the stretch of a foliation and geometric superrigidity we consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type such objects can be considered as compact models for the geometry of the symmetric space based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions to achieve this we consider the heat flow equation along the leaves of a foliation a bochner formula on foliations and a geometric invariant for foliations with leafwise riemannian metrics called the stretch we obtain as applications a metric rigidity theorem for foliations and a rigidity type result for riemannian manifolds whose geometry is only partially symmetric

hitchin s self duality equations on complete riemannian manifolds let m be a complete riemannian manifold let g be a real semisimple noncompact algebraic group and let k be its maximal compact subgroup let p be a principal g bundle over m fix a reduction of the structure group g of p to k and consider the space d of smooth g connections on p d is an affine space modeled on the vector space 21 ad p of 1 forms on m with values in the adjoint bundle associated to p for any connection d e d we have a decomposition d d 0 where d is a connection preserving the k structure and 0 is a 1 form on m with values in the orthogonal complement in ad p of ad pk here px is the k bundle defined by the reduction of structure group i f d is a flat connection we have

arithmeticity of holonomy groups of lie foliations the point of this paper is to prove arithmeticity of a class of finitely generated groups that arise naturally in a purely geometric context namely we prove arithmeticity of the holonomy group of a lie foliation of a compact manifold assuming the existence of a metric for which the leaves are symmetric spaces of nonpositive curvature and rank at least 2 this provides a partial answer to the question raised a number of years ago by a haefliger 13 of determining the possible holonomy groups of lie foliations to indicate the nature of the questions we are considering let us first recall that if n is a riemannian manifold a fundamental question is to understand the relationship of the topology of n and the geometry of n or for many purposes the geometry of n there has been a wide variety of work on this type of question from a variety of viewpoints in particular we can ask how the geometry of n or n controls the nature of rc1 n one can ask a more general question namely to understand to what extent the geometry of n controls the actions of rc1 n on a manifold t a standard construction associates to every such action a foliation of a manifold m which has all leaves locally isometric to n in which t appears naturally as a submanifold of m transverse to the foliation and the 7 n action on t controls the way in which the leaves of this foliation are tied together we can thus reinterpret our last question and broaden its context as well by formulating the following question determine the extent to which the geometry of a simply connected riemannian manifold x controls the way in which the leaves of a foliation are tied together if we assume the leaves are all locally isometric to x in general it is of course difficult to understand actions of a group on a manifold and correspondingly difficult to understand foliations with leaves locally isometric to a given x it is natural to begin an investigation of such actions with the case in which the action on t preserves some geometric structure e g a riemannian metric in terms of the associated foliation this becomes a special case of the

monotone methods for higher order partial difference equations abstract a nonlinear difference equation of the order n m in two independent variables nx myu x y u together with appropriate initial and periodic boundary conditions is considered several comparison results for these problems are established these results are then used to provide sufficient conditions for the convergence of monotone iterative methods numerical illustrations dwelling upon the importance of our results are also included

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

convexity at infinity in cartan hadamard manifolds and applications to the asymptotic dirichlet and plateau problems we study the asymptotic dirichlet and plateau problems on cartan hadamard manifolds satisfying the so called strict convexity abbr sc condition the main part of the paper consists in studying the sc condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point in particular we are able to verify the sc condition on manifolds whose curvature lower bound can go to infty and upper bound to 0 simultaneously at certain rates or on some manifolds whose sectional curvatures go to infty faster than any prescribed rate these improve previous results of anderson borbely and ripoll and telichevsky we then solve the asymptotic plateau problem for locally rectifiable currents with mathbb z 2 multiplicity in acartan hadamard manifold satisfying the sc condition given any compact topologically embedded k 1 dimensional submanifold of partial infty m 2 le k le n 1 as the boundary data we also solve the asymptotic plateau problem for locally rectifiable currents with mathbb z multiplicity on any rotationally symmetric manifold satisfying the sc condition given a smoothly embedded submanifold as the boundary data these generalize previous results of anderson bangert and lang moreover we obtain new results on the asymptotic dirichlet problem for a large class of pdes in particular we are able to prove the solvability of this problem on manifolds with super exponential decay to infty of the curvature

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

the poisson boundary of teichmuller space abstract for a markov operator on teichmuller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated markov chain in the thurston compactification and show that the poisson boundary of the markov operator can be identified with the space of projective measured foliations the approach consists in using the authors results on the poisson boundary of the mapping class group in combination with a discretization procedure based on a harnack inequality for markov operators on teichmuller space

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

dirichlet conditions at infinity for parabolic and elliptic equations we investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in rn r under specific assumptions we establish existence of solutions satisfying prescribed conditions at infinity depending on the direction along which infinity is approached moreover the large time behavior of such solutions is addressed we consider also elliptic equations in rn with similar conditions at infinity

martin points on open manifolds of non positive curvature the martin boundary of a cartan hadamard manifold describes a fine geometric structure at infinity which is a sub space of positive harmonic functions we describe conditions which ensure that some points of the sphere at infinity belong to the martin boundary as well in the case of the universal cover of a compact manifold with ballmann rank one we show that martin points are generic and of full harmonic measure the result of this paper provides a partial answer to an open problem of s t yau

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

the dirichlet problem for p harmonic functions with respect to the mazurkiewicz boundary and new capacities in this paper we develop the perron method for solving the dirichlet problem for the analog of the p laplacian i e for p harmonic functions with mazurkiewicz boundary values the setting conside

minimal graphic functions on manifolds of nonnegative ricci curvature we study minimal graphic functions on complete riemannian manifolds with nonnegative ricci curvature euclidean volume growth and quadratic curvature decay we derive global bounds for the gradients for minimal graphic functions of linear growth only on one side then we can obtain a liouville type theorem with such growth via splitting for tangent cones of at infinity when in contrast we do not impose any growth restrictions for minimal graphic functions we also obtain a liouville type theorem under a certain nonradial ricci curvature decay condition on in particular the borderline for the ricci curvature decay is sharp by our example in the last section 2014 wiley periodicals inc

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

hausdorff dimension of harmonic measures on negatively curved manifolds we show by probabilistic means that harmonic measures on manifolds whose curvature is sandwiched between two negative constants have positive hausdorff dimensions a lower bound for harmonic measures of open sets is derived as well we end with the results concerning the hausdorff dimension of harmonic measures on universal covers of compact negatively curved manifolds

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

asymptotic behavior of solutions for a lotka volterra mutualism reaction diffusion system with time delays this paper is to investigate the asymptotic behavior of solutions for a time delayed lotka volterra n species mutualism reaction diffusion system with homogeneous neumann boundary condition it is shown under a simple condition on the reaction rates that the system has a unique bounded time dependent solution and a unique constant positive steady state solution and for any nontrivial nonnegative initial function the corresponding time dependent solution converges to the constant positive steady state solution as time tends to infinity this convergence result implies that the trivial steady state solution and all forms of semitrivial steady state solutions are unstable and moreover the system has no nonconstant positive steady state solution a condition ensuring the convergence of the time dependent solution to one of nonnegative semitrivial steady state solutions is also given

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

interior gradient estimates for mean curvature equations the interior gradient estimate for the prescribed mean curvature equation has been extensively studied see 9 and the references therein for high order mean curvature equations it has also been obtained in 11 18 in most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion from the view point of partial differential equations such estimate should hold for equations with similar structural conditions in 13 6 the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions different proofs for the gradient estimate for mean curvature equations have been given in 1 3 in this note we show that for curvature and hessian equations the interior gradient estimate can be obtained very easily our proof which is based on suitable choice of auxiliary functions is elementary and avoids geometric computations on the graph of solutions the technique in this note has actually been widely used in literature see e g 9

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

asymptotic behavior of solutions for a cooperation diffusion model with a saturating interaction this paper is concerned with a lotka volterra cooperation diffusion model with a saturating interaction term for one species the goal of the paper is to investigate the asymptotic behavior of the time dependent solution in relation to the corresponding steady state solutions under homogeneous neumann boundary condition some simple and easily verifiable conditions are given to the rate constants so that for every nontrivial nonnegative initial function the corresponding time dependent solution converges to one of the nonnegative constant steady state solutions as time tends to infinity this convergence result leads to the existence and uniqueness of a positive or nonnegative steady state solution and the global asymptotic stability of a given nonnegative constant steady state solution in terms of ecological dynamics it also gives some coexistence permanence and extinction results for the model

central limit theorem in negative curvature we prove a central limit theorem for the distance of the brownian point on the universal cover of a compact negatively curved riemannian manifold the technical point is a contraction property for the leafwise brownian motion along the stable foliation

