second eigenvalue of schrodinger operators and mean curvature let m be a compact immersed submanifold of the euclidean space the hyperbolic space or the standard sphere for any continuous potential q on m we give a sharp upper bound for the second eigenvalue of the operator q in terms of the total mean curvature of m and the mean value of q moreover we analyze the case where this bound is achieved as a consequence of this result we obtain an alternative proof for the alikakos fusco conjecture concerning the stability of the interface in the allen cahn reaction diffusion model

euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

rotational surfaces in mathbb l 3 and solutions of the nonlinear sigma model the gauss map of non degenerate surfaces in the three dimensional minkowski space are viewed as dynamical fields of the two dimensional o 2 1 nonlinear sigma model in this setting the moduli space of solutions with rotational symmetry is completely determined essentially the solutions are warped products of orbits of the 1 dimensional groups of isometries and elastic curves in either a de sitter plane a hyperbolic plane or an anti de sitter plane the main tools are the equivalence of the two dimensional o 2 1 nonlinear sigma model and the willmore problem and the description of the surfaces with rotational symmetry a complete classification of such surfaces is obtained in this paper indeed a huge new family of lorentzian rotational surfaces with a space like axis is presented the description of this new class of surfaces is based on a technique of surgery and a gluing process which is illustrated by an algorithm

euler lagrange equations for variational problems on space curves we derive the euler lagrange equations for a large class of variational problems on curves our result generalizes a recent result obtained in the literature moreover it is simple and self contained it directly yields euler lagrange equations in the form of equilibrium equations for the internal force and moment

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

studies in numerical nonlinear instability ii a new look at u t uu x 0 abstract it is shown that in general leap frog schemes any particular unstable solution behaves as an attractor of other solutions for a leap frog discretization of u t uu x 0 a particular kind of unstable solution is constructed which generically attracts any other solution estimates of the overflow time are presented and related to the notions of stability threshold and restricted stability

regular tessellations of surfaces and p q 2 triangle groups the purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the schwarz triangle groups of type p q 2 these tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five platonic solids a regular tessellation of type p q a p q pattern for short on a surface m is a tessellation of m by p sided faces such that the valence of each vertex is q a more precise definition appears in 1 no global symmetry is assumed a p q pattern on m is said to be geometric if m admits a riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are 2 7 q if the curvature happens to he zero then we further insist that the area of each face be 1 it is natural to call two p q patterns on m

curve straightening and the palais smale condition this paper considers the negative gradient trajectories associated with the modified total squared curvature functional f k2 iv ds the focus is on the limiting behavior as m tends to zero from the positive side it is shown that when m 0 spaces of curves exist in which some trajectories converge and others diverge in one instance the collection of critical points splits into two subsets as m tends to zero the critical curves in the first subset tend to the critical points present when m 0 meanwhile all the critical points in the second subset have lengths that tend to infinity it is shown that this is the only way the palais smale condition fails in the present context the behavior of the second class of critical points supports the view that some of the trajectories are dragged all the way to infinity when the curves are rescaled to have constant length the euler figure eight emerges as a critical point at infinity it is discovered that a reflectional symmetry need not be preserved along the trajectories there are examples where the length of the curves along the same trajectory is not a monotone fiunction of the flow time it is shown how to determine the elliptic modulus of the critical curves in all the standard cases the modulus p must satisfy 2e p k p 1 gl l when the space is limited to curves of fixed length l and the endpoints are separated by the vector g

hamilton equations for elasticae in the euclidean 3 space abstract the variational problem on spatial curves defined by the integral of the squared curvature whose solutions are the elasticae or nonlinear splines is analyzed from the hamiltonian point of view using a procedure developed by munoz masqueand pozo coronado j munoz masque l m pozo coronado j phys a 31 1998 6225 6242 the symmetry of the problem under rigid motions is then used to reduce the euler lagrange equations to a first order dynamical system

a variational characterization and geometric integration for bertrand curves in this paper we introduce a class of functionals in the three dimensional semi euclidean space rq3 having an energy density that depends only on curvature and whose moduli space of trajectories consists of lw curves i e curves with curvature and torsion for which there are three real constants such that with 2 2 0 this family of curves includes plane curves helices curves of constant curvature curves of constant torsion lancret curves also called generalized helices and bertrand curves we present an algorithm to construct bertrand curves in rq3 by using an arclength parametrized curve in a totally umbilical surface s2 s12 or h2 and prove that every bertrand curve in rq3 can be obtained in this way a second algorithm is presented for the construction of lw curves by using a curve of constant slope in the ruled surface s whose directrix is a certain curve with non zero curvature and whose rulings are generated by its modified darboux vector field

classification of rotational surfaces with constant skew curvature in 3 space forms abstract in this paper we classify the rotational surfaces with constant skew curvature in 3 space forms we also give a variational characterization of the profile curves of these surfaces as critical points of a curvature energy involving the exponential of the curvature of the curve finally we provide a converse process to produce all rotational surfaces with constant skew curvature based on the evolution of a critical curve under the flow of its binormal vector field with prescribed velocity

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

local geometric invariants of integrable evolution equations the integrable hierarchy of commuting vector fields for the localized induction equation of 3d hydrodynamics and its associated recursion operator are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept out surface

unit speed stationary points of the acceleration we consider the variational problem associated to the l2 norm of the angular acceleration for variations with constant length this is a classical problem related to recent models for the spinning relativistic particles both massive and massless and to bernoulli s elastica problem its interest arises also in computer graphics and in trajectory planning problems for rigid body motion explicit integration of the euler lagrange equations is considerably difficult in general we show how one can use a geometric integration procedure to achieve this goal for unit speed critical points in real space forms from the euler lagrange equations we obtain the geometric integration critical curves are a family of ordinary helices in a real space form of dimension 3 at most moreover ordinary helices in real space forms can be characterized in turn as geodesics in certain constant mean curvature flat cylinders this will allow us to obtain the analytical integration of the euler lagrange equations we also d

curve straightening in closed euclidean submanifolds the flow in the negative direction of the gradient vector field associated with the functional total squared geodesic curvature k2ds is the so called curvestraightening flow this paper will consider spaces of closed curves in closed euclidean submanifolds it will define these spaces of curves as submanifolds of certain hilbert manifolds representing all curves the main result will then be to show the existence of a particular set of functionals defined on the entire hilbert manifold which have the following four properties 1 the directional derivatives of these functionals may be computed by solving an initial value problem for a system of ordinary differential equations 2 by introducing a suitable hilbert space basis for the sobolev spaces used the gradients may be effectively computed but of course not explicitly computed except in very special cases 3 the gradients span the space normal to the tangent space of the space of closed curves 4 despite the fact that these gradients in general are not given explicitly it is nevertheless possible to compute the projection onto the tangent space to the space of closed curves in particular we do this for the gradient of k2ds when all details are worked out this gives us an algorithm which we supply for finding critical points in the space of closed curves it is not known if the trajectories actually always converge to critical points if the functional is modified to include a multiple of the length so the functional becomes k2 ds then the above convergence is known for 0 the motivating application for the curvestraightening flow is the possibility of using it to find non trivial closed periodic geodesics note that if 0 then a closed geodesic is a global minimum for any geodesics are critical but there are also other critical points the so called elastic curves the paper concludes by deriving the second variation formula for k2 ds along closed geodesics the quadratic functional associated with the second derivative is shown to be positive definite even for non zero along some closed geodesics in some particular manifolds of interest

how paper folds bending with local constraints a variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry this framework is applied to describe the patterns of a folded sheet of paper the unstretchability of paper implies a constraint on the surface metric bending is penalized by an energy quadratic in mean curvature the local lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet the framework is illustrated by examining the deformation of a flat sheet into a generalized cone

on modulational instabilities in discretisations of the korteweg devries equation abstract w l briggs a c newell and t sarie j comput phys 51 83 1983 have studied nonlinear instabilities in discretisations of the equation u 1 u u u x 0 where u is a constant they demonstrated that solutions of leap frog discretisations of this equation could be destabilised by a dynamical process which can localise noise on the spatial grid d m sloan and a r mitchell j comput phys 67 1986 examined these instabilities by considering perturbations of the basic solution which take the form of side band fourier modes here this latter work is extended to cover analogous discretisations of the korteweg de vries equation

total positive curvature of circular dna the properties of double stranded dna and other chiral molecules depend on the local geometry i e on curvature and torsion yet the paths of closed chain molecules are globally restricted by topology when both of these characteristics are to be incorporated in the description of circular chain molecules e g plasmids it is shown to have implications for the total positive curvature integral for small circular micro dnas it follows as a consequence of fenchel s inequality that there must exist a minimum length for the circular plasmids to be double stranded it also follows that all circular micro dnas longer than the minimum length must be concave a result that is consistent with typical atomic force microscopy images of plasmids predictions for the total positive curvature of circular micro dnas are given as a function of length and comparisons with circular dnas from the literature are presented

hamiltonians for curves we examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion to do this we tailor the theory of deformations to the frenet serret frame of the curve the euler lagrange equations describing equilibrium are obtained noether s theorem is exploited to identify the constants of integration of these equations as the casimirs of the euclidean group in three dimensions while this system appears not to be integrable in general it is in various limits of interest let the energy density be given as some function of the curvature and torsion f i ff is a linear function of either of its arguments but otherwise arbitrary we claim that the first integral associated with rotational invariance permits the torsion to be expressed as the solution of an algebraic equatio ni n terms ofthe bending curvature the first integral associated with translational invariance can then be cast as a quadrature for or for

environmental bias and elastic curves on surfaces the behavior of an elastic curve bound to a surface will reflect the geometry of its environment this may occur in an obvious way the curve may deform freely along directions tangent to the surface but not along the surface normal however even if the energy itself is symmetric in the curve s geodesic and normal curvatures which control these modes very distinct roles are played by the two if the elastic curve binds preferentially on one side or is itself assembled on the surface not only would one expect the bending moduli associated with the two modes to differ binding along specific directions reflected in spontaneous values of these curvatures may be favored the shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of lagrange multipliers to the darboux frame associated with the curve the forces transmitted to the surface along the surface normal will be determined features associated with a number of different energies both of physical relevance and of mathematical interest are described the conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined

critical curves for a santalo problem in 3 space forms abstract l santalo 1956 studied in 15 the minimization of frenet first curvature dependent energies for spaces of curves constrained to lie in a surface of the euclidean 3 space in this paper we analyze an extension of that problem by considering energy functionals determined by lagrangians which depend not only on the curvature but also on the torsion acting on spaces of surface contained clamped curves of real 3 space forms

null scrolls as solutions of a sigma model the two dimensional o 2 1 nonlinear sigma model with boundary is considered we calibrate the size of its space of field configurations by exhibiting new and wide classes of solutions we first construct solutions by evolving under a certain group of transformations free elastic curves in any surface either riemannian or lorentzian of constant curvature furthermore we show that any null scroll can provide a solution of this sigma model this surprising phenomenon which obviously has no euclidean counterpart guarantees the existence of an ample class of solutions which are generated by null or lightlike curves evolving through null ruling flows

constant mean curvature invariant surfaces and extremals of curvature energies abstract we determine the extremal curves of curvature energy functionals which generalize a variational problem studied by blaschke in r 3 the generalization is made by extending both the lagrangian energy itself and the ambient space to riemannian and lorentzian 3 space forms then we show that constant mean curvature cmc invariant surfaces in 3 space forms can be constructed locally as the evolution of the above extremals under their binormal flow with appropriate velocity moreover we see that these surfaces are intrinsically described by a warping function satisfying an ermakov milne pinney equation finally we use the previous findings to extend known results on isometric deformations of cmc surfaces and the lawson s correspondence of cmc cousins to our background spaces

equivariant constrained willmore tori in the 3 sphere in this paper we study equivariant constrained willmore tori in the 3 sphere these tori admit a 1 parameter group of mobius symmetries and are critical points of the willmore energy under conformal variations we show that the spectral curve associated to an equivariant torus is given by a double covering of mathbb c and classify equivariant constrained willmore tori by the genus g of their spectral curve in this case the spectral genus satisfies g le 3

strange actions of groups on spheres in fs we investigated certain topological analogs of schottky groups called admissible actions and their compatibility with various structures on spheres we constructed an action f 2 s 3 s 3 which was not topologically conjugate to a uniformly quasiconformal action also there was an example f r 2 s 3 s 3 r sufficiently large which was smooth and uniformly quasiconformal but not topologically conjugate to a conformal action and we gave examples of admissible actions on higher dimensional spheres but did not analyse the structures preserved

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

the influence of differencing and cfl number on implicit time dependent non linear calculations abstract this paper is concerned with the stability and precision of transient implicit finite difference algorithms the test problem is the burgers equation the influence of space discretisation and initial values is studied linear analysis provides incomplete stability conditions for both continuous and weak solutions and we derive a simple criterion of augmented stability taking into account the non linearity of the problem the real stability limits appear to be a combination of the linear stability analysis criterion and of this augmented stability criterion

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

extremal curves for weighted elastic energy in surfaces of 3 space forms a variational problem closely related to the bending energy of curves contained is surfaces of real 3 space forms is considered we seek curves in a surface which are critical for the elastic energy when this is weighted by the total squared normal curvature energy under two different sets of constraints clamped curves and one free end curves of constant length we start by deriving the first variation formula and the corresponding euler lagrange equations and natural boundary conditions of these energies and characterize critical geodesics we show how surfaces locally foliated by critical geodesics can be found by using the fundamental theorem of submanifolds in order to find explicit solutions we classify complete rotation surfaces in a real space form for which every parallel is critical

parametric excitation of computational modes inherent to leap frog scheme applied to the korteweg de vries equation abstract the numerical instability in the usage of the leap frog scheme applied to the korteweg de vries equation is studied the leap frog scheme generates computational modes which are not included in the original partial differential equation it is shown that the numerical instability comes from the parametric excitation of the computational modes by physical waves a method to suppress the computational modes is given the method enables us to make long time integration of the korteweg de vries equation even if the amplitude of wave is large

on otsuki tori and their willmore energy abstract otsuki tori in the three dimensional unit sphere are studied from calculations of the willmore energy the lower and upper bound estimates for the otsuki tori are established invariance of the otsuki tori under antipodal maps is also considered and in such cases better lower bound estimates of their willmore energies are obtained

existence of free nonclosed euler bernoulli elastica the total squared curvature functional f y s k2 ds has appeared in many different contexts in differential geometry f has emerged in conjunction with the development of flow methods used to locate closed geodesics see l s since geodesics are global minima of f it is natural to attempt to introduce a hilbert manifold of curves in which the negative gradient flow may be used to locate the geodesics

the hamiltonian theory of elastica the total squared curvature of the curve 7 for a brief historical discussion see truesdell 7 it is worth noting here though that euler solved the problem for curves in the plane the problem of elastica has recently generated renewed interest notable recent works include that by langer and singer 5 who were able to integrate the frenet equations for elastica with fixed arclength in spaces of constant curvature with the use of killing fields griffiths 4 section ii b iii derived the same integrals for the euler lagrange equations for elastica in r3 bryant and griffiths 2 performed marsden weinstein reduction on elastica in the plane using the griffiths formalism however none of these works has fully revealed the role played by the euclidean symmetry for elastica in space in this paper we find what this role is for elastica in r3 langer and singer showed that the euler lagrange equa tions are

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

grid resonances focusing and benjamin feir instabilities in leapfrog time discretizations abstract w l briggs a c newell and t sarie j comput phys 50 83 1983 studied the solutions of a class of discretizations of u t au x uu x 0 where a is a constant they discovered that the numerical solution of the discretization becomes unbounded despite the fact that theoretically the solution should remain bounded they attributed this anomalous behavior to a focusing mechanism in this paper we make use of a multiple scales analysis and show that the instabilities are caused by a resonance effect introduced by the discretizations the contributions from the space and time discretizations are analysed separately and in detail thus the structure of the focusing mechanism becomes transparent

extendability of kirchhoff elastic rods in complete riemannian manifolds the kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods and is defined to be a solution of the euler lagrange equations associated to the energy with the effect of bending and twisting we consider the initial value problem for the euler lagrange equations in a riemannian manifold in a previous paper the author proved the existence and uniqueness of global solutions of the initial value problem in the case where the ambient space is a space form in the present paper we extend this result to the case where the ambient space is a general complete riemannian manifold this implies that an arbitrary kirchhoff elastic rod of finite length in a complete riemannian manifold extends to that of infinite length

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

on nonlinear instabilities in leap frog finite difference schemes abstract briggs newell and sarie j comput phys 51 83 1983 have discussed a mechanism for the destabilisation of finite difference approximations to nonlinear partial differential equations their ideas were developed using the leap frog approximation to the advection equation here the same situation is examined in a manner which compares the basic solution to a periodic wavetrain an investigation is made into the stability of the basic solution to small disturbances which take the form of side band fourier modes the relation between side band growth and envelope modulation is discussed

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

statistical mechanics of elastica on a plane origin of the mkdv hierarchy in this paper the statistical mechanics of a non stretching elastica in two dimensional space using the path integral method is investigated in the calculation the modified korteweg de vries mkdv hierarchy naturally appeared in the equations including the temperature fluctuation we have classified the moduli of the closed elastica in a heat bath and summed the boltzmann weight with the thermal fluctuation over the moduli due to the bilinearity of the energy functional its exact partition function has been obtained by investigation of the system it is conjectured that an expectation value at a critical point of this system obeys the painleve equation of the first kind and its related equations are extended by the korteweg de vries hierarchy furthermore we also comment on the relation between the mkdv hierarchy and the becchi rouet stora transformation in this system

a criterion for reduction of variables in the willmore chen variational problem and its applications we exhibit a criterion for a reduction of variables for willmorechen submanifolds in conformal classes associated with generalized kaluzaklein metrics on flat principal fibre bundles our method relates the variational problem of willmore chen with an elasticity functional defined for closed curves in the base space the main ideas involve the extrinsic conformal invariance of the willmore chen functional the large symmetry group of generalized kaluza klein metrics and the principle of symmetric criticality we also obtain interesting families of elasticae in both lens spaces and surfaces of revolution riemannian and lorentzian we give applications to the construction of explicit examples of isolated willmore chen submanifolds discrete families of willmore chen submanifolds and foliations whose leaves are willmore chen submanifolds

the weak nonlinear instability of euler explicit scheme for the convective equation abstract the weak nonlinear instability of the euler explicit scheme for the convective equation u uux vxx is studied analytically using perturbation expansion and discrete fourier transform techniques short time weak nonlinear growth of the fourier modes for the noise is found provided that the linear stability conditions are satisfied as long as the perturbation expansion is valid the maximum growth is approximately proportional to the inverse of the grid spacing

exact analysis of nonlinear instability in a discrete burgers equation abstract a family of explicit nonlinear numerical schemes for burgers equation is derived by means of a discrete version of the hopf cole transformation exact nonlinear stability conditions for these schemes are then found and for one particular scheme the exact stability criteria are compared to the conventional linearized stability condition

confinement of semiflexible polymers a variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface as an application the confinement of a closed polymer loop of fixed length 2 r within a spherical cavity of smaller radius r 0 is considered it is shown that an infinite number of distinct periodic completely attached equilibrium states exist labeled by two integers n 2 3 4 and p 1 2 3 the number of periods of the polar and azimuthal angles respectively small loops oscillate about a geodesic circle n 2 p 1 is the stable ground state states with higher n exhibit instabilities if r 2r 0 new states appear as oscillations about a doubly covered geodesic circle the state n 3 p 2 replaces the twofold as the ground state in a finite band of values of r with increasing r loop states make a transition from oscillatory and orbital behavior on crossing the poles returning to oscillation upon collapse to a multiple cover of a geodesic circle signaled respectively by an increase in p and an increase in n the force transmitted to the surface does not increase monotonically with loop size but does asymptotically it behaves discontinuously where n changes the contribution to energy from geodesic curvature is bounded in large loops the energy becomes dominated by a state independent contribution proportional to the loop size the energy gap between the ground state and excited states disappears

knot complexity and related observables from path integrals for semiflexible polymers we analyse earlier obtained numerical results for the knot complexity and the topological persistence length i e the minimal number of steps on the lattice needed for the knot to be formed with help of well known path integrals for semiflexible chains in addition we provide an estimate for the number of different knots having knot diagrams with exactly n crossings analytical results obtained are in satisfactory agreement with the available numerical data

