construction of constant mean curvature n noids from holomorphic potentials using a weierstrass type representation of constant mean curvature surfaces we give a general method for constructing constant mean curvature n noids of genus 0 from holomorphic potentials where n 3 the ends of these surfaces are embedded and asymptotically approach delaunay surfaces while the surfaces are in general not even almost embedded in particular a 3 parameter family of constant mean curvature trinoids is constructed

a short proof of the c 0 alpha regularity of viscosity subsolutions for superquadratic viscous hamilton jacobi equations and applications abstract recently i capuzzo dolcetta f leoni and a porretta obtained a very surprising regularity result for fully nonlinear superquadratic elliptic equations by showing that viscosity subsolutions of such equations are locally holder continuous and even globally if the boundary of the domain is regular enough the aim of this paper is to provide a simplified proof of their results together with an interpretation of the regularity phenomena some extensions and various applications

hamiltonian mappings and circle packing phase spaces we introduce three area preserving maps with phase space structures which resemble circle packings each mapping is derived from a kicked hamiltonian system with one of the three different phase space geometries planar hyperbolic or spherical and exhibits an infinite number of coexisting stable periodic orbits which appear to pack the phase space with circular resonances

real forms of complex surfaces of constant mean curvature it is known that complex constant mean curvature cmc for short immersions in c 3 are natural complexifications of cmc immersions in r 3 in this paper conversely we consider real form surfaces of a complex cmc immersion which are defined from real forms of the twisted sl 2 c loop algebra asl 2 and classify all such surfaces according to the classification of real forms of asl 2 there are seven classes of surfaces which are called integrable surfaces and all integrable surfaces will be characterized by the lorentz harmonicities of their gauss maps into the symmetric spaces s 2 h 2 s 1 1 or the 4 symmetric space sl 2 u 1 we also give a unification to all integrable surfaces via the generalized weierstrass type representation

lipschitz regularity results for nonlinear strictly elliptic equations and applications abstract most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable subquadratic for instance for equations with superquadratic growth power in gradient one usually uses weak bernstein type arguments which require regularity and or convex type assumptions on the gradient nonlinearity in this article we obtain new lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the hamiltonian with respect to the gradient variable using some ideas coming from ishii lions method we use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation

actions of loop groups on harmonic maps we describe a general framework in which subgroups of the loop group agin act on the space of harmonic maps from s2 to gln this represents a simplification of the action considered by zakharov mikhailov shabat zm zs in that we take the contour for the riemann hilbert problem to be a union of circles however it reduces the basic ingredient to the well known birkhoff decomposition of agin s and this facilitates a rigorous treatment we give various concrete examples of the action and use these to investigate a suggestion of uhlenbeck uh that a limiting version of such an action completion gives rise to her fundamental process of adding a uniton it turns out that this does not occur because completion preserves the energy of harmonic maps however in the special case of harmonic maps from s2 to complex projective space we describe a modification of this completion procedure which does indeed reproduce adding a uniton one aspect of the theory of symmetry groups for diffierential equations is the idea of proliferation of solutions by this is meant that starting with an obvious solution application of symsmetries can lead to further perhaps less obvious solutions in this paper we shall apply the general framework described by zakharov et al zm zs to the harmonic map equation the principal chiral or sigma model of mathematical physics to put this into a wider context we recall that a fundamental discovery in the study of the korteweg de vries and related equations was that the space of solutions admits a certain infinite dimensional symmetry group historically the first manifestation of this via noether s theorem was the appearance of infinitely many conservation laws later this led to an elegant algebraic description of solutions of the equation as the points of an orbit of a representation of an infinite dimensional group see for example wi for a brief description of some of these ideas the existence of previously unnoticed symmetries hidden symmetries for other equations became an interesting possibility and attempts have been made so far with much less success than in the case of the kdv equation to unearth them one such example is the harmonic map equation for maps from a surface d into a compact lie group g or homogeneous space g h the basic observation po uh zm zs which introduces an infinite dimensional group is that harmonic maps from d to g correspond to certain received by the editors february 9 1990 1980 mathematics subject classification 1985 revision primary 58e20 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

risk sensitive production planning of a stochastic manufacturing system this paper is concerned with long run average risk sensitive control of production planning in a manufacturing system with machines that are subject to breakdown and repair by using a logarithmic transformation it is shown that the associated hamilton jacobi bellman equation has a viscosity solution the risk sensitive control problem has a dynamic stochastic game interpretation finally a limiting problem is obtained when the rates of machine breakdown and repair go to infinity

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

holder estimates for degenerate elliptic equations with coercive hamiltonians we prove a priori estimates and regularity results for some quasi linear degenerate elliptic equations arising in optimal stochastic control problems our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally holder continuous and solutions to be locally lipschitz continuous we also give an existence result for the associated dirichlet problem

gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems abstract we obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the hamiltonian is a sum of a sublinear and a superlinear part in the sense of barles and souganidis 2001 we use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations our results apply to some asymmetric systems where some equations contain a sublinear hamiltonian whereas the others contain a superlinear one moreover we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

spaces of complex null geodesics in complex riemannian geometry the notion of a conlplex rienlanmianl 11 mcanifold meaning a complex 1 manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point is briefly developed it is shown that provided ii 4 every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal bundle type in an associated space of comple x liull geodesics this relationship between local geometry and global complex analysis is stable under deformations introduction the present work provides a general framework analogous to but distinct from penrose s twistor correspondence penrose 15 atiyah et al 2 for the study of real analytic pseudo riemannian geometry on manifolds whose dimension exceeds three in which points of the original manifold correspond to compact complex submanifolds quadrics in an associated complex manifold the natural intermediate step in the construction is to consider complexification of the geometry leading to the notion of a complex riemannian manifold implicitly present for instance in the work of penrose in fact it is really the conformal structure that is of fundamental importance and so having briefly introduced holomorphic metrics in chapter i we proceed to establish the fundamental facts of holomorphic conformal geometry in chapter ii finally in chapter iii we develop our generalized twistor correspondence a brief historical note is in order a special case of this correspondence where the complex riemannian manifold is complex euclidean 4 space proved to be useful for the study of yang mills fields issenberg et at 7 witten 18 several years ago in the wake of this discovery the present author first sketched his ideas on the subject in 1979 in the informal twistor newsletter of the oxford mathematical institute a more fully developed version became the author s ph d thesis lebrun 10 which forms the basis of much of the present more refined paper several results are completely new however such as the theorems in 111 4 5 in dimension four some recent results have shown these notions to be of genuine physical interest within the last year a beguiling theory of the dirac manin 13 and wave lebrun 11 operators fitting neatly within this general framework has emerged for the present however we will examine only the big picture relegating this arguably more beautiful restricted case to treatment elsewhere received by the editors march 10 1982 1980 mathematics subject classification primary 32g10 53a30 secondary 32d15 53c15 53c22 83c99 d1983 american mathematical society 0002 9947 82 00001440 06 75

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

on codings and dynamics of planar piecewise rotations abstract in this paper we investigate codings and dynamics of the piecewise rotations belonging to a sub class of piecewise isometries we show that the irrational set is empty for some piecewise rational rotations under some assumptions while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space we further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation

global stability of a class of discontinuous dual billiards an infinite parameter family of discontinuous area preserving maps is studied using geometrical methods necessary and sufficient conditions are determined for the existence of some bounding invariant sets which guarantee global stability it is shown that under some additional constraints all orbits become periodic most of them lyapounov stable and with a maximal period in any bounded domain of phase space this yields a class of maps acting on a discrete phase space

dynamical behaviors and codings of invertible planar piecewise isometric systems abstract in this paper the authors discuss some important sets including singular sets exceptional sets irrational sets etc generated by invertible planar piecewise isometric maps they first extend the coding map onto the entire phase space and characterize singular sets and exceptional sets by the number of codings then they reveal the relation among these sets and show that an irrational set is contained in an exceptional set and they confirm the existence of admissible rational codings under some conditions including a necessary and sufficient condition and a sufficient condition they also discuss dynamical properties of the discontinuity line segments which determine the dynamics of a piecewise isometric system finally they list some open problems for further research

coarse classification of constant mean curvature cylinders we give a coarse classification of constant mean curvature cmc immersions of cylinders into r 3 via the loop group method particularly for this purpose we consider double loop groups and a new type of potentials which are meromorphic 1 forms on riemann surfaces

hamiltonians and zero curvature equations for integrable partial differential equations we discuss the relationship between two approaches to integrable partial differential equations one using formal affine lie algebras and the other banach lie groups in the first approach integrability of the equations follows from commutativity of hamiltonian flows on the lie algebra while in the second it follows from commutativity of certain flows induced by an action on the banach lie group we show that these two methods are essentially equivalent since one can calculate one type of the flows from the other the classes of solutions encompassed by the two methods however vary significantly we demonstrate this relationship specifically with the nonlinear schrodinger equation

harmonic maps and classical surface theory in minkowski 3 space harmonic maps from a surface s with nondegenerate prescribed and induced metrics are characterized showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the riemannian case moreover holomorphic quadratic differentials are shown to arise as naturallv on surfaces of constant h or k in m3 as on their counterparts in e3 the connection between the sine gordon sinh gordon and cosh gordon equations and harmonic maps is explained various local and global results are established for surfaces in m3 with constant h or constant k 0 in particular the gauss map of a spacelike or timelike surface in m3 is shown to be harmonic if and only if h is constant also k is shown to assume values arbitrarilv close to h2 on any entire spacelike surface in m3 with constant h except on a hyperbolic cylinder

adjoint and compensated compactness methods for hamilton jacobi pde we investigate the vanishing viscosity limit for hamilton jacobi pde with nonconvex hamiltonians and present a new method to augment the standard viscosity solution approach the main idea is to introduce a solution e of the adjoint of the formal linearization and then to integrate by parts with respect to the density e this procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique