the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

algebro geometric solutions of 2 1 dimensional coupled modified kadomtsev petviashvili equations new 2 1 dimensional integrable coupled modified kadomtsev petviashvili mkp equations are proposed with the help of known 1 1 dimensional soliton equations the 2 1 dimensional coupled mkp equations are decomposed into systems of solvable ordinary differential equations the abel jacobi coordinates are introduced to straighten the flows from which new algebro geometric solutions of the 2 1 dimensional mkp equation and algebro geometric solutions of the 2 1 dimensional coupled mkp equations are obtained in terms of the riemann theta functions

3 manifolds with nonnegative ricci curvature for a complete noncompact 3 manifold with nonnegative ricci curvature we prove that either it is diffeomorphic to 3 or the universal cover splits this confirms milnor s conjecture in dimension 3

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

dualities for ising networks in this letter we study the equivalence between planar ising networks and cells in the positive orthogonal grassmannian we present a microscopic construction based on amalgamation which establishes the correspondence for any planar ising network the equivalence allows us to introduce two recursive methods for computing correlators of ising networks the first is based on duality moves which generate networks belonging to the same cell in the grassmannian this leads to fractal lattices where the recursion formulas become the exact renormalization group equations of the effective couplings the second we use an amalgamation in which each iteration doubles the size of the seed lattice this leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites

constant mean curvature spheres in rm sol _3 based on work of daniel and mira we prove that for each h0 there exists a unique up to ambient isometry immersed constant mean curvature h sphere s h in rm sol 3 where rm sol 3 is the non abelian solvable 3 dimensional lie group equipped with its usual left invariant metric that makes it into a model space for one of the eight thurston geometries see bonahon or scott for a discussion of the thurston geometries that can occur for compact 3 manifolds since this proof also demonstrates that s h has index one the results in daniel mira imply s h is embedded and the left invariant gauss map of s h is a diffeomorphism one key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature h graphs in rm sol 3 which depend only on any fixed positive lower bound for h by the work in daniel mira the existence of the spheres s h for all h0 implies the existence for all h geq 0 of a complex holomorphic only when h 0 quadratic differential q h defined on all constant mean curvature h surfaces m in rm sol 3 and which for h0 vanishes on an open set of m if and only if m is contained in some left translation of s h furthermore when q h is not identically zero on m it has isolated zeros on m and negative index at these zeros

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

a 2 1 dimensional derivative toda equation in the context of the kaup newell spectral problem a 2 1 dimensional derivative toda equation is derived from the lax triad composed of the kaup newell the negative kaup newell and a discrete spectral problem two integrable hamiltonian systems and an integrable symplectic map are derived from the lax triad they are straightened out in the jacobi variety of the associated hyperelliptic curve a finite genus solution is obtained through a technique based on the riemann jacobi inversion theorem besides explicit solutions are calculated for other associated integrable models including the mkp equation and the nkn equation

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

neumann type integrable reduction for nonlinear evolution equations in 1 1 and 2 1 dimensions a family of new neumann type systems is given in view of the nonlinearization technique realizing the variable separation of the modified jaulent miodek hierarchy and a new coupled modified kadomtsev petviashvili equation on the symplectic submanifold tsn 1 by two casimir functions and a special solution of the lenard eigenvalue equation we deduce the lax moser matrix of the neumann type systems that yields integrals of motion and the constrained hamiltonians whose vector fields are tangent to tsn 1 based on the dirac poisson bracket and a lax equation on tsn 1 a new systematic way is proposed to prove the liouville integrability of a family of neumann type systems synchronously the dirac poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and neumann type systems and then we point out that compatible solutions of neumann type systems yield the finite parametric solutions of 1 1 and 2 1 dimensional integrable nonlinear

lax representation r matrix method and separation of variables for the neumann type restricted flow the r matrix method and the method of separation of variables for restricted flows with neumann constraints are considered restricted jaulent miodek flows are taken as illustrative examples lax representations are obtained from the adjoint representation of the corresponding soliton hierarchy the r matrix formula based on this lax representation is derived and the separation of variables for the hamilton jacobi equation are performed

some algebro geometric solutions for the coupled modified kadomtsev petviashvili equations arising from the neumann type systems a treatment is described for getting some algebro geometric solutions of the coupled modified kadomtsev petviashvili cmkp equations and a hierarchy of 1 1 dimensional integrable nonlinear evolution equations inlees by using the neumann type systems through three steps i according to the nonlinearization of lax pair the cmkp equations and the 1 1 dimensional inlees are reduced to a family of compatible neumann type systems on a symplectic submanifold whose involutive solutions give rise to the finite parametric solutions of the inlees in both 2 1 and 1 1 dimensions ii from the holomorphic differentials and the abel map on a hyperelliptic curve of riemman surface the abel jacobi variables are introduced to straighten out the neumann type flows the 1 1 and 2 1 dimensional flows giving the abel jacobi solutions iii based on the riemann theorem and the trace formulas the jacobi inversion is applied to the straightened flows for getting some new algebro geometric solutions of the cmkp equations and the 1 1 dimensional integrable hierarchy

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

finite gap solutions of 2 1 dimensional integrable nonlinear evolution equations generated by the neumann systems each soliton equation in the korteweg de vries kdv hierarchy the 2 1 dimensional breaking soliton equation and the 2 1 dimensional caudrey dodd gibbon kotera sawada cdgks equation are reduced to two or three neumann systems on the tangent bundle tsn 1 of the unit sphere sn 1 the lax moser matrix for the neumann systems of degree n 1 is deduced in view of the mckean trubowitz identity and a bilinear generating function whose favorite characteristic accounts for the problem of the genus of riemann surface matching to the number of elliptic variables from the lax moser matrix the constrained hamiltonians in the sense of dirac poisson bracket for all the neumann systems are written down in a uniform recursively determined by integrals of motion the involution of integrals of motion and constrained hamiltonians is completed on tsn 1 by using a lax equation and their functional independence is displayed over a dense open subset of tsn 1 by a direct calculation which contribute to the liouville integ

the positive orthogonal grassmannian and loop amplitudes of abjm in this paper we study the combinatorics associated with the positive orthog onal grassmannian ogk and its connection to abjm scattering amplitudes we present a canonical embedding of ogk into the grassmannian gr k 2k from which we deduce the canonical volume form that is invariant under equivalence moves remarkably the canon ical forms of all reducible graphs can be converted into irreducible ones with products of d log forms unlike n 4 super yang mills here the jacobian plays a crucial role to ensure the d log form of the reduced representation furthermore in the positive region we identify the functional map that arises from the triangle equivalence move as a 3 string scattering s matrix which satises the tetrahedron equations by zamolodchikov implying 2 1 dimensional integrability we study the solution to the bcfw recursion relation for loop amplitudes and demonstrate the presence of all physical singularities as well as the absence of all spurious ones the on shell diagram solution to the loop recursion relation exhibits manifest two site cyclic symmetry and reveals that to all loop four and six point amplitudes only have logarithmic singularities

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

a 3 sx 3 matrix spectral problem for akns hierarchy and its binary nonlinearization abstract a 3 3 matrix spectral problem for akns soliton hierarchy is introduced and the corresponding bargmann symmetry constraint involving lax pairs and adjoint lax pairs is discussed an explicit new poisson algebra is proposed and thus the liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system the obtained nonlinearized lax systems in which the nonlinearized spatial system is intimately related to stationary akns flows lead to a sort of new involutive solutions to each akns soliton equation therefore the binary nonlinearization theory is successfully extended to a case of 3 3 matrix spectral problem for akns hierarchy

quasi periodic solutions of the modified kadomtsev petviashvili equation the 2 1 dimensional modified kadomtsev petviashvili equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of lax pairs abel jacobi coordinates are introduced to straighten the flows from which quasi periodic solutions of the modified kadomtsev petviashvili equation are obtained in terms of riemann theta functions

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process