critical tori for mean curvature energies in killing submersions abstract we study surface energies depending on the mean curvature in total spaces of killing submersions which extend the classical notion of willmore energy based on a symmetry reduction procedure we construct vertical tori critical for these mean curvature energies these vertical tori are based on closed curves critical for curvature energy functionals in riemannian 2 space forms the binormal evolution of these critical curves in riemannian 3 space forms generates rotational tori solutions for an ample family of weingarten surfaces therefore we also introduce some correspondence results between these two types of tori and illustrate their relation

relativistic particles with rigidity generating non standard examples of willmore chen hypersurfaces we study a natural extension to higher dimensions of the nambu goto polyakov action in particular those dynamical objects evolving with so 3 symmetry in four dimensions we show that this problem is strongly related to that of relativistic particles with rigidity of order three in a hyperbolic plane the moduli space of solitonic solutions is completely determined in terms of the so called rotation number a quantization principle for closed solutions is also obtained and this gives a rational one parameter family of willmore chen hypersurfaces in the standard conformal structure of dimension four moreover these are the first non standard examples of this kind of hypersurfaces

extrinsic upper bounds for λ1 si m est compact alors il existe c r tel que 1 m c n m hu 2 volm pour toutes les sous varietes m compactes connexes a n dimensions isometriquement immergees de m 1 m est la premiere valeur propre du laplacien de m

area constrained planar elastica we determine the equilibria of a rigid loop in the plane subject to the constraints of fixed length and fixed enclosed area rigidity is characterized by an energy functional quadratic in the curvature of the loop we find that the area constraint gives rise to equilibria with remarkable geometrical properties not only can the euler lagrange equation be integrated to provide a quadrature for the curvature but in addition the embedding itself can be expressed as a local function of the curvature the configuration space is shown to be essentially one dimensional with surprisingly rich structure distinct branches of integer indexed equilibria exhibit self intersections and bifurcations a gallery of plots is provided to highlight these findings perturbations connecting equilibria are shown to satisfy a first order ode which is readily solved we also obtain analytical expressions for the energy as a function of the area in some limiting regimes

the geometry of the plate ball problem a ball rolls without slipping between two horizontal plates separated by the distance equal to the diameter of the ball it is assumed that the lower plate is fixed and that the ball is rolled through the horizontal movement of the upper plate the problem is to transfer the ball from a given initial position and a given initial orientation to a prescrived final position and final orientation along a certain path

a finite element model of two layer ocean circulation a mixed finite element formulation is applied to solve the quasi geostrophic equations in a two layer ocean domain enclosed by solid boundaries a combination of leap frog and schemes is used to integrate the equations in time wood s formula of the wall vorticity is derived in our formulation for the case of square elements adjacent to the boundaries long term computations with and without lumping have been carried out

elastic circles in 2 spheres by solving the euler lagrange equations we determine the elastic curves in the two dimensional sphere which are circular at rest we characterize the family of closed critical curves by a rational condition and the existence of closed elastic curves for the different possible values of their curvature at rest is proved in the final part of the paper we utilize a numerical approach to get a better understanding of the space of closed critical curves in this manner we analyse their shape uniqueness and minimizing properties

rotational surfaces of constant astigmatism in space forms abstract a surface in a riemannian space is called surface of constant astigmatism if the difference between the principal radii of curvature at each point is a constant function in this paper we give a classification of all rotational surfaces of constant astigmatism in space forms we also prove that generating curves of such surfaces are critical points of a variational problem for a curvature energy using the description of these curves we locally construct all rotational surfaces of constant astigmatism as the associated binormal evolution surfaces from the generating curves

critical curves for the total normal curvature in surfaces of 3 dimensional space forms abstract a variational problem closely related to the bending energy of curves contained in surfaces of real 3 dimensional space forms is considered we seek curves in a surface which are critical for the total normal curvature energy and its generalizations we start by deriving the first variation formula and the corresponding euler lagrange equations of these energies and apply them to study critical special curves geodesics asymptotic lines lines of curvature on surfaces then we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear weingarten surface finally we give some classification and existence results for this family of rotation surfaces

deformations of surfaces associated with integrable gauss mainardi codazzi equations using the formulation of the immersion of a two dimensional surface into the three dimensional euclidean space proposed recently a mapping from each symmetry of integrable equations to surfaces in r3 can be established we show that among these surfaces the sphere plays a unique role indeed under the rigid su 2 rotations all integrable equations are mapped to a sphere furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine gordon equation are homeomorphic to a sphere we also find some new weingarten surfaces arising from the deformations of the modified kurteweg de vries and of the nonlinear schrodinger equations surfaces can also be associated with the motion of curves we study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity

the elastica problem under area constraint we show that the elastic energy e gamma of a closed curve gamma has a minimizer among all plane simple regular closed curves of given enclosed area a gamma and that the minimum is attained for a circle the proof is of a geometric nature and deforms parts of gamma in a finite number of steps to construct some related convex sets with smaller energy

geometrical equilibrium of curves a showcase of helical numerical solutions in this work we plot loop configurations that minimize energy functionals depending on geometrical invariants of the loop itself in particular we consider a family of functionals including curvature and torsion terms both linear and quadratic such that their combinations produce geometrical invariants in 1 noether s theorem was advantageously used to identify the constants of integration of the euler lagrange equations describing the equilibrium of loops for this family of energy functionals those authors demonstrated the integrability of these functionals by means of quadratures in this work we follow that approach to realize numeric calculations and to show plots of numerical solutions of the relevant equations for equilibrium of loops as these were presented but not studied in 1 we then show as the main result of this work a representative catalogue of such solutions in euclidean three dimensional space

runge kutta smoother for suppression of computational mode instability of leapfrog scheme abstract the runge kutta smoother is applied to suppress nonlinear numerical instabilities in the leapfrog scheme for time integration of the korteweg de vries equation the accuracy of integration is compared with that by the use of the second order smoother the runge kutta smoother enables us to make long time integration of the korteweg de vries equation for large amplitudes

a linear focusing mechanism for dispersive and non dispersive wave problems a linear focusing mechanism for a wave packet propagation in a non periodic domain is explained here using fourier laplace spectral theory the global analysis method is used here to obtain numerical properties at each node of the full domain in this work we show the spectacular growth of an error packet at a particular node and wavenumber a phenomenon termed as focusing which depends on a cfl number n c and b numerical method used for discretization in this study we have shown the focusing phenomenon for the numerical solution of the 1d convection equation and linearized rotating shallow water equation

hamiltonian point of view of non euclidean geometry and elliptic functions abstract this paper offers a new way of looking at the classical geometries and the theory of elliptic functions through hamiltonian systems on lie groups in particular the paper shows that i the classical models of non euclidean geometries are canonically induced by bi invariant sub riemannian metrices on lie groups which act by left actions on the underlying space ii there is a class of canonical variational problems on lie groups g whose projections on homogeneous spaces g k generalize euler s elasticae and include all curves of constant curvature and all functions of weierstrass iii complex lie groups unify non euclidean geometries and complex elasticae offer a distinctive look at the elliptic functions

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

equivariant willmore surfaces in conformal homogeneous three spaces abstract the complete classification of homogeneous three spaces is well known for some time of special interest are those with rigidity four which appear as riemannian submersions with geodesic fibres over surfaces with constant curvature consequently their geometries are completely encoded in two values the constant curvature c of the base space and the so called bundle curvature r in this paper we obtain the complete classification of equivariant willmore surfaces in homogeneous three spaces with rigidity four all these surfaces appear by lifting elastic curves of the base space once more the qualitative behaviour of these surfaces is encoded in the above mentioned parameters c r the case where the fibres are compact is obtained as a special case of a more general result that works via the principle of symmetric criticality for bundle like conformal structures in circle bundles however if the fibres are not compact a different approach is necessary we compute the differential equation satisfied by the equivariant willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the euler lagrange equation of 4 r 2 elasticae in surfaces with constant curvature c we also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature it allows us to give explicit parametrizations of willmore surfaces and willmore tori in those conformal homogeneous 3 spaces