rigidity theorems for submetries in positive curvature abstract we derive general structure and rigidity theorems for submetries f m x where m is a complete riemannian manifold with sectional curvature sec m 1 when applied to a non trivial riemannian submersion it follows that diam x 2 in case of equality there is a riemannian submersion s m from a unit sphere and as a consequence f is known up to metric congruence a similar rigidity theorem also holds in the general context of riemannian foliations

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

a morse theoretical approach for the boundary mean curvature problem on b4 abstract in this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4 dimensional ball due to the existence of critical point at infinity the standard variational methods cannot be applied to overcome this difficulty we prove that in a neighborhood of critical points at infinity a morse lemmas at infinity reduction holds then develop a whole morse theory of this noncompact variational problem in particular we establish under generic boundary condition morse inequalities at infinity which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated euler lagrange functional as further application of this morse theoretical approach we prove more existence results and extend a topological invariant introduced by a bahri

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

on a conjecture of anosov abstract in this paper we prove that for every bumpy finsler n sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 there exist 2 n 1 2 prime closed geodesics this gives a confirmed answer to a conjecture of anosov 1 in 1974 for a generic case

on the average indices of closed geodesics on positively curved finsler spheres in this paper we prove that on every finsler n sphere s n f for n 6 with reversibility and flag curvature k satisfying frac lambda lambda 1 2 k le 1 either there exist infinitely many prime closed geodesics or there exist frac n 2 2 closed geodesics possessing irrational average indices if in addition the metric is bumpy then there exist n 3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite

multi peak solutions for a singularly perturbed semilinear elliptic problem this paper is concerned with the existence of multi peak solutions of the following problem formula where is a smooth bounded domain in rn n 2 e 0 is a small positive number and p 2 2nn 2 for n 3 p 2 for n 2 using a variational method we construct positive solutions concentrating near given saddle points of the distance function dist an explicit expression for the dominant parts of the solutions is also obtained

the index growth and multiplicity of closed geodesics abstract in the recent paper 31 of long and duan 2009 we classified closed geodesics on finsler manifolds into rational and irrational two families and gave a complete understanding on the index growth properties of iterates of rational closed geodesics this study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible including riemannian finsler sphere and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 3 dimensional manifold in this paper we study the index growth properties of irrational closed geodesics on finsler manifolds this study allows us to extend results in 31 of long and duan 2009 on rational and in 12 of duan and long 2007 39 of rademacher 2010 and 40 of rademacher 2008 on completely non degenerate closed geodesics on spheres and cp 2 to every compact simply connected finsler manifold then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 4 dimensional manifold

conformal transformation of metrics on the n sphere abstract the purpose of this work is to discuss some new results on the scalar curvature problem in dimension n 3 we give precise estimates on the losses of compactness and we prove the existence of at least a solution under the assumption that the prescribed function is flat near its critical points

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

rigidity of group actions on solvable lie groups we establish analogs of the three bieberbach theorems for a lattice gamma in a semidirect product mathsf k rtimes mathsf k where mathsf s is a connected simply connected solvable lie group and mathsf k is a compact subgroup of its automorphism group we first prove that the action of gamma on mathsf s is metrically equivalent to an action of gamma on a supersolvable lie group the latter is shown to be determined by gamma itself up to an affine diffeomorphism then we characterize these lattices algebraically as polycrystallographic groups furthermore we realize any polycrystallographic group gamma as a lattice in a semidirect product mathsf s rtimes mathsf f with mathsf f being a finite group whose order is bounded by a constant only depending on the dimension of mathsf s this generalization of the first bieberbach theorem is used to obtain a partial generalization of the third one as well finally we show for any torsion free closed subgroup upsilon subset mathsf k rtimes mathsf k that the quotient mathsf s upsilon is the total space of a vector bundle over a compact manifold b where b is the quotient of a solvable lie group by a torsion free polycrystallographic group

