serrin s overdetermined problem and constant mean curvature surfaces for all n 9 we find smooth entire epigraphs in rn namely smooth domains of the form x rn xn f x1 xn 1 which are not half spaces and in which a problem of the form u f u 0 in has a positive bounded solution with 0 dirichlet boundary data and constant neumann boundary data on this answers negatively for large dimensions a question by berestycki caffarelli and nirenberg in 1971 serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball in analogy to a famous result by alexandrov that states that an embedded compact surface with constant mean curvature cmc in euclidean space must be a sphere in lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given cmc surface where serrin s overdetermined problem is solvable

the classification of complete stable area stationary surfaces in the heisenberg group h1 abstract we prove that any c 2 complete orientable connected stable area stationary surface in the sub riemannian heisenberg group h 1 is either a euclidean plane or congruent to the hyperbolic paraboloid t x y

on stable constant mean curvature surfaces with free boundary ros and vergasta geom dedicata 56 19 33 1995 proved among others interesting results a theorem which states that an immersed orientable compact stable constant mean curvature surface sigma with free boundary in a closed ball b subset mathbb r 3 must be a planar equator a spherical cap or a surface of genus 1 with at most two boundary components in this article by using a modified hersch type balancing argument we complete their work by proving that sigma cannot have genus 1

homoclinic snaking in bounded domains homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side on the real line this process continues forever in finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions with periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states however with non neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state this behavior shown here in detail for the swift hohenberg equation explains the phenomenon of snaking without bistability recently observed in simulations of binary fluid convection by mercader et al phys rev e 80 025201 2009

stability and eigenvalue estimates of linear weingarten hypersurfaces in a sphere abstract let m be an n dimensional compact hypersurface without boundary in a unit sphere s n 1 1 m is called a linear weingarten hypersurface if c r d h e 0 where c d and e are constants with c 2 d 2 0 r and h denote the scalar curvature and the mean curvature of m respectively by the gauss equation we can rewrite the condition c r d h e 0 as n 1 e h 2 a h b where h 2 is the 2nd mean curvature a b and e are constants such that a 2 e 2 0 when e 0 it reduces to the constant mean curvature case in this paper we obtain some stability results about linear weingarten hypersurfaces which generalize the stability results about the hypersurfaces with constant mean curvature or with constant scalar curvature we show that linear weingarten hypersurfaces satisfying n 1 h 2 a h b where a and b are constants can be characterized as critical points of the functional m a n h d v for volume preserving variations we prove that such a linear weingarten hypersurface is stable if and only if it is totally umbilical and non totally geodesic we also obtain optimal upper bounds for the first and second eigenvalues of the jacobi operator of linear weingarten hypersurfaces

asymptotic solutions of stationary patterns in convection reaction diffusion systems we study and map the possible stationary patterns that emerge in a convection reaction diffusion crd system using a learning polynomial kinetics we classify the patterns according to the kinetic model oscillatory bistable or intermediate the instability nature of the bounded system convective or absolute the applied boundary conditions and the system length this analysis presents a unifying approach to various pattern inducing mechanisms such as difici differential flow induced chemical instability which predicts moving patterns in systems with wide difference of convective rates and differential capacity patterns which predicts stationary patterns in cross flow reactors with a large heat capacity previous studies of crd systems have considered only oscillatory kinetics nonlinear analysis which follows the front motion by approximating its velocity accounts for the stability of the stationary whether spatially periodic or other patterns the most dominant state is the large amplitude stationary spatially periodic pattern with oscillatory kinetics these emerge in the convectively unstable domain above the amplification threshold the domain of absolute instability which is determined analytically for unbounded systems is divided in the bounded system into two subdomains with moving difici waves or stationary patterns with bistable kinetics the large amplitude stationary patterns can be sustained only within a narrow subdomain but other stationary patterns that incorporate several fronts upstream and an almost homogeneous tail downstream can be sustained as well with intermediate kinetics the large amplitude axisymmetric stationary patterns may coexist with small amplitude stationary nonaxisymmetric patterns

a geometric approach to the classification of the equilibrium shapes of self gravitating fluids the classification of the equilibrium shapes that a self gravitating fluid can take in a riemannian manifold is a classical problem in mathematical physics in this paper it is proved that the equilibrium shapes are isoparametric submanifolds some geometric properties of them are also obtained e g classification and existence for some riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold our approach to the problem is geometrical and allows to study the equilibrium shapes on general riemannian spaces

