symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

hamilton s gradient estimates and a monotonicity formula for heat flows on metric measure spaces abstract in this paper we extend the hamilton s gradient estimates hamilton 1993 and a monotonicity formula of entropy ni 2004 for heat flows from smooth riemannian manifolds to non smooth metric measure spaces with appropriate riemannian curvature dimension condition

the limits of stochastic integrals of differential forms this paper is concerned with the integration of 1 forms against the markov stochastic process associated with a second order elliptic differential operator in divergence form it focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity this extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure applications are given for example it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved cartan hadamard manifold has an asymptotic direction recovering and strengthening the previous arguments of pratt sullivan and others the approach can also be used to construct a levy area for such processes to study the thinness of sets for the elliptic operator and probably has wider applications

remarks on non compact gradient ricci solitons in this paper we show how techniques coming from stochastic analysis such as stochastic completeness in the form of the weak maximum principle at infinity parabolicity and lp liouville type results for the weighted laplacian associated to the potential may be used to obtain triviality rigidity results and scalar curvature estimates for gradient ricci solitons under lp conditions on the relevant quantities

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

the first sign change of fourier coefficients of cusp forms let f sum no a n q n a n real be an arbitrary nonzero cusp form of even integral weight k geq 2 on gamma 0 n square free n we obtain an effective bound that the first sign change of a n occurs

harnack inequality for quasilinear elliptic equations on riemannian manifolds abstract we study viscosity solutions to degenerate and singular elliptic equations div f u u u h of p laplacian type on riemannian manifolds where an even function f c 1 r c 2 0 is supposed to be strictly convex on 0 under the assumption that either f c 2 r or its convex conjugate f c 2 r with some structural condition we establish a locally uniform abp type estimate and the krylov safonov type harnack inequality on riemannian manifolds with the use of an intrinsic geometric quantity to the operator here the c 2 regularities of f and f account for degenerate and singular operators respectively

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

elliptic gradient estimates for a nonlinear heat equation and applications abstract in this paper we study elliptic gradient estimates for a nonlinear f heat equation which is related to the gradient ricci soliton and the weighted log sobolev constant of smooth metric measure spaces precisely we obtain hamilton s and souplet zhang s gradient estimates for positive solutions to the nonlinear f heat equation only assuming the bakry emery ricci tensor is bounded below as applications we prove parabolic liouville properties for some kind of ancient solutions to the nonlinear f heat equation some special cases are also discussed

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

fourier coefficients of cusp forms of half integral weight let mathfrak f be a cusp form of half integral weight whose fourier coefficients mathfrak a mathfrak f n are all real we study the sign change problem of mathfrak a mathfrak f n when n runs over some specific sets of integers lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs these give an improvement on some recent results of bruinier and kohnen modular forms on schiermonnikoong cambridge university press cambridge 57 66 2008 and kohnen int j number theory 6 1255 1259 2010

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

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

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

periodicity of signs of fourier coefficients of eta quotients we study the periodicity of signs of fourier coefficients of the function d f qd rd where is a positive integer f q n 1 1 qn and rd z as an application we will prove the periodicity of signs for the crank differences conjectured by g e andrews and r lewis and for the weighted counts of certain types of partitions

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

on the first sign change of hecke eigenvalues of newforms let f be a non zero cusp form of even integral weight k 2 and level n with real fourier coefficients a n n n using a classical theorem of landau together with the facts that the hecke l function of f is entire and the rankin selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the petersson norm of f one easily shows that there are infinitely many n n such that a n 0 as well as infinitely many n with a n 1 siegel s argument does not apply and the result above is not known now suppose that n is squarefree and f is a newform which is a normalized hecke eigenform so a 1 1 and a n is the n th hecke eigenvalue of f we shall prove

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

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

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

stochastic completeness for graphs with curvature dimension conditions abstract we prove pointwise gradient bounds for heat semigroups associated to general possibly unbounded laplacians on infinite graphs satisfying the curvature dimension condition cd k using gradient bounds we show stochastic completeness for graphs satisfying the curvature dimension condition

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

maximum principles and gradient ricci solitons abstract it is shown that the omori yau maximum principle holds true on complete gradient shrinking ricci solitons both for the laplacian and the f laplacian as an application curvature estimates and rigidity results for shrinking ricci solitons are obtained furthermore applications of maximum principles are also given in the steady and expanding situations

boundary behavior of harmonic functions on manifolds in this paper we mainly set up a kind of representation theorem of harmonic functions on manifolds with ricci curvature bounded below and study non tangential limits of harmonic functions

localized upper bounds of heat kernels for diffusions via a multiple dynkin hunt formula we prove that for a general diffusion process certain assumptions on its behavior only within a fixed open subset of the state space imply the existence and sub gaussian type off diagonal upper bounds of the global heat kernel on the fixed open set the proof is mostly probabilistic and is based on a seemingly new formula which we call a multiple dynkin hunt formula expressing the transition function of a hunt process in terms of that of the part process on a given open subset this result has an application to heat kernel analysis for the liouville brownian motion the canonical diffusion in a certain random geometry of the plane induced by a massive gaussian free field

a new topological approach to the l uniqueness of operators and the l1 uniqueness of fokker planck equations abstract the usual semigroups of kernels on a polish space e are in general not strongly continuous on l e with respect to the norm topology we introduce a new topology on l e such that they become c 0 semigroups for which we can establish a simplified hille yosida theorem the new topology will allow us to introduce the uniqueness of pre generator on l e which turns out to be equivalent to the l 1 uniqueness of the associated fokker planck equation among many others and it is intimately related with the liouville properties for l 1 harmonic functions the uniqueness of several second order elliptic differential operators in l are studied 1 one dimensional diffusion operators a x f b x f 2 schrodinger operators 1 2 v 3 multi dimensional diffusion generator 1 2

geometry of manifolds with densities abstract we study the geometry of complete riemannian manifolds endowed with a weighted measure where the weight function is of quadratic growth assuming the associated bakry emery curvature is bounded from below we derive a new laplacian comparison theorem and establish various sharp volume upper and lower bounds we also obtain some splitting type results by analyzing the busemann functions in particular we show that a complete manifold with nonnegative bakry emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions while some of our results are new even for the gradient ricci solitons the novelty here is that only a lower bound of the bakry emery curvature is involved in our analysis

the heat kernel of a schrodinger operator with inverse square potential we consider the schrodinger operator h v x with radial potential v which may have singularity at 0 and a quadratic decay at infinity first we study the structure of positive harmonic functions of h and give their precise behavior second under quite general conditions we prove an upper bound for the correspond heat kernel p x y t of the type 0 p x y t ct n 2u min x t u min y t u t 2exp x y 2ct for all x y rn and t 0 where u is a positive harmonic function of h third if u2 is an a2 weight on rn then we prove a lower bound of a similar type

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

counting homomorphisms onto finite solvable groups we present a method for computing the number of epimorphisms from a finitely presented group g to a finite solvable group which generalizes a formula of

positive solutions to schrodinger equations and geometric applications a variant of li tam theory which associates to each end of a complete riemannian manifold a positive solution of a given schrodinger equation on the manifold is developed it is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant sobolev inequality consequently a finiteness result for the number of ends follows in the case when the sobolev inequality is of particular type the finiteness result is proven directly as an application an estimate on the number of ends for shrinking gradient ricci solitons and submanifolds of euclidean space is obtained

uniqueness for the heat equation in riemannian manifolds abstract we investigate uniqueness in suitable weighted lebesgue spaces of solutions to the heat equation in geodesically complete riemannian manifolds

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

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

harnack inequality for degenerate and singular operators of p laplacian type on riemannian manifolds we study viscosity solutions to degenerate and singular elliptic equations of p laplacian type on riemannian manifolds the krylov safonov type harnack inequality for the p laplacian operators with 1 p infty is established on the manifolds with ricci curvature bounded from below based on abp type estimates we also prove the harnack inequality for nonlinear p laplacian type operators assuming that a nonlinear perturbation of ricci curvature is bounded below

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

implications of energy conditions on standard static space times abstract in the framework of standard static space times we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories we concentrate our study on questions about the sub harmonicity of the warping function the scalar curvature map conformal hyperbolicity conjugate points and the time like diameter of this class of space times

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

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

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

on complete gradient shrinking ricci solitons in this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons based on this we prove that a complete noncompact gradient shrinking ricci soliton has at most euclidean volume growth this latter result can be viewed as an analog of the well known volume comparison theorem of bishop that a complete noncompact riemannian manifold with nonnegative ricci curvature has at most euclidean volume growth

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

some liouville type theorems for harmonic functions on finsler manifolds abstract in this paper we give some liouville type theorems for l p p r harmonic resp subharmonic superharmonic functions on forward complete finsler manifolds moreover we derive a gradient estimate for harmonic functions on a closed finsler manifold as an application one obtains that any harmonic function on a closed finsler manifold with nonnegative weighted ricci curvature ric n n n must be constant

crowns and factorization of the probabilistic zeta function of a finite group abstract we employ the concept of crown to study the generalized euler factors of the probabilistic zeta function of finite groups

oscillations of fourier coefficients of modular forms a p 2p cos 0 p since we know the truth of the ramanujan petersson conjecture it follows that the 0 p s are real inspired by the sato tate conjecture for elliptic curves serre 14 conjectured that the 0 p s are uniformly distributed in the interval 0 rc with respect to the 1 measure sin2odo following serre we shall refer to this as the sato tate r conjecture there being no room for confusion it has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the fourier coefficients in particular the following is implied by the sato tate conjecture theorem 1 for any normalized hecke eigenform

geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature we apply alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature we obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point moreover we show the volume doubling property and the poincare inequality on such graphs the quadratic volume growth of these graphs implies the parabolicity finally we prove the polynomial growth harmonic function theorem analogous to the case of riemannian manifolds

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

heat kernel on smooth metric measure spaces and applications we derive a harnack inequality for positive solutions of the f heat equation and gaussian upper and lower bound estimates for the f heat kernel on complete smooth metric measure spaces with bakry emery ricci curvature bounded below both upper and lower bound estimates are sharp when the bakry emery ricci curvature is nonnegative the main argument is the de giorgi nash moser theory as applications we prove an l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness theorem for f heat equations when f has at most linear growth we also obtain eigenvalues estimates and f green s function estimates for the f laplace operator

the diffusion phenomenon for dissipative wave equations in metric measure spaces abstract we study the long time behavior of the solution to a type of dissipative wave equation where the operator in the equation is time dependent and the solution is defined on a metric measure space x m satisfying appropriate conditions the operator is assumed to be self adjoint and is related to a time dependent dirichlet form we link hyperbolic pdes with the firmly established theories for parabolic pdes in metric measure spaces and dirichlet forms subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation we present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works

curvature dimension inequalities on sub riemannian manifolds obtained from riemannian foliations part ii using the curvature dimension inequality proved in part i we look at consequences of this inequality in terms of the interaction between the sub riemannian geometry and the heat semigroup p t corresponding to the sub laplacian we give bounds for the gradient entropy a poincare inequality and a li yau type inequality these results require that the gradient of p t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

holder estimates and regularity for holomorphic and harmonic functions in this paper we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions moreover if we further assume that the volume decays at most polynomially at a singular point

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

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

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

on the geometry of semiclassical limits on dirichlet spaces abstract this paper is a contribution to semiclassical analysis for abstract schrodinger type operators on locally compact spaces let x be a metrizable separable locally compact space let be a radon measure on x with a full support let t x y p t x y be a strictly positive pointwise consistent heat kernel and assume that the generator h p 0 of the corresponding self adjoint contraction semigroup in l 2 x induces a regular dirichlet form then given a function 0 1 0 such that the limit lim t 0 p t x x t exists for all x x we prove that for every potential w x r one has lim t 0 t tr e t h p w e w x lim t 0 p t x x t d x for the schrodinger type operator h p w provided w satisfies very mild conditions at that are essentially only made to guarantee that the sum of quadratic forms h p w t is self adjoint and bounded from below for small t and to guarantee that e w x lim t 0 p t x x t d x the proof is probabilistic and relies on a principle of not feeling the boundary for p t x x in particular this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete riemannian manifolds and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs