hamilton differential harnack inequality and w entropy for witten laplacian on riemannian manifolds abstract in this paper we prove the hamilton differential harnack inequality for positive solutions to the heat equation of the witten laplacian on complete riemannian manifolds with the c d k m condition where m n and k 0 are two constants moreover we introduce the w entropy and prove the w entropy formula for the fundamental solution of the witten laplacian on complete riemannian manifolds with the c d k m condition and on compact manifolds equipped with k m super ricci flows

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

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

data based stochastic model reduction for the kuramoto sivashinsky equation national science foundation dms 1217065 dms 1418775 dms 1419044 u s department of energy de ac02 05ch11231 national science foundation

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

harmonic maps and the schoen conjecture we show that every quasisymmetric homeomorphism of the circle h 2 admits a harmonic quasiconformal extension to the hyperbolic plane h 2 this proves the schoen conjecture

singularities of lagrangian mean curvature flow zero maslov class case we study singularities of lagrangian mean curvature flow in n when the initial condition is a zero maslov class lagrangian we start by showing that in this setting singularities are unavoidable more precisely we construct lagrangians with arbitrarily small lagrangian angle and lagrangians which are hamiltonian isotopic to a plane that nevertheless develop finite time singularities under mean curvature flow

the conjugate heat equation and ancient solutions of the ricci flow abstract we prove gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the ricci flow as a consequence for dimension 4 and higher we show that the backward limit of type i solutions of the ricci flow must be a non flat gradient shrinking ricci soliton this extends perelman s previous result on backward limits of solutions in dimension 3 in which case the curvature operator is nonnegative it follows from hamilton ivey curvature pinching estimate as an application this also addresses an issue left in naber 2010 23 where naber proves the interesting result that there exists a type i dilation limit that converges to a gradient shrinking ricci soliton but that soliton might be flat the gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest

cyclic surfaces and hitchin components in rank 2 we prove that given a hitchin representation in a real split rank 2 group g 0 there exists a unique equivariant minimal surface in the corresponding symmetric space as a corollary we obtain a parametrization of the hitchin components by a hermit ian bundle over teichmuller space the proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of g 0 some partial extensions of the construction hold for cyclic bundles in higher rank

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

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

methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions the equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions the relationship between the landau equations of the seventh order which were reduced from the navier stokes equations by applying these two methods is clarified how we should define the disturbance amplitude to obtain the complete equivalence between them is discussed

spacelike convex surfaces with prescribed curvature in 2 1 minkowski space abstract we prove existence and uniqueness of solutions to the minkowski problem in any domain of dependence d in 2 1 dimensional minkowski space provided d is contained in the future cone over a point namely it is possible to find a smooth convex cauchy surface with prescribed curvature function on the image of the gauss map this is related to solutions of the monge ampere equation det d 2 u z 1 z 1 z 2 2 on the unit disc with the boundary condition u d for a smooth positive function and a bounded lower semicontinuous function we then prove that a domain of dependence d contains a convex cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function is in the zygmund class moreover in this case the surface of constant curvature k contained in d has bounded principal curvatures for every k 0 in this way we get a full classification of isometric immersions of the hyperbolic plane in minkowski space with bounded shape operator in terms of zygmund functions of d finally we prove that every domain of dependence as in the hypothesis of the minkowski problem is foliated by the surfaces of constant curvature k as k varies in 0

liouville theorems and gradient estimates for a nonlinear elliptic equation in this paper we establish gradient estimates for positive solutions to the equation 0 1 fup u on any smooth metric measure space whose m bakry emery curvature is bounded from below by m 1 k with k 0 these estimates imply liouville theorems for 0 1 when p 1 our main theorem reduces to the gradient estimate of wang 2010 9 as applications several harnack inequalities are obtained

heat flows on hyperbolic spaces in this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere s n 1 n 3 can be extended to the n dimensional hyperbolic space such that the heat flow starting with this extension converges to a quasi isometric harmonic map this implies the schoen li wang conjecture that every quasiconformal map of s n 1 n 3 can be extended to a harmonic quasi isometry of the n dimensional hyperbolic space

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

hamilton s gradient estimates and liouville theorems for porous medium equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold of dimension n in this paper we derive a local gradient estimate for positive solutions of porous medium equation t u u 1 1 1 2 n 1 on m 0 we also obtain a theorem of liouville type for positive solutions of the porous medium equation

gradient estimates for ut δf u on manifolds and some liouville type theorems abstract in this paper we first prove a localized hamilton type gradient estimate for the positive solutions of porous media type equations u t f u with f u 0 on a complete riemannian manifold with ricci curvature bounded from below in the second part we study fast diffusion equation fde and porous media equation pme u t u p p 0 and obtain localized hamilton type gradient estimates for fde and pme in a larger range of p than that for aronson benilan estimate harnack inequalities and cauchy problems in the literature applying the localized gradient estimates for fde and pme we prove some liouville type theorems for positive global solutions of fde and pme on noncompact complete manifolds with nonnegative ricci curvature generalizing yau s celebrated liouville theorem for positive harmonic functions

cubic differentials and hyperbolic convex sets we give a parameterization of the moduli space of marked gromov hyperbolic convex domains of rp by the space of bounded holomorphic cubic differentials on the disk

defect mediated stability an effective hydrodynamic theory of spatiotemporal chaos abstract spatiotemporal chaos stc exhibited by the kuramoto sivashinsky ks equation is investigated analytically and numerically an effective stochastic equation belonging to the kpz universality class is constructed by incorporating the chaotic dynamics of the small ks system in a coarse graining procedure the bare parameters of the effective theory are computed approximately stability of the system is shown to be mediated by space time defects that are accompanied by stochasticity the method of analysis and the mechanism of stability may be relevant to a class of stc problems

fixed points of compositions of earthquakes let s be a closed surface of genus at least 2 and consider two measured geodesic laminations that fill s right earthquakes along these laminations are diffeomorphisms of the teichmuller space of s we prove that the composition of these earthquakes has a fixed point in the teichmuller space another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic ads manifold the proof uses some estimates from the geometry of those ads manifolds

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

gradient estimate for the heat kernel of a complete riemannian manifold and its applications abstract we derive some higher order gradient estimates for the heat kernels on complete manifolds applying these results we show that the riesz transform on a complete manifold with nonnegative ricci curvature is of weak type 1 1 in addition we consider the boundedness of the riesz potential on a complete manifold with nonnegative ricci curvature

equivariant maps into anti de sitter space and the symplectic geometry of ℍ ℍ given two fuchsian representations l and r of the fundamental group of a closed oriented surface s of genus 2 we study the relation between lagrangian submanifolds of m h 2 l 1 s h 2 r 1 s and equivariant embeddings of s into anti de sitter space where l r is the corresponding representation into psl 2 r psl 2 r it is known that if is a maximal embedding then its gauss map takes values in the unique minimal lagrangian submanifold ml of m we show that given any equivariant embedding its gauss map gives a lagrangian submanifold hamiltonian isotopic to ml conversely any lagrangian submanifold hamiltonian isotopic to ml is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of anti de sitter space

harnack estimates for a heat type equation under the ricci flow abstract in this paper we consider the gradient estimates for a positive solution of the nonlinear parabolic equation t u t u h u p on a riemannian manifold whose metrics evolve under the ricci flow two harnack inequalities and other interesting results are obtained

gradient estimates on the weighted p laplace heat equation in this paper by a regularization process we derive new gradient estimates for positive solutions to the weighted p laplace heat equation when the m bakry emery curvature is bounded from below by k for some constant k 0 when the potential function is constant which reduce to the gradient estimate established by ni and kotschwar for positive solutions to the p laplace heat equation on closed manifolds with nonnegative ricci curvature if k 0 and reduce to the davies hamilton and li xu s gradient estimates for positive solutions to the heat equation on closed manifolds with ricci curvature bounded from below if p 2

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

pulse dynamics in low reynolds number interfacial hydrodynamics experiments and theory abstract we analyze interaction of nonlinear pulses in active dispersive dissipative nonlinear media a particular example of such media is a viscous thin film coating a vertical fibre experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place in such events larger pulses catch up with smaller ones and annihilate them however for certain flow conditions and after a certain distance from the inlet no more coalescence is observed and the flow is described by quasi equilibrium solitary pulses interacting continuously with each other through attractions and repulsions and eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole this experimental study represents the first evidence of formation of bound states in low reynolds number interfacial hydrodynamics to gain theoretical insight into the interaction of the pulses and formation of bound states we derive a weakly nonlinear model for the flow the generalized kuramoto sivashinsky gks equation that retains the fundamental mechanisms of the wave evolution namely dominant nonlinearity instability stability and dispersion much like in the experiments the spatio temporal evolution of the gks equation is dominated by quasi stationary solitary pulses which continuously interact with each other through coalescence events or attractions repulsions to understand the latter case we utilize a weak interaction theory for the solitary pulses of the gks equation the theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses or alternatively the evolution of the separation distances by analyzing the fixed points of this system we obtain bound states of interacting pulses for two pulses we provide a criterion for the existence of a countable infinite or finite number of bound states depending on the strength of the dispersive term in the equation the interaction theory and resulting bound states are corroborated by computations of the full equation we also find qualitative agreement between the theory and the experiments

connections of berry and hannay type for moving lagrangian submanifolds abstract berry s phase is the holonomy of the natural connection on the canonical circle bundle over projectivized quantum hilbert space given a symplectic manifold p a classical limit of this phase is constructed as the holonomy of a berry connection over a finite codimensional isodrastic foliation defined by constancy of action integrals on the space of lagrangian submanifolds in p equipped with smooth densities of total measure 1 if the densities are determined by a kahler metric compatible with the symplectic structure the curvature of the berry connection at the lagrangian submanifold l is given by a simple formula involving curvature of l in particular the berry connection defines a homotopy invariant for isodrastic loops of minimal lagrangian submanifolds in a simply connected kahler manifold a similar invariant constructed by the author for loops of symplectomorphisms is also a special case of the classical berry phase a classical analogue of berry s phase was discovered by hannay for moving families of completely integrable systems following berry and hannay we interpret hannay s angles as the holonomy of a hannay connection on a bundle of tori over the isodrastic foliation on the space of lagrangian toral layers consisting of lagrangian tori with a flat affine structure and an extension of this structure to the first infinitesimal neighborhood finally we show that the hannay angles are the derivatives with respect to action variables of the classical berry phase

immersed hypersurfaces with constant weingarten curvature in a recent paper korevaar 1 5 used the alexandrov reflection principle to show that closed embedded hypersurfaces in n 1 lhn 1 or the upper hemisphere of s 1 are umbilic spheres provided a certain function f of the principal curvatures t t 22 is constant he only had to assume that f is positive on the positive cone 212i 0 vi and that f is elliptic on the component f of 21 f 0 which contains here f is said to be elliptic if 0f d21 0 for all i 1 inn this generalization of earlier sphere theorems see 9 and 10 for references cannot be extended to hypersurface immersions in view of recent counterexamples 1 11 however assuming additional curvature conditions walter derived in 10 global results for hypersurface immersions in a space n l c of constant curvature c which have a constant higher mean curvature function h here h is the r th symmetric function of the principal curvatures it was shown that such hypersurfaces are of constant mean curvature h1 provided they have nonnegative sectional curvature and non negative principal curvatures as a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified here we show that it is not necessary to assume all principal curvatures to be non negative moreover we extend waiter s result to general symmetric functions f f 2 let m be a smooth connected and compact manifold without boundary then we have the following result

additive global noise delays turing bifurcations we apply a stochastic center manifold method to the calculation of noise induced phase transitions in the stochastic swift hohenberg equation this analysis is applied to the reduced mode equations that result from fourier decomposition of the field variable and of the temporal noise the method shows a pitchfork bifurcation at lower perturbation order but reveals a novel additive noise induced postponement of the turing bifurcation at higher order good agreement is found between the theory and the numerics for both the reduced and the full system the results are generalizable to a broad class of nonlinear spatial systems

new stochastic mode reduction strategy for dissipative systems we present a new methodology for studying non hamiltonian nonlinear systems based on an information theoretical extension of a renormalization group technique using a modified maximum entropy principle we obtain a rigorous dimensionally reduced description for such systems the neglected degrees of freedom by this reduction are replaced by a systematically defined stochastic process under a constraint on the second moment this then forms the basis of a computationally efficient method numerical computations for the generalized kuramoto sivashinsky equation support our method and reveal that the long time underlying stochastic process of the fast unresolved modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

singularity of mean curvature flow of lagrangian submanifolds in this article we study the tangent cones at first time singularity of a lagrangian mean curvature flow if the initial compact submanifold 0 is lagrangian and almost calibrated by re in a calabi yau n fold m and t 0 is the first blow up time of the mean curvature flow then the tangent cone of the mean curvature flow at a singular point x 0 t is a stationary lagrangian integer multiplicity current in r 2n with volume density greater than one at x 0 when n 2 the tangent cone is a finite union of at least two 2 planes in r 4 which are complex in a complex structure on r 4

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

gradient estimates for a simple elliptic equation on complete non compact riemannian manifolds in this paper we study the local gradient estimate for the positive solution to the following equation u aulog u bu 0i nm where a 0 b are real constants m is a complete non compact riemannian manifold our result is optimal in the sense when m g is a complete non compact expanding gradient ricci soliton by definition m g is called an expanding gradient ricci soliton if for some constant c 0 it satisfies that rc cg d 2 f

polynomial quadratic differentials on the complex plane and light like polygons in the einstein universe abstract we construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light like polygons in the 2 dimensional einstein universe as an application we find a class of minimal lagrangian maps between ideal polygons in the hyperbolic plane

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

harmonic maps between 3 dimensional hyperbolic spaces we prove that a quasiconformal map of the sphere s 2 admits a harmonic quasi isometric extension to the hyperbolic space h 3 thus confirming the well known schoen conjecture in dimension 3

maximal surfaces and the universal teichmuller space we show that any element of the universal teichmuller space is realized by a unique minimal lagrangian diffeomorphism from the hyperbolic plane to itself the proof uses maximal surfaces in the 3 dimensional anti de sitter space we show that in adsn 1 any subset e of the boundary at infinity which is the boundary at infinity of a space like hypersurface bounds a maximal space like hypersurface in ads3 if e is the graph of a quasi symmetric homeomorphism then this maximal surface is unique and it has negative sectional curvature as a by product we find a simple characterization of quasi symmetric homeomorphisms of the circle in terms of 3 dimensional projective geometry

collisions of particles in locally ads spacetimes i local description and global examples we investigate 3 dimensional globally hyperbolic ads manifolds or more generally constant curvature lorentz manifolds containing particles i e cone singularities along a graph we impose physically relevant conditions on the cone singularities e g positivity of mass angle less than 2 on time like singular segments we construct examples of such manifolds describe the cone singularities that can arise and the way they can interact the local geometry near the vertices of we then adapt to this setting some notions like global hyperbolicity which are natural for lorentz manifolds and construct some examples of globally hyperbolic ads manifolds with interacting particles

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

gradient estimates for doubly nonlinear diffusion equations abstract in this short note we obtain some gradient estimates for the positive solution to the doubly nonlinear diffusion equation on closed riemannian manifold with ricci curvature bounded below by a non positive constant as applications we also derive corresponding harnack inequalities

aronson benilan estimates for the fast diffusion equation under the ricci flow abstract we study the fast diffusion equation with a linear forcing term 0 1 u t div u p 1 u r u under the ricci flow on a complete manifold m such that m r 2 has bounded curvature and nonnegative isotropic curvature where 0 p 1 and r r x t is the evolving scalar curvature of m at time t we prove aronson benilan and li yau hamilton type differential harnack estimates for positive solutions of 0 1 in addition we use similar method to prove certain li yau hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in cao and hamilton 2009 cao 2008 and kuang and zhang 2008 to the noncompact setting

spacelike hypersurfaces with prescribed boundary values and mean curvature we consider the boundary value problem for the mean curvature operator in minkowski space and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions our main results hold for non constant mean curvature and make no assumptions about the smoothness of the boundary or boundary data

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

a new framework for extracting coarse grained models from time series with multiscale structure in many applications it is desirable to infer coarse grained models from observational data the observed process often corresponds only to a few selected degrees of freedom of a high dimensional dynamical system with multiple time scales in this work we consider the inference problem of identifying an appropriate coarse grained model from a single time series of a multiscale system it is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback furthermore we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

the calabi homomorphism lagrangian paths and special lagrangians let mathcal o be an orbit of the group of hamiltonian symplectomorphisms acting on the space of lagrangian submanifolds of a symplectic manifold x omega we define a functional mathcal c mathcal o rightarrow mathbb r for each differential form beta of middle degree satisfying beta wedge omega 0 and an exactness condition if the exactness condition does not hold mathcal c is defined on the universal cover of mathcal o a particular instance of mathcal c recovers the calabi homomorphism if beta is the imaginary part of a holomorphic volume form the critical points of mathcal c are special lagrangian submanifolds we present evidence that mathcal c is related by mirror symmetry to a functional introduced by donaldson to study einstein hermitian metrics on holomorphic vector bundles in particular we show that mathcal c is convex on an open subspace mathcal o subset mathcal o as a prerequisite we define a riemannian metric on mathcal o and analyze its geodesics finally we discuss a generalization of the flux homomorphism to the space of lagrangian submanifolds and a lagrangian analog of the flux conjecture

gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on riemannian manifolds abstract we derive the gradient estimates and harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations t u x t h x t u x t 0 and u b u hu 0 on riemannian manifolds we also obtain a theorem of liouville type for positive solutions of the nonlinear elliptic equation

harmonic diffeomorphisms of the hyperbolic plane in this paper we consider the dirichlet problem at infinity for harmonic maps between the poincare model d of the hyperbolic plane h2 and solve this when given boundary data are c4 immersions of d oo the boundary at infinity of d to d oo also we present a construction of nonconformal harmonic diffeomorphisms of d and give a complete description of the boundary behavior including their first derivatives