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

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

effective category and measure in abstract complexity theory strong variants of the operator speed up theorem operator gap theorem and compression theorem are obtained using an effective version of baire category theorem it is also shown that all complexity classes of recursive predicates have effective measure zero in the space of recursive predicates and on the other hand the class of predicates with almost everywhere complexity above an arbitrary recursive threshold has recursive measure one in the class of recursive predicates

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

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

sensitivity analysis for boundary element error estimation and mesh refinement the subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points the direct differentiation approach is considered here and the analysis is performed analytically since only the collocation points are perturbed the shape of the body and the corresponding discretization remain unaltered this aspect makes the present work quite different in spirit with respect to earlier analyses on shape sensitivities sensitivities of approximate bem solutions with respect to the positions of collocation points are shown to be related to the residual of hypersingular integral equations numerical results confirm that the present approach can be seen as the analytical counterpart of an adaptive scheme for mesh refinement presented by the same author in some recent papers some other advantages of the present approach over the former one are also outlined

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

a new boundary element technique without domain integrals for elastoplastic solids summary a simple idea is proposed to solve boundary value problems for elastoplastic solids via boundary elements namely to use the green s functions corresponding to both the loading and unloading branches of the tangent constitutive operator to solve for plastic and elastic regions respectively in this way domain integrals are completely avoided in the boundary integral equations though a discretization of the region where plastic flow occurs still remains necessary to account for the inhomogeneity of plastic deformation the elastoplastic analysis reduces in essence to a straightforward adaptation of techniques valid for anisotropic linear elastic constitutive equations the loading branch of the elastoplastic constitutive operator may be viewed formally as a type of anisotropic elastic law numerical examples using j2 flow theory with linear hardening demonstrate that the proposed method retains all the advantages related to boundary element formulations is stable and performs well the method presented is for simplicity developed for the associative flow rule however a full derivation of green s function and boundary integral equations is also given for the general case of non associative flow rule it is shown that in the non associative case a domain integral unavoidably arises in the formulation copyright 2005 john wiley sons ltd

sensitivity analysis and optimal design in charge transport problems the focus of this paper is the accurate and efficient determination of sensitivities of electrostatic potential and charge density in charge transport problems and the use of these sensitivities to carry out optimal design direct differentiations of the boundary integral formulation of poisson s equation for charge conservation and of the non linear partial differential equation for current continuity are carried out to obtain equations satisfied by the sensitivities methods for solving the sensitivity equations are discussed a numerical implementation of the methods is validated through several examples it is demonstrated that the design sensitivity coefficients dscs of the quantities of interest in charge transport are obtained accurately and that optimal design problems can be solved efficiently using these dscs

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

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

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

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

design sensitivity coefficients for elasto viscoplastic problems by boundary element methods this paper is concerned with accurate and efficient determination of design sensitivity coefficients dscs for solid mechanics problems with small strains and rotations but with material non linearities present elasto plastic or elasto viscoplastic problems this approach is based on direct differentiation dda of the relevant derivative boundary element method dbem formulation of the problem analytical differentiation of the dbem equations leads to singular integral equations for the dscs with weakly logarithmically for 2 d singular kernels which are easy to deal with also stress components and their sensitivities are obtained on the boundary of a body with great accuracy these quantities are typically difficult to obtain accurately from finite element methods fem a computer program for general two dimensional plane strain and plane stress problems has been developed based on the above formulation numerical results are presented for some sample problems and these are compared against direct solutions the agreement between the dbem and direct solutions is excellent for these examples

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

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

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

turning point solutions for thin shell vibrations abstract asymptotic solutions of the equations representing the free vibration of thin shells of revolution with a first order turning point are developed these solutions are uniform and fully approximate namely satisfy the accuracy of the theory of thin shells three categories of the generalized function are defined h h 1 2 3 4 and j in terms of which a singular membrane solution and four bending solutions can be expressed respectively in particular the second category of the generalized function is first obtained which is a generalization of the new solution of the related equation this new solution is found from the modification of the laplace transform method

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

on the topological size of p m complete degrees abstract all polynomial many one degrees are shown to be of second baire category in the superset topology when witness functions are allowed to run in 2loghn time for any h any improvement of this result for the complete p m degrees of re exp or np implies p np or the nonisomorphism of the np complete sets

flaw identification using the boundary element method in this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two dimensional structures an introduction to inverse analysis is given with special reference to methods of flaw identification along with a brief review of the optimization methods employed both the standard boundary element and the dual boundary element method are presented with the dual boundary element method proposed as the basis for the new formulation the flaw identification method is presented along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location

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

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

if not empty np p is topologically large abstract in the classical cantor topology or in the superset topology np and consequently classes included in np are meagre however in a natural combination of the two topologies we prove that np p if not empty is a second category class while np complete sets form a first category class these results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies p immune sets in np np simple sets p bi immune sets and np effectively simple sets are all second category if not empty it is shown that if c is any of the above second category classes then for all b np there exists an a c such that a is arbitrarily close to b infinitely often

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

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

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

category and measure in complexity classes this paper presents resource bounded category and resource bounded measure two new tools for computational complexity theory and some applications of these tools to the structure theory of exponent

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

optimum shape design and positioning of features using the boundary integral equation method a design optimization procedure is developed using the boundary integral equation bie method for linear elastostatic two dimensional domains optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure a fully analytical approach is adopted for the design sensitivity analysis where the bie is implicitly differentiated the ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables how the advantages of the bie method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted a mathematical programming approach with the penalty function method is used for solving the overall optimization problem the procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures

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

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

is independence an exception abstract godel s incompleteness theorem asserts that any sufficiently rich sound and recursively axiomatizable theory is incomplete we show that in a quite general topological sense incompleteness is a rather common phenomenon with respect to any reasonable topology the set of true and unprovable statements of such a theory is dense and in many cases even corare

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