li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

first eigenvalue of the p laplacian under integral curvature condition abstract we give various estimates of the first eigenvalue of the p laplace operator on closed riemannian manifold with integral curvature conditions

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

fixed point theory for a class of generalized nonexpansive mappings in this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior

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

edelstein s method and fixed point theorems for some generalized nonexpansive mappings abstract a new condition for mappings called condition c which is more general than nonexpansiveness was recently introduced by suzuki t suzuki fixed point theorems and convergence theorems for some generalized nonexpansive mappings j math anal appl 340 2008 1088 1095 following the idea of kirk and massa theorem in w a kirk s massa remarks on asymptotic and chebyshev centers houston j math 16 1990 364 375 we prove a fixed point theorem for mappings with condition c on a banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact this covers a result obtained by suzuki t suzuki fixed point theorems and convergence theorems for some generalized nonexpansive mappings j math anal appl 340 2008 1088 1095 we also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of banach spaces satisfying property d consequently we extend the results in t suzuki fixed point theorems and convergence theorems for some generalized nonexpansive mappings j math anal appl 340 2008 1088 1095 to many other banach spaces

fixed point theorems and convergence theorems for some generalized nonexpansive mappings we introduce some condition on mappings the condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness we present fixed point theorems and convergence theorems for mappings satisfying the condition

reconstruction of dynamical and geometrical properties of chaotic attractors from threshold crossing interspike intervals we reconstruct the largest lyapunov exponent and fractal dimension of a chaotic attractor using thresholdcrossing interspike intervals alone we show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor we also give an explanation of this possibility based on the concept of instantaneous frequency s1063 651x 98 51207 x pacs number s 05 45 1b the problem of extracting information about a dynamical system from experimental data is of great importance for all those who study the systems of natural origin that may be thought of as black boxes the traditional way of obtaining experimental data is to fix the sampling stepdt and store the values of some state variable s t corresponding to the time moments idt the fundamental works 1 state that the delay method applied to such time series often allows one to reconstruct a set being topologically equivalent to the initial system attractor on the base of these works a series of methods have been developed allowing one to evaluate attractor dimension 2 and the largest lyapunov characteristic exponent lce 3 to detect unstable equilibrium points 4 and unstable periodic orbits 5 and finally to create in some cases global models 6 or different predictors 7 another way of getting experimental data which is often preferred by biologists is to generate spike train data 8 one imposes a certain condition on the value of state variable s t and records the interspike intervals isi ti between the time moments for which the condition is fulfilled there are two general methods for recording spike train data the first one is to generate a spike when s t crosses some threshold level u in one direction and measure isi as an alternative one may measure the time intervals between the local maxima of s t fig 1 a the values of ti can be treated as the times of trajectory return to the surface defined by the condition imposed to the initial s t the condition s t 5u means the definition of a plane in the phase space while the condition s t 50 s t being a realization x j t o f a dynamical system dx w dt5f

fixed point theorems for some generalized multivalued nonexpansive mappings abstract in this paper we introduce a condition on multivalued mappings which is a multivalued version of condition c defined by garcia falset et al 2011 3 it is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition our results generalize the results in lim 1974 lami dozo 1973 kirk and massa 1990 garcia falset et al 2011 dhompongsa et al 2009 and abkar and eslamian 2010 4 5 6 3 7 8 and many others

an ultrafilter approach to locally almost nonexpansive maps abstract let k be a bounded closed convex subset of a banach space x and suppose f k k is locally almost nonexpansive in the sense of nussbaum it is shown that the mapping id f is demiclosed if x is either uniformly convex or satisfies the opial property these facts are known but the ultrapower approach given here is new in fact we give ultrapower characterizations of locally almost nonexpansive maps and of the opial property finally we obtain a new demiclosedness result for the class of locally almost pointwise contractive mappings

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

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

an eigenvalue pinching theorem in 1958 lichnerowicz 7 showed that for a compact n dimensional riemannian manifold m whose ricci curvature is bounded below by n 1 the first non zero eigenvalue 21 of the laplacian satisfies 2 t n if in fact 21 n obata proved that m must be isometric to the standard sphere a natural question is do there exist constants c n 1 depending only on n such that if m is as above and c n n 21 n then m must be diffeomorphic to a sphere here by combining the works of gromov 3 berard and meyer 1 and grove and shiohama 4 we show

manifolds of positive ricci curvature with almost maximal volume 10 in this note we consider complete riemannian manifolds with ricci curvature bounded from below the well known theorems of myers and bishop imply that a manifold mn with ric n 1 satisfies diam mn diam sn l vol mn vol sn l it follows from ch that equality in either of these estimates can be achieved only if mn is isometric to sn 1 the natural conjecture is that a manifold mn with almost maximal diameter or volume must be a topological equivalent to sn with respect to diameter this is true only if mn satisfies some additional assumptions see an 0 gp e with respect to volume however no extra restriction is necesary

regularity of kahler ricci flows on fano manifolds in this paper we will establish a regularity theory for the kahler ricci flow on fano n manifolds with ricci curvature bounded in l p norm for some p n using this regularity theory we will also solve a long standing conjecture for dimension 3 as an application we give a new proof of the yau tian donaldson conjecture for fano 3 manifolds the results have been announced in 45

convergence of iterative algorithms for multivalued mappings in banach spaces abstract we show strong and weak convergence for mann iteration of multivalued nonexpansive mappings t in a banach space furthermore we give a strong convergence of the modified mann iteration which is independent of the convergence of the implicit anchor like continuous path z t t u 1 t t z t

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

dynamic reconstruction of chaotic systems from inter spike intervals using least squares support vector machines this work presents a methodology for dynamic reconstruction of chaotic systems from inter spike interval isi time series obtained via integrate and fire if models in this methodology least squares support vector machines lssvms have been employed for approximating the dynamic behaviors of the systems under investigation higher generalization capability and avoidance of local minima constitute the main reasons behind the choice of lssvms as the approximation tool simulation results have shown that established lssvm models possess great potential for the reconstruction of chaotic dynamics in other words they are able to estimate some dynamic invariants of the underlying chaotic systems as well as they can accurately predict short term evolution within the horizon of predictability moreover lssvm models maintain their reconstruction performance even in the case of the existence of noisy data

geometric and spectral estimates based on spectral ricci curvature assumptions we obtain a bonnet myers theorem under a spectral condition a closed riemannian manifold mn g for which the lowest eigenvalue of the ricci tensor is such that the schr odinger operator n 2 is positive has finite fundamental group as a continuation of our earlier results we obtain isoperimetric inequalities from a kato condition on the ricci curvature furthermore we obtain the kato condition for the ricci curvature under purely geometric assumptions

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

manifolds with lower ricci and l sectional curvature control suppose that a sequence of riemannian manifolds with ricci curvature k 2 converges to a gromov hausdorff limit x we show that if the amount of sectional curvature below k of the limiting manifolds approaches 0 in a suitable l 1 sense then x is an alexandrov space of curvature k as applications we present several generalizations of classical theorems

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

zhong yang type eigenvalue estimate with integral curvature condition we prove a sharp zhong yang type eigenvalue lower bound for closed riemannian manifolds with control on integral ricci curvature

harnack inequality heat kernel bounds and eigenvalue estimates under integral ricci curvature bounds abstract let m n g i j be a complete riemannian manifold we prove that for any p n 2 when k p 1 is small enough some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equation u t u a u log u on geodesic balls b o r in m with 0 r 1 we can also derive the gradient estimates for any solutions to the above nonlinear parabolic equation along the ricci flow on a closed manifold without any curvature conditions as its application we derive some parabolic type gradient estimates for a positive solution u x t of the heat equation u t u moreover our estimate is stronger than zhang and zhang s estimate see remark 2 in section 4 those results are generalizations of li yau hamilton li xu type gradient estimates under the integral ricci curvature bounds by utilizing the gradient estimates of the heat equation we obtain harnack inequalities the upper bound and the lower bound for the heat kernel eigenvalue estimate and the lower bound of green s function on riemannian manifolds under the integral ricci curvature bounds

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

local sobolev constant estimate for integral ricci curvature bounds abstract we obtain a local sobolev constant estimate for integral ricci curvature which enables us to extend several important tools such as the maximal principle the gradient estimate the heat kernel estimate and the l 2 hessian estimate to manifolds with integral ricci lower bounds without the non collapsing conditions

large manifolds with positive ricci curvature the main purpose of this paper is to show that an n dimensional manifold with ricci curvature greater or equal to n 1 which is close in the gromov hausdor topology to the unit n sphere has volume close to that of the sphere this shows the converse of the theorem in c1 namely together with c1 it shows that an n manifold with ricci curvature greater or equal to n 1 is close to the sphere if and only if the volume is close to that of the sphere in particular by p such a manifold is homeomorphic to a sphere further as an application of this and the result of c1 we prove a radius theorem saying that if an n manifold with ricci curvature greater or equal to n 1 has radius almost equal to then the volume is close to that of the sphere in order to obtain these results we further develop and apply the estimates of c1 whereas the main concern in c1 were with the large scale geometry the main concern of this paper is with the small scale geometry let n be the volume of the round n sphere sn with sectional curvature one

lim s theorems for multivalued mappings in cat 0 spaces let x be a complete cat 0 space we prove that if e is a nonempty bounded closed convex subset of x and t e k x a nonexpansive mapping satisfying the weakly inward condition i e there exists p e such that p 1 tx ie x x e 0 1 then t has a fixed point in banach spaces this is a result of lim on asymptotic centers and fixed points of nonexpansive mappings canad j math 32 1980 421 430 the related result for unbounded r trees is given

banach ultrapowers and multivalued nonexpansive mappings the banach ultrapower construction is applied in fixed point theory for multivalued mappings we introduce the notion of ultra asymptotic centers and use it to remove the separability assumption from the results of dominguez benavides lorenzo ramirez 2004 and dhompongsa kaewcharoen kaewkhao 2006

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

conformal metric sequences with integral bounded scalar curvature let m g be a smooth compact riemiannian manifold without boundary and g k be a metric conformal to g suppose text vol m g k vert r k vert l p m g k c where r k is the scalar curvature and p frac n 2 we will use the 3 circles theorem to study the bubble tree convergence of g k