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

rigidity of invariant convex sets in symmetric spaces the main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have zariski dense stabilizer

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

asymptotic dirichlet problems for harmonic functions on riemannian manifolds we define the asymptotic dirichlet problem and give a sufficient condition for solving it this proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete riemannian manifolds 0 introduction in this paper we will prove the existence of nonconstant bounded harmonic functions on certain classes of noncompact riemannian manifolds by defining and solving an asymptotically defined dirichlet problem for harmonic functions the motivation comes from the classical uniformization theorem of riemann surfaces which says that a simply connected riemann surface is biholo morphic to the riemann sphere s2 the complex plane c or the unit disk d this is a geometric theorem but its original proof due to koebe relies heavily on function theory the function theoretic interpretation of this theorem is the following among the noncompact simply connected surfaces c is characterized by the fact that it admits no nonconstant bounded harmonic functions and d by that it admits nonconstant bounded harmonic functions the geometric aspect of the uniformization theorem could be roughly stated as follows let m be a simply connected riemann surface equipped with a complete riemannian metric with gaussian curvature km if km c 0 then m is biholo morphic to s2 if km c 0 then m is biholomorphic to d if km is close to zero then m is biholomorphic to c the following precise version is due to greene and wu gw1 p 120

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

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

poisson boundary of a relativistic diffusion in this article we study the asymptotic behaviour of a random motion in minkowski spacetime representing the random evolution of an object or signal traveling at a speed strictly less than the speed of the light introduced by dudley in his article ark mat 6 241 268 1966 we determine its invariant algebra and give an explicit description of the poisson boundary of its differential generator

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

uniqueness of positive solutions to the rotationally symmetric p harmonic map equations abstract let and be positive solutions on 0 to the rotationally symmetric p harmonic map equation on model manifolds m f and m g where f is assumed to be sufficiently large near infinity and g y 0 for y 0 we show that if and have the same limit at infinity then on 0

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

bounded harmonic functions on riemannian manifolds of nonpositive curvature certain general conditions are put forth on a complete simply connected riemannian manifold of nonpositive curvature which guarantee that they support nontrivial bounded harmonic functions this result includes the cartan hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains mathfrak r i n n and mathfrak r ii n n 2 as special cases

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

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

limit sets of discrete groups of isometries of exotic hyperbolic spaces let be a geometrically finite discrete group of isometries of hyperbolic space hf where f r c h or o in which case n 2 we prove that the critical exponent of equals the hausdorff dimension of the limit sets and that the smallest eigenvalue of the laplacian acting on square integrable functions is a quadratic function of either of them when they are sufficiently large a generalization of hopf ergodicity theorem for the geodesic flow with respect to the bowen margulis measure is also proven

unbounded positive entire solutions of rotationally symmetric harmonic map equations abstract we study the existence of unbounded positive entire c 2 solutions of the rotationally symmetric harmonic map equations using the existence result we solve the dirichlet problem at infinity for any nonnegative boundary value at infinity

construction of harmonic diffeomorphisms and minimal graphs we study complete minimal graphs in x which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon in we give necessary and sufficient conditions on the lengths of the sides of the polygon and all inscribed polygons in that ensure the existence of such a graph we then apply this to construct entire minimal graphs in x that are conformally the complex plane c the vertical projection of such a graph yields a harmonic diffeomorphism from c onto disproving a conjecture of rick schoen and s t yau

superrigidity for irreducible lattices and geometric splitting we prove general superrigidity results for actions of irreducible lattices on cat 0 spaces first in terms of the ideal boundary and then for the intrinsic geometry including for infinite dimensional spaces in particular one obtains a new and self contained proof of margulis superrigidity theorem for uniform irreducible lattices in non simple groups the proofs rely on simple geometric arguments including a splitting theorem which can be viewed as an infinite dimensional and singular generalization of the lawson yau gromoll wolf theorem appendix a gives a very elementary proof of commensurator superrigidity appendix b proves that all our results also hold for certain non uniform lattices

intrinsic parameter calibration procedure for a high distortion fish eye lens camera with distortion model and accuracy estimation this paper presents a calibration procedure for a fish eye lens a high distortion lens mounted on a ccd tv camera the method is designed to account for the differences in images acquired via a distortion free lens camera setup and the images obtained by a fish eye lens camera the calibration procedure essentially defines a mapping between points in the world coordinate system and their corresponding point locations in the image plane this step is important for applications in computer vision which involve quantitative measurements the objective of this mapping is to estimate the internal parameters of the camera including the effective focal length one pixel width on the image plane image distortion center and distortion coefficients the number of parameters to be calibrated is reduced by using a calibration pattern with equally spaced dots and assuming a pin hole model camera behavior for the image center thus assuming negligible distortion at the image distortion center our method employs a non linear transformation between points in the world coordinate system and their corresponding location on the image plane a lagrangian minimization method is used to determine the coefficients of the transformation the validity and effectiveness of our calibration and distortion correction procedure are confirmed by application of this procedure on real images

minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain hadamard surfaces we construct harmonic diffeomorphisms from the complex plane bf c onto any hadamard surface mathbb m whose curvature is bounded above by a negative constant for that we prove a jenkins serrin type theorem for minimal graphs in mathbb m times mathbb r over domains of mathbb m bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in mathbb m times mathbb r with the conformal structure of bf c

stability of steady solutions of burgers s equation we develop tools to discuss the stability of viscous shock profiles for conservation laws the technique is based on replacing the far field regions by boundary conditions that describe the effect of the outer solution on the solution in the shock layer region we apply the technique to burgers s equation and prove stability for perturbations that are not localized in space

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

dirichlet problem at infinity on gromov hyperbolic metric measure spaces we study the asymptotic dirichlet problem for p harmonic functions in a very general setting of gromov hyperbolic metric measure spaces

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

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

hierarchical block matching method for fast rotation of binary images in this paper we develop a block matching method for fast rotation of binary images we define coarse and fine blocks to extract bit patterns of an original image and calculate their predrawn mapping patterns pmps using the given rotation angle the traditional calculation for rotation of images can be thus replaced with simple matching of bit patterns of blocks and drawing of their two dimensional 2 d pmps at the output plane a scheme of overlapping blocks is also used to solve problems of hole generation and topology variation usually occurring in rotation experimental results demonstrate that our proposed method performs best in terms of rotation speed compared to other algorithms

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

stability of viscous shocks on finite intervals consider the cauchy problem for a system of viscous conservation laws with a solution consisting of a thin viscous shock layer connecting smooth regions we expect the time dependent behavior of such a solution to involve two processes one process consists of the large scale evolution of the solution this process is well modeled by the corresponding inviscid equations the other process is the adjustment in shape and position of the shock layer to the large scale solution the time scale of the second process is much faster than the first 1 v compared to 1 the second process can be divided into two parts adjustment of the shape and of the position during this adjustment the end states are essentially constant in order to answer the question of stability we have developed a technique where the two processes can be separated to isolate the fast process we consider the region in the vicinity of the shock layer the equations are augmented with special boundary conditions that reflect the slow change of the end states we show that for the isolated fast process the perturbations decay exponentially in time

double line image rotation this paper proposes a fast algorithm for rotating images while preserving their quality the new approach rotates images based on vertical or horizontal lines in the original image and their rotated equation in the target image the proposed method is a one pass method that determines a based line equation in the target image and extracts all corresponding pixels on the base line floating point multiplications are performed to calculate the base line in the target image and other line coordinates are calculated using integer addition or subtraction and logical justifications from the base line pixel coordinates in the target image to avoid a heterogeneous distance between rotated pixels in the target image each line rotates to two adjacent lines the proposed method yields good performance in terms of speed and quality according to the results of an analysis of the computation speed and accuracy

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

on a new gap phenomenon in riemannian geometry abstract this note announces some theorems that show that the flat metric of euclidean space has no nearby nonflat complete metric with the property that its curvature keeps a sign and is uniformly small in a sense made precise here

euler and navier stokes equations on the hyperbolic plane we show that nonuniqueness of the leray hopf solutions of the navier stokes equation on the hyperbolic plane 2 observed by chan and czubak is a consequence of the hodge decomposition we show that this phenomenon does not occur on n whenever n 3 we also describe the corresponding general hamiltonian framework of hydrodynamics on complete riemannian manifolds which includes the hyperbolic setting

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

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

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

the ergodic theory of discrete isometry groups on manifolds of variable negative curvature this paper studies the ergodic theory at infinity of an arbitrary discrete isometry group acting on any hadamard manifold h of pinched variable negative curvature most of the results obtained by sullivan in the constant curvature case are generalized to the case of variable curvature we describe connections between measures supported on the limit set of dynamics of the geodesic flow and the geometry of m h we explore the relationship between the growth exponent of the group the hausdorff dimension of the limit set and the topological entropy of the geodesic flow the equivalence of various descriptions of an analogue of the hopf dichotomy is proved as applications we settle a question of j feldman and m ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points at the end of this paper we apply our results to the study of some rigidity problems more applications to mostow rigidity for discrete subgroups of rank 1 noncompact semisimple lie groups with infinite covolume will be published in subsequent papers by the author notations h is a hadamard manifold with pinched sectional curvature k 2 k k 1 k2 k1 0 is a non elementary discrete isometry group acting on h freely and properly discontinuously which is called fuchsian h is the ideal boundary of h m h is a complete riemannian manifold sh or sm is the unit tangent bundle g is the geodesic flow on sh or sm v t is the unique geodesic in sh resp sm with the initial velocity v 0 v sh resp sm v resp v is the asymptotic class of the geodesic v t resp v t l is the limit set of on h l is the radical limit set of in l is the nonwandering set of the geodesic flow on sm received by the editors may 7 1995 1991 mathematics subject classification primary 58f17 secondary 58f11 58f15 20h10 research at msri supported by nsf grant dms 8505550 also partially supported by nsf grant dms 9403870 and sfb 170 c 1996 american mathematical society

the radial part of brownian motion on a manifold a semimartingale property on montre que la discrepance entre les deux membres de la formule de ito forme un processus aleatoire monotone

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

the space of complete embedded maximal surfaces with isolated singularities in the 3 dimensional lorentz minkowski space we prove that a complete embedded maximal surface in 3 dx12 dx22 dx32 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane and so it is asymptotic to a spacelike plane or a half catenoid we show that the moduli space of entire maximal graphs over x3 0 in with n 1 2 singular points and vertical limit normal vector at infinity is a 3n 4 dimensional differentiable manifold the convergence in means the one of conformal structures and weierstrass data and it is equivalent to the uniform convergence of graphs on compact subsets of x3 0 moreover the position of the singular points in 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology we also introduce the moduli space of marked graphs with n 1 singular points a mark in a graph is an ordering of its singularities which is a n 1 sheeted covering of we prove that identifying marked graphs differing by translations rotations about a vertical axis homotheties or symmetries about a horizontal plane the corresponding quotient space is an analytic manifold of dimension 3n 1 this manifold can be identified with a spinorial bundle associated to the moduli space of weierstrass data of graphs in

the limiting angle of certain riemannian brownian motions let m be a cartan hadamard manifold of dimension d 3 let p m and x exp r x x be geodesic polar coordinates with pole p and let x be the brownian motion on m let sectm x denote the sectional curvature of any plane section in mx we prove that for each c 2 there is a 0 1 such that if l2r x 2 sectm x cr x 2 for all x in the complement of a compact set then limt xt exists a s and defines a nontrivial invariant random variable the dirichlet problem at infinity and a conjecture of greene and wu are also discussed

a discontinuous galerkin method for viscous compressible multifluids abstract we present a generalized discontinuous galerkin method for a multicomponent compressible barotropic navier stokes system of equations the system presented has a functional viscosity which depends on the pressure p p i of the flow with the density and the local concentration i high order runge kutta time discretization techniques are employed and different methods of dealing with arbitrary coupled boundary conditions are discussed analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact analytic test cases finally several examples comparisons generalizations and physical applications are presented

computation and stability of fluxons in a singularly perturbed sine gordon model of the josephson junction new numerical results are reported regarding the solutions of a singularly perturbed sine gordon equation modeling magnetic flux quanta fluxons in long josephson tunnel junctions with nonzero surface impedance previous authors hampered by an inadequate numerical shooting method conjectured that the fluxon branch of solitary wave solutions terminates at a critical value of the bias current in the present paper a new numerical boundary value solver is developed and then used to show that the critical value of the bias current corresponds not to a termination point but to a turning point in the bifurcation diagram plotting a suitable norm of the solution against the bias current we find that the solution curve turns back at the critical value and then oscillates via a sequence of turning points to a limiting value of the bias current below the value at the first turning point these conclusions are confirmed a posteriors by adapting current numerical procedures described in the literature e g

zero dissipation limit of solutions with shocks for systems of hyperbolic conservation laws we consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers these interactions generate four different wave patterns initial layers shock layers diffusion waves and coupling waves we study the propagation and interactions of the four wave patterns by a detailed pointwise analysis

on the convergence to steady state of solutions of nonlinear hyperbolic parabolic systems the cauchy problem for time dependent quasi linear partial differential equations is considered and the nonlinear stability of steady state solutions is investigated

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