heat equations on minimal submanifolds and their applications soit m n une sous variete a n dimensions immergee de facon minimale de m n l l 1 soit d m un domaine compact on demontre des theoremes de comparaisons pour les noyaux de la chaleur h x y t et k x y t pour les conditions aux limites de dirichet et de neumann

the stable topology of self dual moduli spaces soit m une 4 variete de riemann compacte connexe orientee et soit g un groupe de lie connexe simple on fixe un entier q m q alors l application inclusion i induit un epimorphisme sur les groupes respectifs d homologie et d homotopie a toutes les dimensions inferieures ou egales a q

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

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

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

differential harnack inequalities and perelman type entropy formulae for subelliptic operators abstract in this paper under the generalized curvature dimension inequality recently introduced by f baudoin and n garofalo we obtain differential harnack inequalities theorem 2 1 for the positive solutions to the schrodinger equation associated to subelliptic operator with potential as applications of the differential harnack inequality we derive the corresponding parabolic harnack inequality theorem 4 1 also we define the perelman type entropy associated to subelliptic operators and derive its monotonicity theorem 5 3

a sub riemannian curvature dimension inequality volume doubling property and the poincare inequality let mathbb m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l satisfying l1 0 and which is symmetric with respect to mu we show that if l satisfies with a non negative curvature parameter the generalized curvature inequality introduced by the first and third named authors in http arxiv org abs 1101 3590 then the following properties hold the volume doubling property the poincare inequality the parabolic harnack inequality the key ingredient is the study of dimension dependent reverse log sobolev inequalities for the heat semigroup and corresponding non linear reverse harnack type inequalities our results apply in particular to all sasakian manifolds whose horizontal webster tanaka ricci curvature is nonnegative all carnot groups of step two and to wide subclasses of principal bundles over riemannian manifolds whose ricci curvature is nonnegative

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

volume and distance comparison theorems for sub riemannian manifolds abstract in this paper we study global distance estimates and uniform local volume estimates in a large class of sub riemannian manifolds our main device is the generalized curvature dimension inequality introduced by the first and the third author in 3 and its use to obtain sharp inequalities for solutions of the sub riemannian heat equation as a consequence we obtain a gromov type precompactness theorem for the class of sub riemannian manifolds whose generalized ricci curvature is bounded from below in the sense of 3

weighted littlewood paley inequalities for heat flows in rcd spaces abstract we establish inequalities on vertical littlewood paley square functions for heat flows in the weighted l 2 space over metric measure spaces satisfying the rcd 0 n condition with n 1 and the maximum volume growth assumption in the noncompact setting the later assumption can be removed by showing that the volume of the ball growths at least linearly the estimates are sharp on the growth of the 2 heat weight and the 2 muckenhoupt weight considered the p muckenhoupt weight and the p heat weight are also compared for all p 1

stochastic analysis on sub riemannian manifolds with transverse symmetries we prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub riemannian manifolds with tranverse symmetries this representation is obtained from the study of bochner weitzenbock type formulas for sub laplacians on 1 forms as a consequence we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions the results are new even in the case of the heisenberg group which is the simplest example of a sub riemannian manifold with transverse symmetries

compact manifolds with a little negative curvature on etend le theoreme de bochner et le theoreme de myers pour des p formes en donnant des obstructions topologiques pour des varietes admettant des metriques avec une petite courbure negative

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

generalized ricci curvature bounds for three dimensional contact subriemannian manifolds measure contraction property is one of the possible generalizations of ricci curvature bound to more general metric measure spaces in this paper we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

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

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

log sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality let m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l which is symmetric with respect to mu we assume that l satisfies a generalized curvature dimension inequality as introduced by baudoin garofalo cite bg1 our goal is to discuss functional inequalities for mu like the poincare inequality the log sobolev inequality or the gaussian logarithmic isoperimetric inequality