on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

reductions of minimal lagrangian submanifolds with symmetries let m be a fano manifold equipped with a kahler form omega in 2 pi c 1 m and k a connected compact lie group acting on m as holomorphic isometries in this paper we show the minimality of a k invariant lagrangian submanifold l in m with respect to a globally conformal kahler metric is equivalent to the minimality of the reduced lagrangian submanifold l 0 l k in a kahler quotient m 0 with respect to the hsiang lawson metric furthermore we give some examples of kahler reductions by using a circle action obtained from a cohomogenenity one action on a kahler einstein manifold of positive ricci curvature applying these results we obtain several examples of minimal lagrangian submanifolds via reductions

the motion of closed hypersurfaces in the central force fields this paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential physically this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential the evolution of such surface has been investigated by schnurer and smoczyk 20 to study its motion we introduce a quasi linear degenerate hyperbolic equation which describes the motion of the surfaces extrinsically our main results show the large time existence of such cauchy problem and the stability with respect to small initial data when the radially symmetric potential function v 1 the local existence and stability results have been obtained by notz 18 the proof is based on a new nash moser iteration scheme

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

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

curvature decay estimates of graphical mean curvature flow in higher codimensions we derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions to the best of our knowledge this is the first such estimates without assuming smallness of first derivatives of the defining map an immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat riemann surfaces

second variation and stabilities of minimal lagrangian submanifolds in kahler manifolds first we derive a new second variation formula which holds for any minimal lagrangian submanifold in k ihler manifolds using this we prove that any minimal lagrangian submanifold in kfihler manifolds with nonpositive chern class is stable we also prove that in kfihler manifolds with positive chern class any stable minimal lagrangian submanifold l satisfies that h l 0 next we introduce the notion of hamiltonian stability either local or global of minimal lagrangian submanifolds in einstein kfihler manifolds using our second variation formula we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in einstein k ihler manifolds this criterion is satisfied for all real forms which are einstein of irreducible hermitian symmetric spaces e g the cannonical rp c cp it is also satisfied by the clifford torus t c cp

escape from the unstable equilibrium in a random process with infinitely many interacting particles we consider a one dimensional version of the model introduced in ref 1 at each site of z there is a particle with spin 1 particles move according to the stirring process and spins change according to the glauber dynamics in the hydrodynamical limit with the stirring process suitably speeded up the local magnetic density m r is proven in ref 1 to satisfy the reaction diffusion equation 8tmt r 1 2 rmt r v mt v m 89 4 and fl 0 c and fl being determined by the parameters of the glauber dynamics in the present paper we consider an initial state with zero magnetization mo r o we then prove that at long times before taking the hydrodynamical limit the evolution departs from that predicted by and that the microscopic state becomes a nontrivial mixture of states with different magnetizations

angle theorems for the lagrangian mean curvature flow we prove that symplectic maps between riemann surfaces l m of constant nonpositive and equal curvature converge to minimal symplectic maps if the lagrangian angle alpha for the corresponding lagrangian submanifold in the cross product space l times m satisfies text osc alpha le pi if one considers a 4 dimensional kahler einstein manifold overline m of nonpositive scalar curvature that admits two complex structures j k which commute and assumes that l subset overline m is a compact oriented lagrangian submanifold w r t j such that the kahler form overline kappa w r t k restricted to l is positive and text osc alpha le pi then l converges under the mean curvature flow to a minimal lagrangian submanifold which is calibrated w r t overline kappa

on donaldson s flow of surfaces in a hyperkahler four manifold we prove some basic properties of donaldson s flow of surfaces in a hyperkahler 4 manifold when the initial submanifold is symplectic with respect to one kahler form and lagrangian with respect to another we show that certain kinds of singularities cannot form and we prove a convergence result under a condition related to one considered by m t wang for the mean curvature flow

a note on singular time of mean curvature flow we show that mean curvature flow of a compact submanifold in a complete riemannian manifold cannot form singularity at time infinity if the ambient riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions

the dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions let be a bounded c2 domain in n and m be a continuous map the dirichlet problem for the minimal surface system asks whether there exists a lipschitz map f m with f and with the graph of f a minimal submanifold in n m for m 1 the dirichlet problem was solved more than 30 years ago by jenkins and serrin 12 for any mean convex domains and the solutions are all smooth this paper considers the dirichlet problem for convex domains in arbitrary codimension m we prove that if m satisfies 8n sup d2 2 sup d 1 then the dirichlet problem for is solvable in smooth maps here is the diameter of such a condition is necessary in view of an example of lawson and osserman 13 in order to prove this result we study the associated parabolic system and solve the cauchy dirichlet problem with as initial data 2003 wiley periodicals inc

on the second boundary value problem for lagrangian mean curvature flow abstract we consider a fully nonlinear parabolic equation with nonlinear neumann type boundary condition and show the long time existence and convergence of the flow finally we apply this study to the boundary value problem for minimal lagrangian graphs

hamiltonian stability of the gauss images of homogeneous isoparametric hypersurfaces i the image of the gauss map of any oriented isopara metric hypersurface of the unit standard sphere s n 1 1 is a mini mal lagrangian submanifold in the complex hyperquadric qn c in this paper we determine the hamiltonian stability of all com pact minimal lagrangian submanifolds embedded in complex hy perquadrics which are obtained as the images of the gauss map of homogeneous isoparametric hypersurfaces in the unit spheres

an iterated implicit complex map abstract iterated complex maps defined by a root of a biquadratic equation g z z 0 are studied numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the riemann sphere an analogy with hamiltonian systems is discussed

twistors 4 symmetric spaces and integrable systems an order four automorphism of a lie algebra gives rise to an integrable system introduced by terng we show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a riemannian symmetric space as applications we find that surfaces with holomorphic mean curvature in 4 dimensional real or complex space forms constitute an integrable system as do hamiltonian stationary lagrangian surfaces in 4 dimensional hermitian symmetric spaces this last providing a conceptual explanation of a result of helein romon

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

deforming symplectomorphisms of complex projective spaces by the mean curvature flow we apply the mean curvature flow to deform symplectomorphisms of cp n in particular we prove that for each dimension n there exists a constant explicitly computable such that any pinched symplectomorphism of cp n is symplectically isotopic to a biholomorphic isometry

symbolic dynamics and rotation numbers abstract in the space of binary sequences minimal sets that is sets invariant under the shift operation that have no invariant proper subsets are investigated in applications such as a piecewise linear circle map and the smale horseshoe in a mapping of the annulus each of these sets is invariant under the mapping these sets can be assigned a unique rotation number equal to the average of the number of ones in the sequences one special minimal set is the optimal set for which the convergence of the running average to the rotation number is faster than for any other minimal set these sequences are instrumental in analytically solving for the width of the parameter intervals for which members of a one parameter family of piecewise linear critical circle maps have rational rotation number

mean curvature flow of surfaces in einstein four manifolds let be a compact oriented surface immersed in a four dimensional kahler einstein manifold m w we consider the evolution of in the direction of its mean curvature vector it is proved that being symplectic is preserved along the flow and the flow does not develop type i singularity when m has two parallel kahler forms w and w that determine different orientations and is symplectic with respect to both w and w we prove the mean curvature flow of exists smoothly for all time in the positive curvature case the flow indeed converges at infinity

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

mean curvature flow of monotone lagrangian submanifolds we use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone lagrangian submanifolds in mathbb c n

minimal surfaces in kahler surfaces and ricci curvature soit m une surface immergee dans une surface de kahler n soient p le nombre de points tangents complexes et q le nombre de points tangents anticomplexes denombres selon leur multiple cite on deduit une version locale de p q x m x ou x m est la caracteristique d euler de m et x la caracteristique d euler du fibre normal de m dans n

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

long time existence and convergence of graphic mean curvature flow in arbitrary codimension let f 1 2 be a map between compact riemannian manifolds of constant curvature this article considers the evolution of the graph of f in 1 2 by the mean curvature flow under suitable conditions on the curvature of 1 and 2 and the differential of the initial map we show that the flow exists smoothly for all time at each instant t the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity this also provides a regularity estimate for lipschitz initial data

hamiltonian minimal submanifolds in complex hyperbolic space with so n symmetry in this paper we construct complete hamiltonian minimal submanifolds in complex hyperbolic space with so n symmetry which are not minimal in the usual sense

homotopy of area decreasing maps by mean curvature flow abstract let f m n be a smooth area decreasing map between two riemannian manifolds m g m and n g n under weak and natural assumptions on the curvatures of m g m and n g n we prove that the mean curvature flow provides a smooth homotopy of f to a constant map

on the existence of hamiltonian stationary lagrangian submanifolds in symplectic manifolds let m omega be a compact symplectic 2n manifold and g a riemannian metric on m compatible with omega for instance g could be k ahler with k ahler form omega consider compact lagrangian submanifolds l of m we call l it hamiltonian stationary or it h minimal if it is a critical point of the volume functional mathop rm vol nolimits g under hamiltonian deformations computing mathop rm vol nolimits g l using g vert l it is called it hamiltonian stable if in addition the second variation of mathop rm vol nolimits g under hamiltonian deformations is nonnegative our main result is that if l is a compact hamiltonian stationary lagrangian in mathbin bbb c n which is it hamiltonian rigid then for any m omega g as above there exist compact hamiltonian stationary lagrangians l in m contained in a small ball about some p in m and locally modelled on t l for small t0 identifying m near p with mathbin bbb c n near 0 if l is hamiltonian stable we can take l to be hamiltonian stable applying this to known examples l in mathbin bbb c n shows that there exist families of hamiltonian stable hamiltonian stationary lagrangians diffeomorphic to t n and to cal s 1 times cal s n 1 mathbin bbb z 2 and with other topologies in every compact symplectic 2n manifold m omega with compatible metric g

projections of vector addition system reachability sets are semilinear abstract the reachability sets of vector addition systems of dimension six or more can be non semilinear this may be one reason why the inclusion problem as well as the equality problem for reachability sets of vector addition systems in general is undecidable even though the reachability problem itself is known to be decidable we show that any one dimensional projection of the reachability set of an arbitrary vector addition system is semilinear and hence simple

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

on the regularity of hamiltonian stationary lagrangian submanifolds abstract we prove a morrey type theorem for hamiltonian stationary lagrangian submanifolds of c n if a c 1 lagrangian submanifold is a critical point of the volume functional under hamiltonian variations then it must be real analytic locally a hamiltonian stationary lagrangian submanifold is determined geometrically by harmonicity of its lagrangian phase function or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the hamiltonian stationary lagrangian submanifolds locally our result shows that morrey s theorem for minimal submanifolds admits a complete fourth order analogue we establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with c 1 1 norm below a dimensional constant and to c 1 1 potential functions under certain convexity conditions whose lagrangian phase functions are weakly harmonic

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

the extension and convergence of mean curvature flow in higher codimension in this paper we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a riemannian manifold with codimension d 1 which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to le ysum 12 and the authors 25 26 using the extension theorem we prove two convergence theorems for the mean curvature flow of closed submanifolds in r n d under suitable integral curvature conditions

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

mean curvature flow of spacelike graphs we prove the mean curvature flow of a spacelike graph in 1 2 g1 g2 of a map f 1 2 from a closed riemannian manifold 1 g1 with ricci1 0 to a complete riemannian manifold 2 g2 with bounded curvature tensor and derivatives and with sectional curvatures satisfying k2 k1 remains a spacelike graph exists for all time and converges to a slice at infinity we also show with no need of the assumption k2 k1 that if k1 0 or if ricci1 0 and k2 c c 0 constant any map f 1 2 is trivially homotopic provided f g2 0 and in case k2 0 this largely extends some known results for ki constant and 2 compact obtained using the riemannian structure of 1 2 and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo riemannian setting then in the riemannian one

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

an algorithm for the general petri net reachability problem an algorithm is presented for the general petri net reachability problem it is based on a generalization of the basic reachability tree construction which is made symmetric with respect to the ini

finite time singularities for lagrangian mean curvature flow given any embedded lagrangian on a four dimensional compact calabi yau we find another lagrangian in the same hamiltonian isotopy class that develops a finite time singularity under mean curvature flow this contradicts a weaker version of the thomas yau conjecture regarding long time existence and convergence of lagrangian mean curvature flow

the degenerate special lagrangian equation abstract this article introduces the degenerate special lagrangian equation dsl and develops the basic analytic tools to construct and study its solutions the dsl governs geodesics in the space of positive graph lagrangians in c n existence of geodesics in the space of positive lagrangians is an important step in a program for proving existence and uniqueness of special lagrangians moreover it would imply certain cases of the strong arnold conjecture from hamiltonian dynamics we show the dsl is degenerate elliptic we introduce a space time lagrangian angle for one parameter families of graph lagrangians and construct its regularized lift the superlevel sets of the regularized lift define subequations for the dsl in the sense of harvey lawson we extend the existence theory of harvey lawson for subequations to the setting of domains with corners and thus obtain solutions to the dirichlet problem for the dsl in all branches moreover we introduce the calibration measure which plays a role similar to that of the monge ampere measure in convex and complex geometry the existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the dsl have a well defined length in the space of positive lagrangians

mean curvature flow of compact spacelike submanifolds in higher codimension we prove the longtime existence for mean curvature flow of a smooth n dimensional spacelike submanifold of an n m dimensional manifold whose metric satisfies the timelike curvature condition

symbolic dynamics of order preserving orbits abstract the maps we consider are roughly those that can be obtained by truncating non invertible maps to weakly monotonic maps they have a flat piece the binary sequences that correspond to order preserving orbits are shown to satisfy a minimax principle which was already known for order preserving orbits with rational rotation number the converse is also proven all minimax orbits are order preserving with respect to some rotation number for certain families of such circle maps one can solve exactly for the parameter values for which the map has a specified rotation number rho for rho rational we obtain the endpoints of the resonance intervals recursively these parameter values can be organized in a natural way as the nodes of a farey tree we give some applications of the ideas discussed

radial solutions of a fourth order hamiltonian stationary equation abstract we consider smooth radial solutions to the hamiltonian stationary equation which are defined away from the origin we show that in dimension two all radial solutions on unbounded domains must be special lagrangian in contrast for all higher dimensions there exist non special lagrangian radial solutions over unbounded domains moreover near the origin the gradient graph of such a solution is continuous if and only if the graph is special lagrangian

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

hamiltonian minimal lagrangian submanifolds in kaehler manifolds with symmetries abstract by making use of the symplectic reduction and the cohomogeneity method we give a general method for constructing hamiltonian minimal lagrangian submanifolds in kaehler manifolds with symmetries as applications we construct infinitely many nontrivial complete hamiltonian minimal lagrangian submanifolds in c p n and c n

supercritical behavior of an ordered trajectory the behavior of a particle undergoing a discrete stepped motion around a circle is investigated for the case in which the average number of steps per revolution is the golden mean the new features of this study are that the map in question is not invertible and that consequently the orbit is bunched into intervals within the circle because of this bunching one can calculate the motion in some detail

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem