sturm and lott villani in the case n to the bakry emery approach in 4 the rcd k property is defined in three equivalent ways and several properties of rcd k spaces including the regularization properties of the heat flow the connections with the theory of dirichlet forms and the stability under tensor products are provided in 4 only finite reference measures m have been considered the goal of this paper is twofold on one side we extend these results to general finite spaces on the other we remove a technical assumption appeared in 4 concerning a strengthening of the cd k condition this more general class of spaces includes euclidean spaces endowed with lebesgue measure complete noncompact riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower ricci curvature bounds

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

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

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

kahler non collapsing eigenvalues and the calabi flow abstract we first prove a new compactness theorem of kahler metrics which confirms a prediction in 17 then we establish several eigenvalue estimates along the calabi flow combining the compactness theorem and these eigenvalue estimates we generalize the method developed for the kahler ricci flow in 22 to obtain several new small energy theorems of the calabi flow

relating notions of convergence in geometric analysis abstract we relate l p convergence of metric tensors or volume convergence to a given smooth metric to intrinsic flat and gromov hausdorff convergence for sequences of riemannian manifolds we present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal g j f j 2 g 0 to a fixed manifold we then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold m are bounded 1 1 j g 0 g j k g 0 and either the volumes converge vol j m vol 0 m or the metric tensors converge in the l p sense then the riemannian manifolds m g j converge in the measured gromov hausdorff and volume preserving intrinsic flat sense to m g 0

escaping geodesics in riemannian surfaces with variable negative curvature abstract in this paper we give a lower bound for the visual hausdorff dimension of the geodesics escaping through different ends of riemannian surfaces with pinched negative curvature this allows to show that in any riemannian surface with pinched negative curvature and infinite area there is a large set of geodesics escaping to infinity

irreducible representations of infinite dimensional lie algebras abstract some results concerning the operatorially irreducible topologically irreducible operatorially and topologically irreducible representations of a class of infinite dimensional lie algebras by bounded operators in a complex banach space are obtained

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

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

twisted and conical kahler ricci solitons on fano manifolds in this paper we first study the relationship between the existence of twisted kahler ricci solitons and the properness of modified twisted k energy approximating by a sequence of smooth twisted kahler ricci solitons we obtain an existence result of conical kahler ricci solitons

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

joint spectral radius operator semigroups and a problem of w wojtynski abstract we investigate the connections between the invariant subspace problem for operator semigroups and the joint spectral radius as a consequence it is proved that any quasinilpotent lie algebra of compact operators on a banach space is triangularizable we extend the berger wang formula to precompact sets of essentially scalar operators and prove the continuity of the joint spectral radius on them

invariant subspaces of operator lie algebras and lie algebras with compact adjoint action it is proved that a lie algebra of compact operators with a non zero volterra ideal is reducible has a nontrivial invariant subspace a number of other criteria of reducibility for collections of operators is obtained the results are applied to the structure theory of lie algebras of compact operators and normed lie algebras with compact adjoint action

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

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

an asymptotic formula for the commutators in what follows we shall describe some asymptotic formulas for the commutators defined by the values of the analytic functional calculus of some commutingn tuple of bounded operators in a complex banach space

jost maps ball homogeneous and harmonic manifolds abstract given a real number e 0 small enough an associated jost map j e between two riemannian manifolds is defined then we prove that connected riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball i e for which the identity is a j e map are ball homogeneous in the analytic case we characterize such manifolds in terms of the euclidean laplacian and we show that they have constant scalar curvature under some restriction on the ricci curvature we prove that riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic

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

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

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

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