injectivity radius and diameter of the manifolds of flags in the projective planes the manifolds of flags in the projective planes are among the very few compact manifolds that are known to admit metrics with positive sectional curvature they also arise as isoparametric hypersurfaces in spheres we show how their appearance in these two fields is related and study the global geodesic geometry of these spaces in detail

an o 3 1 nonlinear sigma model and the ablowitz ladik hierarchy a solvable nonlinear 2d field model is proposed which is shown to be closely connected with the nonlinear schrodinger chain the conservation laws and soliton solutions for the corresponding field equations are obtained

isometric actions on spheres with an orbifold quotient we classify representations of compact connected lie groups whose induced action on the unit sphere has the orbit space isometric to a riemannian orbifold

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

on the quasi periodic solutions to the discrete nonlinear schrodinger equation the authors have obtained the quasi periodic solutions to the discrete non linear schrodinger equation by a variant of a method due to date and tanaka 1976 it is shown explicitly that the non linear field variables at the different lattice points can be determined in a recursive fashion in terms of combinations of riemann s theta functions depending on lattice position and time

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

bifurcation from infinity in a nonlinear elliptic equation involving the limiting sobolev exponent we consider an elliptic pde with critical nonlinearity and additonal subcritical and super quadratic nonlinearity with dirichlet boundary conditions on a 3 dimensional smooth and bounded domain we prove the existence of at least p 1 solutions where p is the ljusternik schnirelman category of the domain we prove additional results concerning double peaked solutions on a ringshaped domain in particular we characterize the points at which the solutions blow up as the ringshaped domain becomes thinner or thicker

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

existence results for the fractional nirenberg problem abstract we consider the fractional nirenberg problem on the standard sphere s n with n 4 using the theory of critical points at infinity we establish an euler hopf type formula and obtain some existence results for curvature satisfying assumptions of bahri coron type

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

multiplicity result for a nonhomogeneous yamabe type equation involving the kohn laplacian abstract we consider the nonhomogeneous yamabe equation on a bounded set of the heisenberg group h u u q 2 u f where f is a small perturbation in the c 0 sense under suitable hypotheses we will state a multiplicity existence result for positive solutions with zero dirichlet boundary conditions

on the symmetric scalar curvature problem on sn abstract we discuss some existence results dealing with the scalar curvature problem on s n in the presence of various symmetries

finite genus solutions for the ablowitz ladik hierarchy the question of constructing the finite genus quasiperiodic solutions for the ablowitz ladik hierarchy alh is considered by establishing relations between the alh and fay s identity for the functions it is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential difference equations coinciding with the equations of the alh and that the original fay s identity can be rewritten in a form similar to the functional equations representing the alh which have been derived in the previous works of the author this provides an algorithm for obtaining some class of quasiperiodic solutions for the alh which can be viewed as an alternative to the inverse scattering transform or the algebro geometrical approach

explicit solutions for a hierarchy of differential difference equations a new discrete isospectral problem and the corresponding hierarchy of nonlinear differential difference equations are proposed on the basis of the theory of algebraic curves the continuous flow and discrete flow related to the hierarchy of differential difference equations are straightened out using the abel jacobi coordinates the meromorphic function and the baker akhiezer function are introduced on the hyperelliptic curve quasi periodic solutions of the corresponding hierarchy of differential difference equations are constructed with the help of the asymptotic properties and the algebro geometric characters of the meromorphic function the baker akhiezer function and the hyperelliptic curve

on the prescribed q curvature problem on sn in this paper we prescribe a fourth order curvature the q curvature on the standard n sphere n 5 under flatness condition of order n 4 n near each critical point of the prescribed q curvature function we characterize the critical points at infinity of the associated variational problem and we prove new existence results through euler hopf formulae type our argument gives an upper bound on the morse index of the obtained solutions we also give a lower bound on the number of conformal metrics having the same q curvature

critical points at infinity in a fourth order elliptic problem with limiting exponent abstract this paper is concerned with the following nonlinear elliptic problem p 2 u u q u 0 in u u 0 on where is a bounded smooth domain in r n n 6 and q 1 2 n n 4 is the critical sobolev exponent we provide a version of morse lemma at infinity for this problem as application we characterize the critical points at infinity of the associated variational problem we also compute the difference of the topology that these critical points at infinity induce between the level sets of the functional corresponding to the problem

on an elliptic problem with critical nonlinearity in expanding annuli this paper is concerned with an elliptic problem with homogeneous boundary conditions and critical nonlinearity pe u up u 0 onae u 0 on ae whereae x rn e x 1 e are expanding annuli ase 0 n 3 andp 1 2n n 2 is the critical sobolev exponent we compute the difference of topology induced by the critical points at infinity between the level sets of the functional corresponding to pe foresmall enough

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

existence and nonexistence of solutions of nonlinear dirichlet problems with first order terms abstract existence theorems for nonnegative solutions to a class of nonlinear dirichlet problems with first order terms are proved nonexistence results are also discussed depending on the regularity of the coefficient of the first order term the proofs make use of a direct variational approach and of integral identities

riemannian submersions from compact four manifolds we show that under certain conditions a nontrivial riemannian submersion from positively curved four manifolds does not exist this gives a partial answer to a conjecture due to fred wilhelm we also prove a rigidity theorem for riemannian submersions with totally geodesic fibers from compact four dimensional einstein manifolds

blow up and symmetry of sign changing solutions to some critical elliptic equations abstract in this paper we continue the analysis of the blow up of low energy sign changing solutions of semi linear elliptic equations with critical sobolev exponent started in m ben ayed k el mehdi f pacella blow up and nonexistence of sign changing solutions to the brezis nirenberg problem in dimension three ann inst h poincare anal non lineaire in press in addition we prove axial symmetry results for the same kind of solutions in a ball

on a class of singular elliptic problems with nonradial first order terms abstract we investigate existence or nonexistence of solutions to a class of singular dirichlet problems depending in particular on the regularity of the coefficients of the differential equation as well as on geometrical properties of the domain under consideration

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

conformal transformation of metrics on the n ball abstract in this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of r n n 3 under the assumption that the prescribed function is flat near its critical point we give precise estimates on the losses of the compactness and we provide new existence and multiplicity results of bahri coron type

on a fourth order elliptic equation with critical nonlinearity in dimension six abstract this paper is devoted to the study of the following nonlinear elliptic problem 2 u ku 5 u 0 in u u 0 on where k is a positive function and is a bounded and smooth domain in r 6 we prove a version of morse lemma at infinity for this problem using this morse lemma at infinity we characterize the critical points at infinity of the associated functional and we give an existence result

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

classification of pseudo riemannian submersions with totally geodesic fibres from pseudo hyperbolic spaces we classify pseudo riemannian submersions with connected totally geodesic fibres from a real pseudo hyperbolic space onto a pseudo riemannian manifold also we obtain the classification of the pseudo riemannian submersions with para complex connected totally geodesic fibres from a para complex pseudo hyperbolic space onto a pseudo riemannian manifold

on a nirenberg type problem involving the square root of the laplacian abstract through an euler hopf type formula we establish existence result to a nirenberg type problem involving the square root of the laplacian in sphere s 2

algebro geometric solutions of the baxter szegő difference equation we derive theta function representations of algebro geometric solutions of a discrete system governed by a transfer matrix associated with an extension of the trigonometric moment problem studied by szeg and baxter we also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro geometric solutions

algebro geometric constructions of the discrete ablowitz ladik flows and applications resorting to the finite order expansion of the lax matrix the elliptic coordinates are introduced from which the discrete ablowitz ladik equations and the 2 1 dimensional toda lattice are decomposed into solvable ordinary differential equations the straightening out of the continuous flow and the discrete flow is exactly given through the abel jacobi coordinates as an application explicit quasiperiodic solutions for the 2 1 dimensional toda lattice are obtained

index parity of closed geodesics and rigidity of hopf fibrations the diameter rigidity theorem of gromoll and grove 1987 states that a riemannian manifold with sectional curvature 1 and diameter 2 is either homeomorphic to a sphere locally isometric to a rank one symmetric space or it has the cohomology ring of the cayley plane ca the reason that they were only able to recognize the cohomology ring of ca is due to an exceptional case in another theorem gromoll and grove 1988 a riemannian submersion m b b with connected fibers that is defined on the euclidean sphere m is metrically congruent to a hopf fibration unless possibly m b 15 8 we will rule out the exceptional cases in both theorems our argument relies on a rather unusal application of morse theory for that purpose we give a general criterion which allows to decide whether the morse index of a closed geodesic is even or odd

on a nonlinear fourth order elliptic equation involving the critical sobolev exponent in this paper we use an algebraic topological argument due to bahri and coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical sobolev exponent 2u un 4 n 4 in u 0 in u u 0 on where is a bounded domain of rn n 5 with a smooth boundary

closed geodesics on positively curved finsler spheres in this paper we prove that for every finsler n sphere sn f for n3 with reversibility and flag curvature k satisfying either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized poincare map has at least one eigenvalue which is of the form exp pi with an irrational furthermore there always exist three prime closed geodesics on any s3 f satisfying the above pinching condition

on the berger conjecture for manifolds all of whose geodesics are closed we define a besse manifold as a riemannian manifold m g all of whose geodesics are closed a conjecture of berger states that all prime geodesics have the same length for any simply connected besse manifold we firstly show that the energy function on the free loop space of a simply connected besse manifold is a perfect morse bott function with respect to a suitable cohomology secondly we explain when the negative bundles along the critical manifolds are orientable these two general results then lead to a solution of berger s conjecture when the underlying manifold is a sphere of dimension at least four

focal radius rigidity and lower curvature bounds this is the accepted version of the following article luis guijarro and frederick wilhelm focal radius rigidity and lower curvature bounds which has been published in final form at https doi org 10 1112 plms 12113

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

stability of closed geodesics on finsler 2 spheres abstract in this paper we prove that on every finsler 2 dimensional sphere either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics

a prescribed scalar curvature type equation almost critical manifolds and multiple solutions we present an asymptotic analysis for a perturbed prescribed scalar curvature type equation a major consequence is a non existence result in low dimension conversely we prove an existence result in higher dimensions to this aim we develop a general finite dimensional reduction procedure for perturbed variational functionals the general principle can be useful to discuss some other nonlinear elliptic pde with sobolev critical growth in bounded domains

cuplength estimates on lagrangian intersections on demontre l estimation suivante sur des intersections lagrangiennes si l est une sous variete lagrangienne de p avec 2 p l 0 et l est obtenue a partir de l par un diffeomorphisme exact de p alors le nombre d elements de l l est superieure ou egale a la longueur de calotte de p

algebraic topological methods for the supercritical q curvature problem abstract we study the problem of existence of conformal metrics with prescribed q curvature on closed four dimensional riemannian manifolds this problem has a variational structure and in the case of interest here it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated euler lagrange functional the so called true critical points at infinity of the associated variational problem occur using the characterization of the critical points at infinity of the associated variational problem which is established in 42 combined with some arguments from morse theory some algebraic topological methods and some tools from dynamical system originating from conley s isolated invariant sets and isolated blocks theory we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold stable and unstable manifolds of some of the critical points at infinity of the associated euler lagrange functional

existence and multiplicity results for a fourth order mean field equation in this article we consider the following fourth order mean field equation on smooth domain r4 2u keu keuin u u 0on where r and 0 k c2 through a refined blow up analysis we characterize the critical points at infinity of the associated variational problem and compute their contribution of the difference of topology between the level sets of the associated euler lagrange functional we then use topological and dynamical methods to prove some existence and multiplicity results