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

representations of fundamental groups of manifolds with a semisimple transformation group group 1 m and the lie groups that can act on m more precisely let g be a connected semisimple lie group of higher real rank and suppose g acts continuously on a topological manifold m preserving a finite measure the main theme of this paper is that the representation theory of 7r i m in low dimensions is to a large extent controlled by that of g the latter of course being well understood in particular under natural hypotheses e g that the action of g on m is engaging i e there is no loss of ergodicity in passing to finite covers see definition 3 1 below we prove that if g has no nontrivial

asymptotic dirichlet problems in warped products we study the asymptotic dirichlet problem for killing graphs with prescribed mean curvature h in warped product manifolds m times varrho mathbb r in the first part of the paper we prove the existence of killing graphs with prescribed boundary on geodesic balls under suitable assumptions on h and the mean curvature of the killing cylinders over geodesic spheres in the process we obtain a uniform interior gradient estimate improving previous results by dajczer and de lira in the second part we solve the asymptotic dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to infty provided that h satisfies certain bounds with respect to the sectional curvatures of m and the norm of the killing vector field finally we obtain non existence results if the prescribed mean curvature function h grows too fast

on discrete boundary value problems arising in transport phenomena for the discrete boundary value problems arising in transport processes we provide comparison results these results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector the advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems numerical illustration showing the sharpness as well the importance of the obtained results is also included

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

on the differentiability of horocycles and horocycle foliations the curves w i e txs are the leaves of the horocycle foliation wof t s we project the horocycles from s into s to obtain horocycles for vectors in t s similarly we obtain the horocycle foliation of txs again denoted by w an important step in e hopf s proof of the ergodicity of the geodesic flow on a compact surface s of variable negative curvature was to show that the horocycle foliation of t s is c he actually proved 6 that the horocycle foliation is c 1 under the weaker assumption that the curvature of s has bounded derivative and is uniformly bounded away from 0 and oo an immediate consequence is that the horocycles and busemann functions in s

existence and non existence of minimal graphic and p harmonic functions we prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete riemannian manifold m with only one end if m has asymptotically non negative sectional curvature on the other hand we prove the existence of bounded non constant minimal graphic and p harmonic functions on rotationally symmetric cartan hadamard manifolds under optimal assumptions on the sectional curvatures

existence of non trivial harmonic functions on cartan hadamard manifolds of unbounded curvature the liouville property of a complete riemannian manifold m i e the question whether there exist non trivial bounded harmonic functions on m attracted a lot of attention for cartan hadamard manifolds the role of lower curvature bounds is still an open problem we discuss examples of cartan hadamard manifolds of unbounded curvature where the limiting angle of brownian motion degenerates to a single point on the sphere at infinity but where nevertheless the space of bounded harmonic functions is as rich as in the non degenerate case to see the full boundary the point at infinity has to be blown up in a non trivial way such examples indicate that the situation concerning the famous conjecture of greene and wu about existence of non trivial bounded harmonic functions on cartan hadamard manifolds is much more complicated than one might have expected

monotone methods for periodic solutions of second order scalar functional differential equations an iterative method is presented which starting from a lower or from an upper periodic solution provides a monotone sequence converging to a periodic solution of 1 with some restrictions on the growth off the method extends to functional differential equations of type 1 two numerical examples with an a posteriori error analysis are given

generalized harmonic functions of riemannian manifolds with ends we study l harmonic functions solutions of the stationary schrodinger equation on arbitrary noncompact riemannian manifolds with finitely many ends we establish some existence and uniqueness results and obtain sharp dimension estimates for l harmonic functions on such manifolds

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

on asymptotically harmonic manifolds of negative curvature we study asymptotically harmonic manifolds of negative curvature without any cocompactness or homogeneity assumption we show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces in particular we determine the volume entropy the spectrum and the relative densities of visual and harmonic measures on the ideal boundary then we prove an asymptotic analogue of the classical mean value property of harmonic manifolds and we characterize asymptotically harmonic manifolds among cartan hadamard spaces of strictly negative curvature by the existence of an asymptotic equivalent tau u mathrm e er for the volume density of geodesic spheres with tau constant in case dr m is bounded finally we show the existence of a margulis function and explicitly compute it for all asymptotically harmonic manifolds

a new approach to the martin boundary via diffusions conditioned to hit a compact set let l generate a transient diffusion x t on rd and let d be an exterior domain let h be the smallest positive solution of lh 0 in d and h 1 on dd define xh t to be the process x t conditioned to hit dd by doob s h transform theory xh t is also a markov diffusion and its generator lh is defined by lhf l h l hf letting rd be the hitting time of dd define the harmonic measure for xh t on dd starting from x e d by i4x dy p r xh trd e dy let xn n 1 c d be a sequence satisfying limp xn i for which i4h converges weakly call two such sequences xn 1 and x 1 equivalent if lim k limn o x we call the set of equivalence classes thus generated the harmonic measure boundary at infinity for lh this boundary is independent of the particular exterior domain d we prove that the harmonic measure boundary at infinity for lh coincides with the martin boundary for l on rd the formal adjoint of the operator l on rd in the case that l generates a reversible diffusion the martin boundaries of l and l coincide and hence the harmonic measure boundary of lh coincides with the martin boundary for l on rd a similar probabilistic description of the martin boundary for l on rd can be given in the nonreversible case these results are then used to give explicit representations of the martin boundaries of l and l for several classes of diffusion processes

monotone convergence of iterative methods for n p and p n boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for n p and p n boundary value problems converges monotonically an example which illustrates the usefulness of this technique is also included

a fatou theorem for f harmonic functions in this paper we study a class of functions that appear naturally in some equidistribution problems and that we call f harmonic these are functions of the universal cover of a closed and negatively curved manifold which possess an integral representation analogous to the poisson representation of harmonic functions where the role of the poisson kernel is played by a holder continuous kernel more precisely we prove a theorem a la fatou about the nontangential convergence of quotients of such functions from which we deduce some basic properties such as the uniqueness of the f harmonic function on a compact manifold and of the integral representation of f harmonic functions

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

brownian motion and dirichlet problems at infinity we discuss angular convergence of riemannian brownian motion on a cartan hadamard manifold and show that the dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions ce 2 ar x km x a2 vspace 0 5pt 0 and cr x 2 km x 1 r x 2 2 2 respectively

brownian motion harmonic functions and hyperbolicity for euclidean complexes a euclidean complex x is a simplicial complex whose simplices are flat euclidean simplices we construct a natural brownian motion on x and show that if x has nonpositive curvature and satisfies gromov s hyperbolicity condition then with probability one brownian motion tends to a random limit on the gromov boundary applying a combination of geometric and probabilistic techniques we describe spaces of harmonic functions on x

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

potential theory for manifolds with boundary and applications to controlled mean curvature graphs in this paper we characterize the neumann parabolicity of manifolds with boundary in terms of a new form of the classical ahlfors maximum principle and of a version of the so called kelvin nevanlinnaroyden criterion the motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a riemannian product of the type n r in this direction two kind of results will be presented height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold which extend results by a ros and h rosenberg valid for domains of r and slice type results for graphs whose superlevel sets have finite volume finally the use of the ahlfors maximum principle allows us to establish a connection between the neumann parabolicity and the dirichlet parabolicity commonly used in minimal surface theory in particular we will be able to give a deterministic proof of special cases of a result by r neel

periodic solutions of hyperbolic partial differential equations abstract constructive methods by obtaining multiple solutions of first and second order nonlinear hyperbolic periodic boundary value problems are given appropriate partial differential inequalities are developed and are then employed to derive the iterative schemes which yield the desired solutions the advantage of these methods is that the solutions of the given nonlinear problems are constructed as uniform limits of monotone iterates these iterates being solutions of linear problems are explicitly computable

half space theorem embedded minimal annuli and minimal graphs in the heisenberg group we construct a one parameter family of properly embedded minimal annuli in the heisenberg group nil3 endowed with a left invariant riemannian metric these annuli are not rotationally invariant this family of annuli is used to prove a vertical half space theorem which is then applied to prove that each complete minimal graph in nil3 is entire also it is shown that the sister surface of an entire minimal graph in nil3 is an entire constant mean curvature cmc 1 graph in h 2 r and vice versa this gives a classification of all entire cmc 1 graphs in h 2 r finally we construct properly embedded cmc 1 annuli in h 2 r

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

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative