a note on robinson ursescu and lyusternik graves theorem the aim of this note is twofold first we prove an analogue of the well known robinson ursescu theorem on the relative openness with a linear rate restrictive metric regularity of a multivalued mapping second we prove a generalization of graves open mapping theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings the approximated non linear mapping is restricted to a closed convex subset of a banach space

the index of the scattering operator on the positive spectral subspace we construct the scattering operator for a spinor field in a time dependent background by the dyson expansion then we show that the restriction of the scattering operator to the positive spectral subspace with respect to a reference hamiltonian is fredholm the computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero we obtain the index in terms of a cohomological formula by means of the atiyah singer index theorem

eigenfunctions of the laplacian on rotationally symmetric manifolds eigenfunctions of the laplacian on a negatively curved rotationally symmetric manifold m rn ds2 ds2 dr2 f r 2d 2 are constructed explicitly under the assumption that an integral of f r converges this integral is the same one which gives the existence of nonconstant harmonic functions on m

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

on primal regularity estimates for set valued mappings abstract we prove several generalizations of the results in 6 for set valued mappings in some cases we improve also the statements for single valued mappings linear openness of the set valued mapping in question is deduced from the properties of its suitable approximation this approach goes back to the classical lyusternik graves theorem saying that a continuously differentiable single valued mapping between banach spaces is linearly open around an interior point of its domain provided that its derivative at this point is surjective in this paper we consider approximations given by a graphical derivative a contingent variation a strict pseudo h derivative and a bunch of linear mappings

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

keller osserman conditions for diffusion type operators on riemannian manifolds abstract in this paper we obtain essentially sharp generalized keller osserman conditions for wide classes of differential inequalities of the form l u b x f u l u and l u b x f u l u g u h u on weighted riemannian manifolds where l is a non linear diffusion type operator prototypical examples of these operators are the p laplacian and the mean curvature operator the geometry of the underlying manifold is reflected via bounds for the modified bakry emery ricci curvature by growth conditions for the functions b and l a weak maximum principle which extends and improves previous results valid for the laplacian is also obtained geometric comparison results valid even in the case of integral bounds for the modified bakry emery ricci tensor are presented

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

curves of descent steepest descent is central in variational mathematics we present a new transparent existence proof for curves of near maximal slope an influential notion of steepest descent in a nonsmooth setting the existence theory is further amplified for semialgebraic functions prototypical nonpathological functions in nonsmooth optimization such functions always admit nontrivial descent curves emanating from any even critical nonminimizing point we moreover show that curves of near maximal slope of semialgebraic functions have a more classical description as solutions of subgradient dynamical systems

existence and non existence of area minimizing hypersurfaces in manifolds of non negative ricci curvature we study minimal hypersurfaces in manifolds of non negative ricci curvature euclidean volume growth and quadratic curvature decay at infinity by comparison with capped spherical cones we identify a precise borderline for the ricci curvature decay above this value no complete area minimizing hypersurfaces exist below this value in contrast we construct examples

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

eigenvalues estimates on complete noncompact riemannian manifolds and applications we obtain some sharp estimates on the first eigenvalues of complete noncom pact riemannian manifolds under assumptions of volume growth using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures

sobolev spaces adapted to the schrodinger operator with inverse square potential we study the l p theory for the schrodinger operator mathcal l a with inverse square potential a x 2 our main result describes when l p based sobolev spaces defined in terms of the operator mathcal l a s 2 agree with those defined via delta s 2 we consider all regularities 0 s 2 in order to make the paper self contained we also review with proofs multiplier theorems littlewood paley theory and hardy type inequalities associated to the operator mathcal l a

on poincare type inequalities using estimates of the heat kernel we prove a poincare inequality for star shape domains on a complete manifold the method also gives a lower bound for the gap of the first two neumann eigenvalues of a schrodinger operator

heat extensions optimal atomic decompositions and sobolev embeddings in presence of symmetries on manifolds our purpose is twofold we give the heat semi group characterization of spaces of besov and triebel lizorkin types defined on riemannian manifolds of bounded geometry moreover we study boundedness and compactness of sobolev embeddings of the spaces in presence of symmetries to make it possible we first construct an optimal atomic decomposition for the above function spaces

local index theory over etale groupoids we give a superconnection proof of connes index theorem for proper cocompact actions of etale groupoids this includes connes general foliation index theo rem for foliations with hausdorholonomy groupoid

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

the trotter kato product formula for gibbs semigroups the trace norm convergence of the trotter lie product formula has recently been proved for particular classes of gibbs semigroups in the present paper we prove it for the whole generality including generalization of the product formula proposed by kato

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

bar partial torsion for complex manifolds and the adiabatic limit torsion for complex manifolds and the adiabatic limit we consider a complex fibration f m l b and pull back bundles e 1 and e 2 over m using the adiabatic limit idea we compute the metric invariant tp e1 tp e2 where tp e denotes the complex ray singer torsion

stochastic differential equations on noncompact manifolds moment stability and its topological consequences in this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a sde and the topology of the underlying manifold sufficient and necessary conditions are given for the moment stability of a sde in terms of the coefficients finally we prove a vanishing result for the fundamental group of a complete riemannian manifold in terms of purely geometrical quantities

ricci flow under local almost non negative curvature conditions abstract we find a local solution to the ricci flow equation under a negative lower bound for many known curvature conditions under a non collapsing assumption the flow exists for a uniform amount of time during which the curvature stays bounded below by a controllable negative number the curvature conditions we consider include 2 non negative and weakly pic 1 cases of which the results are new we complete the discussion of the almost preservation problem by bamler cabezas rivas wilking and the 2 non negative case generalizes a result in 3d by simon topping to higher dimensions as an application we use the local ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non negative curvature conditions and show that this limit space is bi holder homeomorphic to a smooth manifold

the trotter lie product formula for gibbs semigroups the trace norm and hilbert schmidt norm convergence of the trotter lie formula is proved for some classes of gibbs semigroups

finite propagation speed for solutions of the wave equation on metric graphs abstract we provide a class of self adjoint laplace operators on metric graphs with the property that the solutions of the associated wave equation satisfy the finite propagation speed property the proof uses energy methods which are adaptations of corresponding methods for smooth manifolds

on semiregularity of mappings abstract there are two basic ways of weakening the definition of the well known metric regularity property by fixing one of the points involved in the definition the first resulting property is called metric subregularity and has attracted a lot of attention during the last decades on the other hand the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature we provide a self contained material gathering and extending the existing theory on the topic we demonstrate a clear relationship with other regularity properties for example the equivalence with the so called openness with a linear rate at the reference point is shown in particular cases we derive necessary and or sufficient conditions of both primal and dual type we illustrate the importance of semiregularity in the convergence analysis of an inexact newton type scheme for generalized equations with not necessarily differentiable single valued part

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

a dominated type convergence theorem for the feynman integral a dominated type convergence theorem is proved for the operator valued feynman integral defined via the trotter product formula this definition is intimately related to the one of feynman and the physical setting is the same as in his original paper the convergence result given here provides the final piece of a unifying and rather satisfactory picture of three approaches to the feynman integral

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

lyusternik graves theorems for the sum of a lipschitz function and a set valued mapping in a paper of 1950 graves proved that for a function f acting between banach spaces and an interior point bar x in its domain if there exists a continuous linear mapping a which is surjective and the lipschitz modulus of the difference f a at bar x is sufficiently small then f is linearly open at bar x this is an extension of the banach open mapping principle from continuous linear mappings to lipschitz functions a closely related result was obtained earlier by lyusternik for smooth functions in this paper we obtain lyusternik graves theorems for mappings of the form f f where f is a lipschitz continuous function around bar x and f is a set valued mapping roughly we give conditions under which the mapping f f is linearly open at bar x for bar y provided that for each element a of a certain set of continuous linear operators the mapping f bar x a cdot bar x f is linearly open at bar x for bar y in the case when f is the zero mapping as

kernels of elliptic operators bounds and summability in this paper we shall study the kernel of the resolvent r c a and of some other related functions of a e g e for elliptic operators a on f or more generally for perturbations of elliptic operators it turns out that the resolvent consequently all other related functions are given by an integral kernel which is bounded by a convolution with a radial decreasing l function this result has numerous applications bounds for l spectrum of a closedness semigroup generation essential selfadjointness summability etc in gkl we studied this problem for perturbation of constant coefficient elliptic operators and established the following bound on the kernel r x y of a ir w i c w w ix vi p ici7 1 where h is the radial l function h z i izl s

surfaces with boundary their uniformizations determinants of laplacians and isospectrality let be a compact surface of type g n n 0 obtained by re moving n disjoint disks from a closed surface of genus g assuming 0 while osgood phillips and sarnak o p s3 showed the properness when g 0

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

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

determination of convex functions via subgradients of minimal norm we show in hilbert space setting that any two convex proper lower semicontinuous functions bounded from below for which the norm of their minimal subgradients coincide they coincide up to a constant moreover under classic boundary conditions we provide the same results when the functions are continuous and defined over an open convex domain these results show that for convex functions bounded from below the slopes provide sufficient first order information to determine the function up to a constant giving a positive answer to the conjecture posed in boulmezaoud et al siam j optim 28 3 2049 2066 2018

boundedness of spectral multipliers of generalized laplacians on compact manifolds with boundary consider a second order strongly elliptic negative semidefinite differential operator l may be a system on a compact riemannian manifold overline m with smooth boundary where the domain of l is defined by a coercive boundary condition classically known results and also recent work in duong et al j funct anal 196 443 485 2002 and duong and mcintosh rev math iberoam 15 233 265 1999 establish sufficient conditions for l infty text bmo l continuity of varphi sqrt l where varphi in s 0 1 mathbb r and l is a suitable elliptic operator using a variant of the cheeger gromov taylor functional calculus due to mauceri et al math res lett 16 861 879 2009 and short time upper bounds on the integral kernel of e tl due to greiner arch ration mech anal 41 168 218 1971 we prove that a variant of such sufficient conditions holds for our operator l

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

notes on the feynman integral ii in the setting of cameron and storvick s recent theory our main result establishes the existence of the analytic feynman integral for functions on v dimensional wiener space of the form f x exp ab a s x s x s ds here x is a rv valued continuous function on a b such that x a 0 and a s a s b is a commutative family of real symmetric positive definite matrices such that the square roots of the eigenvalues are functions of bounded variation on a b we obtain the existence theorem just referred to without having to construct special spaces quadratic forms etc to fit the particular problem of interest

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

product formula for imaginary resolvents with application to a modified feynman integral abstract an abstract product formula for imaginary resolvents is proved for a pair of self adjoint operators a b of a complex hilbert space here a is assumed to be nonnegative and the positive part of b is arbitrary while its negative part is small with respect to a in the sense of quadratic forms the proof is somewhat simpler than the author s original one which required both a and b to be nonnegative when specialized this theorem establishes the convergence of the modified feynman integral recently introduced by the author in the most general case for which the schrodinger equation can be solved without ambiguity

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples

a note on the limiting procedures for path integrals limiting procedures for the feynman type path integral are considered the evolution operator is approximated with operators corresponding to the exponential of the hamiltonian s symbol the proof of convergence uses chernoff s theory and its extension on the class of stable operators the approach is especially adequate for the generalized coherent states path integral where decomposition of a hamiltonian into the sums used in the approach based on trotter s formula may be unnatural