harmonic maps and periodic toda systems harmonic maps from the point of view of periodic toda systems are considered the correspondence between certain harmonic immersions of surfaces and solutions to periodic toda systems related to nonexceptional simple lie algebras is constructed

harmonic maps into the orthogonal group and null curves we find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group mathrm o n in terms of free holomorphic data which lead to formulae for all such harmonic maps our work reveals an interesting correspondence between certain harmonic maps and the free weierstrass representation of null curves and minimal surfaces in 3 and 4 space

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

invariant recurrence relations for cp n 1 models in this paper we present invariant recurrence relations for the completely integrable cp n 1 euclidean sigma model in two dimensions defined on the riemann sphere s 2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the su n algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the cp n 1 model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the cp 2 and cp 3 models

on the energy spectrum and parameter spaces of classicalcp n models it is known that a large class of smooth solutions ofcpn models can be constructed starting from holomorphic maps of an algebraic curve into complex projective spaces here we apply results from algebraic geometry to describe the energy spectrum and the parameter spaces for such models

strings harmonic maps and hyperbolic systems abstract we review a number of cases where strings and hyperbolic systems arise in the study of harmonic maps of pseudo riemannian manifolds

a finiteness theorem for harmonic maps into hilbert grassmannians in this article we demonstrate that every harmonic map from a closed riemannian manifold into a hilbert grassmannian has image contained within a finite dimensional grassmannian

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

lie groups and twistor spaces on considere un espace symetrique de riemann simplement connexe m la connexion de levi civita et le fibre torseur metrique 7b f m j m dont les fibres 7b f x m pour x m est constitue de structures complexes orthogonales

harmonic maps of finite uniton number into g2 we establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional lie group g2 in terms of the grassmannian model for the group of based algebraic loops in g2 a description of the frenet frame data for such harmonic maps is given in particular we show that harmonic spheres into g2 correspond to solutions of certain algebraic systems of quadratic and cubic equations

equivariant harmonic maps into homogeneous spaces this paper is about harmonic maps from closed riemann surfaces into homogeneous spaces such as flag manifolds and loop groups it contains the construction of a family of new examples of harmonic maps from t2 s1 s1 into f n or u n that are not holomorphic with respect to any almost complex structure on f n or u n where f n is the quotient of u n by any maximal torus and u n consists of f s1 u n smooth such that f 1 i

the space of almost complex 2 spheres in the 6 sphere the complex dimension of the space of linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is calculated to be d 8 explicit examples of these objects are constructed for every integer value of the degree d 6 d 6 7 furthermore it is shown that when d 6 this space is isomorphic to the group g2 c and when d 7 this space is empty we also show that the dimension of the space of non linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is 2d 5

on algebraic problems behind the brouwer degree of equivariant maps abstract given a finite group g and two unitary g representations v and w possible restrictions on topological degrees of equivariant maps between representation spheres s v and s w are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in s v denoted by v effective applications of these congruences is limited by answers to the following questions i under which conditions is v 1 and ii does there exist an equivariant map with the degree easy to calculate in the present paper we address both questions we show that v 1 for each irreducible non trivial c g module if and only if g is solvable this provides a new solvability criterion for finite groups for non solvable groups we use 2 transitive actions to construct complex representations with non trivial characteristic regarding the second question we suggest a class of norton algebras without 2 nilpotents giving rise to equivariant quadratic maps which admit an explicit formula for the degree

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

classification of conformal minimal immersions of constant curvature from s 2 to hp 2 in this paper we study geometry of conformal minimal two spheres immersed in quaternionic projective spaces we firstly use bahy el dien and wood s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from s 2 to the quaternionic projective space hp 2 then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from s 2 to the quaternionic projective space hp 2

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

on minimal surfaces in a kahler manifold of constant holomorphic sectional curvature this paper studies minimal surfaces in kshler manifolds of constant holomorphic sectional curvature using the technique of the moving frame in particular we provide a classification of the minimal two spheres in cpst complex projective sf space equipped with the fubini study metric this classification can be described as follows to each holomorphic curve in c pt1 classically there is associated a particular framing of c l l called the frenet frame each element of the frenet frame induces a minimal surface in cp the classification theorem states that all minimal surfaces of topological type of the two sphere occur in this manner the theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in kshler manifolds of constant holomorphic sectional curvature together with the riemann roch theorem

on conformal minimal immersions of constant curvature from s 2 to hp n we generalize here our general characterization of the harmonic sequence generated by conformal minimal immersions from s 2 to the quaternionic projective space hp n then we give a classification theorem of linearly full unramified conformal minimal immersions of constant curvature from s 2 to hp n under some conditions

the classification of the surfaces with parallel mean curvature vector in two dimensional complex space forms in this paper we find new surfaces with parallel mean curvature vectors in two dimensional complex space forms that are not of constant kahler angles and then classify all of them the surface can be isometrically deformed preserving both the length of the mean curvature vector and the kahler angle from this classification the deformations appear only when the curvature of the ambiant space is nonpositive

tournaments flags and harmonic maps twistor space constructions have shown that under quite general hypotheses harmonic maps of riemann surfaces into riemannian manifolds can be represented by holomorphic curves with respect to almost complex structures which are often non integrable see for example 19 17 5 for maps into complex grassmannians the almost complex structures are invariant ones on flag manifolds and can be identified with tournaments i e digraphs in which any two nodes are joined by exactly one oriented edge elementary properties of tournaments are summarised in sect i the basic references are moon 14 and reid and beineke 18 emphasis is given to certain types of tournament which are interpreted in sect 2 in terms of the homogeneous structure of complex flag manifolds holomorphic curves in such flag manifolds are described in sect 3 by collections of subbundles of a trivial bundle over a riemann surface whose second fundamental forms define a tournament this approach relies on techniques of burstall and wood 6 and gives rise naturally to harmonic maps into grassmannians when the riemann surface has genus zero a result from algebraic geometry namely the birkhoff grothendieck decomposition theorem for holomorphic vector bundles fits into the scheme and allows one to reconstruct a tournament from a harmonic map into a grassmannian in sect 4 we use such techniques to show that harmonic maps of tl2p 1 into a complex grassmannian can be built up successively from holomorphic data this last result extends results of burstall and wood 6 and is closely related to recent work of uhlenbeck 21 and wolfson 23

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

the topology of rational functions and divisors of surfaces segal studied the homotopy types of the spaces of holomorphic functions of the 2 sphere s 2 of closed surfaces of higher genus and of the spaces of divisors of these surfaces we continue segal s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

quantization of curvature of harmonic two spheres in grassmann manifolds various pinching theorems for curvature of minimal two spheres in grassmann manifolds have been proved in particular we show that when the curvature is large then the minimal map from s2 into g m n must be either holomorphic or antiholomorphic also minimal two spheres of curvature k 2 in g 2 4 have been classified 0 introduction it is known that starting from a harmonic map f from a riemann surface into a complex grassmann manifold one can define sequences of harmonic maps by using the 0 transform and the 0 transform these sequences are called harmonic sequences cw when f is holomorphic the sequence generated by f via the 0 transform is called a pseudoholomorphic sequence and the corresponding maps are called pseudoholomorphic maps in 1980 a m din and w j zakrzewski dz and then in 1983 j eells and j c wood ec showed that all harmonic maps from s2 to the complex projective space cpf are in fact pseudoholomorphic recently f burstall j rawnsley and s salamon announced in brs that all stable harmonic maps from s2 into irreducible hermitian symmetric spaces are either holomorphic or antiholomorphic it is also known that when the target space is a general complex grassmann manifold holomorphic curves do not generate all harmonic maps cw ra on the other hand m rigoli showed in r that for a linearly full nonsingular holomorphic curve from s2 into cpn if its curvature k 4 n then k 4 n is constant then j bolton et al bjrw generalized this result to pseudoholomorphic maps and showed that for a linearly full isometric pseudoholomorphic map of position r from s2 into cpn if its curvature k 4 2r n r n then k 41 2r n r n is constant in this paper we prove various pinching theorems of curvature for harmonic maps and show that for any nonsingular harmonic map from s2 into a complex grassmann manifold if the curvature of its induced metric is large then received by the editors january 18 1988 and in revised form april 11 1988 1980 mathematics subject classification 1985 revision primary 53a10 53c55

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

rigidity of pseudo holomorphic curves of constant curvature in grassmann manifolds rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces we prove that for such curves the curvature must be positive and that all such simply connected curves in cp are generated by veronese curves thus generalizing calabi s counterpart for holomorphic curves in cp we also classify all holomorphic curves from the riemann sphere into g 2 4 whose curvature is equal to 2 into two families which illustrates pseudo holomorphic curves of positive constant curvature in g m n are in general not unitarily equivalent constracting to the fact that generic isometric complex submanifolds in a kaehler manifold are congruent where f x dfi for j 0 1 2 and f m c7 w n are harmonic maps one defines mj x 0 if f is anti holomorphic we call fr a pseudo holomorphic curve of position r generated by when is holomorphic where is called the directrix of fr such curves with the induced metrics from the associated grassmannians form a class of minimal immersions when special ized to g l n i cp a pseudo holomorphic curve of position r is just the r th position vector of a frenet frame of a holomorphic curve projected into cp the importance of such maps comes from the fact that any har monic map from the riemann sphere or any harmonic map of nonzero degree from a torus into cp is pseudo holomorphic 7 furthermore in 6 these curves play a central role when m s defined in the sequence 0 1 are all equal and f s are orthogonal called frenet harmonic sequences for the study of

on the existence and nonexistence of extremal metrics on toric kahler surfaces abstract in this paper we study the existence of extremal metrics on toric kahler surfaces we show that on every toric kahler surface there exists a kahler class in which the surface admits an extremal metric of calabi we found a toric kahler surface of 9 t c 2 fixed points which admits an unstable kahler class and there is no extremal metric of calabi in it moreover we prove a characterization of the k stability of toric surfaces by simple piecewise linear functions as an application we show that among all toric kahler surfaces with 5 or 6 t c 2 fixed points cp 2 3 cp 2 is the only one which allows vanishing futaki invariant and admits extremal metrics of constant scalar curvature

invariant recurrence relations for mathbb c p n 1 models in this paper we present invariant recurrence relations for the completely integrable euclidean sigma model in two dimensions defined on the riemann sphere s2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the and models

isometric copies of c0 and l in duals of banach spaces abstract let x be a banach space then x contains an isometric copy of l if and only if x contains an asymptotically isometric copy of c0

harmonic maps of the two sphere into a complex grassmann manifold ii where qt is a complex valued one form defined up to a factor of absolute value 1 this form qp defines a complex structure on m for x e m the space f x has an orthogonal space f x of dimension n k we denote by f x and f x their corresponding projective spaces of dimensions k 1 and n k 1