spatio temporal structures in a differential flow reactor with cubic autocatalator kinetics abstract a model of a differential flow reactor with gray scott kinetics is considered in which the reactant is immobilised while the autocatalyst is allowed to flow with uniform velocity and to diffuse a linear stability analysis shows that there is a critical flow rate above which the spatially uniform stationary state is convectively unstable a weakly nonlinear analysis close to criticality results in a complex ginzburg landau equation cgle which governs the envelope amplitude of the packet of differential flow induced waves being the first derivation of such a modulation equation for such a problem the analysis of the cgle shows that this bifurcation is supercritical with proparating wave packets developing for flow rates above the critical value and identifies a region of parameter space close to the hopf bifurcation in the kinetic scheme where more complex propagating wave structures can arise the theoretical predictions are confirmed by numerical integrations of the initial value problem which also give results for flow rates well in excess of those amenable to analysis

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

index growth of hypersurfaces with constant mean curvature in this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature cmc 1 in mathbb r n delaunay unduloids when n 3 using the asymptotics result of korevaar kusner and solomon we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

explicit quasi periodic solutions of the kaup newell hierarchy abstract an algebraic curve of arithmetic genus n is introduced with the aid of the lax matrix from which we construct the meromorphic function on the algebraic curve and investigate its properties further we straighten out all the flows associated with the kaup newell hierarchy under the abel jacobi coordinates finally we achieve the explicit quasi periodic solutions for the whole kaup newell hierarchy including the coupled derivative nonlinear schrodinger equations

limit cycles in the presence of convection a traveling wave analysis we consider a diffusion model with limit cycle reaction functions in an unbounded domain diffusion spreads the pattern outwards from the source convection adds instability to the reaction diffusion system the result of this instability is a readiness to create a pattern we choose the lambda omega reaction functions for their simple limit cycle we carry out a transformation of the dependent variables into polar form from this we consider the initiation of the pattern to approximate a traveling wave we carry out numerical experiments to test our analysis these confirm the premise of the analysis that the initiation can be modeled by a traveling wave furthermore the analysis produces a good estimate of the numerical results most significantly we confirm that the pattern consists of two different types

closed hypersurfaces driven by mean curvature and inner pressure we introduce a hyperbolic equation that describes the motion of closed hypersurfaces in a riemannian manifold with surface tension and inner pressure as driving forces in the case of spherical surfaces this equation can be considered as an idealized mathematical model for a moving soap bubble the equation is derived as an euler lagrange equation from a suitable action integral it is a quasi linear degenerate hyperbolic pde of second order that describes the motion of the surfaces extrinsically our main results are the solution of the cauchy problem by means of the nash moser inverse function theorem a continuation criterion and stability estimates 2012 wiley periodicals inc

area stationary surfaces inside the sub riemannian three sphere we consider the sub riemannian metric gh on mathbb s 3 given by the restriction of the riemannian metric of curvature 1 to the plane distribution orthogonal to the hopf vector field we compute the geodesics associated to the carnot caratheodory distance and we show that depending on their curvature they are closed or dense subsets of a clifford torus we study area stationary surfaces with or without a volume constraint in mathbb s 3 g h by following the ideas and techniques by ritore and rosales area stationary surfaces in the heisenberg group mathbb h 1 arxiv math dg 0512547 we introduce a variational notion of mean curvature characterize stationary surfaces and prove classification results for complete volume preserving area stationary surfaces with non empty singular set we also use the behaviour of the carnot caratheodory geodesics and the ruling property of constant mean curvature surfaces to show that the only c2 compact connected embedded surfaces in mathbb s 3 g h with empty singular set and constant mean curvature h such that h sqrt 1 h 2 is an irrational number are clifford tori finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in mathbb s 3 g h

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

canonical structure of soliton equations ii the kaup newell system abstract the canonical structure of the kaup newell system i e the family of the derivative nonlinear schrodinger equation and its generalizations is discussed

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

curvature estimates and sheeting theorems for weakly stable cmc hypersurfaces abstract weakly stable constant mean curvature cmc hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations we establish a pointwise curvature estimate in the non singular dimensions and a sheeting theorem in all dimensions for weakly stable cmc hypersurfaces giving an effective version of the compactness theorem for weakly stable cmc hypersurfaces established in the recent work of the first and third named authors 6 our results generalize the curvature estimate and the sheeting theorem proven respectively by schoen simon yau and schoen simon for strongly stable hypersurfaces

principal bifurcations and symmetries in the emergence of reaction diffusion advection patterns on finite domains pattern formation mechanisms of a reaction diffusion advection system with one diffusivity differential advection and robin boundary conditions of danckwerts type are being studied pattern selection requires mapping the domains of coexistence and stability of propagating or stationary nonuniform solutions which for the general case of far from instability onsets is conducted using spatial dynamics and numerical continuations the selection is determined by the boundary conditions which either preserve or destroy the translational symmetry of the model accordingly we explain the criterion and the properties of stationary periodic states if the system is bounded and show that propagation of nonlinear waves including solitary against the advective flow corresponds to coexisting family that emerges nonlinearly from a distinct oscillatory hopf instability consequently the resulting pattern selection is qualitatively different from the symmetric finite wavenumber turing or hopf instabilities

majoration de la seconde valeur propre d un operateur de schrodinger sur une variete compacte et applications abstract in the first part of this paper we obtain an optimal upper bound for the second eigenvalue of schrodinger operators through using a conformal invariant i e conformal volume in the second part we obtain some applications of this estimate concerning the morse index of constant mean curvature or minimal hypersurfaces the last part deals with some isospectrality problems

applications of compactness results for harmonic maps to stable constant mean curvature surfaces for all lipschitz functions u such that m u da 0 in the above formula and da are the gradient and the area element on m and is the square of the norm of the second fundamental form of we shall call i the index form of the immersion note that i u is the second variation formula of the area for a normal variation leaving constant the volume functional with variational vector field u where is a unit vector field normal to the immersion complete details can be found in bdce in this paper we study stable periodic surfaces i e stable immersions m r3 where is a discrete subgroup of translations in r3 these surfaces are interesting since they are good models for naturally occurring periodic phenomena such as the formation of interfaces in block copolymers see tahh and the references cited there up to now our knowledge of stable immersions in complete flat three manifolds can be resumed in the following result see rr for a sketch of the proof

λ hypersurfaces on shrinking gradient ricci solitons abstract we consider an orientable smooth non totally geodesic hypersurface properly immersed on the cylinder shrinking soliton s 2 k 1 k r l g f where g is the product metric and the weight function f is defined by f p x x 2 4 if has weighted weak index less than or equal to l 1 we show that splits off a linear space also we classify the f stable orientable complete hypersurfaces properly immersed on s 2 k 1 k r g f with f p t t 2 4 obtaining as result s 2 k 1 k t or s 2 k 1 k 1 r

parameter space analysis pattern sensitivity and model comparison for turning and stationary flow distributed waves fds a new type of instability in coupled reaction diffusion advection systems is analysed in a one dimensional domain this instability arising due to the combined action of flow and diffusion creates spatially periodic stationary waves termed flow and diffusion distributed structures fds here we show via linear stability analysis that fds are predicted in a considerably wider domain and are more robust in the parameter domain than the classical turing instability patterns fds also represent a natural extension of the recently discovered flow distributed oscillations fdo nonlinear bifurcation analysis and numerical simulations in one dimensional spatial domains show that fds also have much richer solution behaviour than turing structures in the framework presented here turing structures can be viewed as a particular instance of fds we conclude that fds should be more easily obtainable in chemical systems than turing and fdo structures and that they may play a potentially important role in biological pattern formation

secure steganographic communication algorithm based on self organizing patterns a secure steganographic communication algorithm based on patterns evolving in a beddington de angelis type predator prey model with self and cross diffusion is proposed in this paper small perturbations of initial states of the system around the state of equilibrium result in the evolution of self organizing patterns small differences between initial perturbations result in slight differences also in the evolving patterns it is shown that the generation of interpretable target patterns cannot be considered as a secure mean of communication because contours of the secret image can be retrieved from the cover image using statistical techniques if only it represents small perturbations of the initial states of the system an alternative approach when the cover image represents the self organizing pattern that has evolved from initial states perturbed using the dot skeleton representation of the secret image can be considered as a safe visual communication technique protecting both the secret image and communicating parties

on the stability of hypersurfaces in space forms abstract we define the notion of r s stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a riemannian space form by supposing that such a hypersurface m n is contained either in an open hemisphere of the euclidean sphere or in the hyperbolic space we are able to show that m n is r s stable if and only if m n is a geodesic sphere moreover we obtain a suitable characterization of the r s stability through the analysis of the first eigenvalue of the jacobi operator associated to the corresponding variational problem

pattern selection in a general model of convection diffusion and catalytic reaction abstract we study the process of pattern selection in a large class of heterogeneous models of catalytic reactor by analyzing the behavior of a simple condensed model which obeys reflection and inversion symmetries and which captures the main features of a detailed model the three variable model incorporates a very fast and long ranged variable which may describe the fluid phase in a mixed plug flow or an axial dispersion reactor or the pore phase in a catalytic particle and two variables a medium ranged activator and a localized inhibitor that describe the solid phase patterns are classified according to their symmetry properties patterns may emerge already with simple bistable kinetics in a plug flow reactor i e a single integrodifferential equation pattern selection is determined by the phase planes spanned by the reactor by the ratio of front residence time to the period of oscillations and by the ratio of convection to fluid phase diffusions terms patterns include stationary or oscillatory fronts or pulses antiphase oscillations which appear at the transition from homogeneous solution unidirectional pulses that are the dominant patterns when mixing is weak and stationary or oscillatory waves that emerge with strong fluid mixing the simple form of the model allows us to suggest a general classification of emerging patterns and to identify the corresponding conditions in term of realistic models of reactors

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

a classification theorem for helfrich surfaces in this paper we study the functional mathcal w lambda 1 lambda 2 which is the sum of the willmore energy lambda 1 weighted surface area and lambda 2 weighted volume for surfaces immersed in mathbb r 3 this coincides with the helfrich functional with zero spontaneous curvature our main result is a complete classification of all smooth immersed critical points of the functional with lambda 1 ge 0 and small l 2 norm of tracefree curvature with no assumption on the growth of the curvature in l 2 at infinity this not only improves the gap lemma due to kuwert and schatzle for willmore surfaces immersed in mathbb r 3 but also implies the non existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

min max theory for constant mean curvature hypersurfaces in this paper we develop a min max theory for the construction of constant mean curvature cmc hypersurfaces of prescribed mean curvature in an arbitrary closed manifold as a corollary we prove the existence of a nontrivial smooth closed almost embedded cmc hypersurface of any given mean curvature c moreover if c is nonzero then our min max solution always has multiplicity one

on the entropies of hypersurfaces with bounded mean curvature we are interested in the impact of entropies on the geometry of a hypersurface of a riemannian manifold in particular we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold provided some geometric assumptions are satisfied this comparison will follow from the existence of an embedded tube around such a hypersurface among the consequences of our study of entropies we point out some new answers to a question of do carmo on stable euclidean hypersurfaces of constant mean curvature

dynamics of reaction diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection a one component bistable reaction diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two component activator inhibitor reaction diffusion model with differential advection the effects of asymmetric nonlocal couplings in such a bistable reaction diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling we carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system in a second step a third fast reaction diffusion equation is included which induces the formation of more complex patterns a linear stability analysis predicts traveling waves for asymmetric nonlocal coupling in contrast to a stationary turing patterns for a system with symmetric nonlocal coupling these findings are verified by direct numerical integration of the full equations with nonlocal coupling

isoperimetric regions in cones we consider cones c 0 m n and prove that if the ricci curvature of c is nonnegative then geodesic balls about the vertex minimize perimeter for given volume if strict inequality holds then they are the only stable regions

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

a limit method for solving period problems on minimal surfaces we introduce a new technique to solve period problems on minimal surfaces called limit method if a family of surfaces has weierstras data converging to the data of a known example and this presents a transversal solution of periods then the original family contains a sub family with closed periods

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

stable hypersurfaces with constant scalar curvature it is well known that hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 c of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations

one dimensional spatial structure near a hopf bifurcation at finite wavenumber abstract the theory of the spatial structure in a finite geometry in the weakly non linear regime of an instability to a traveling wave state with x x symmetry is described very close to the threshold a direct expansion in the unstable modes yields a quantitative description of the structure in terms of the reflection coefficient of the ends the calculation of the reflection coefficient and its dependence on fluid and end wall properties is illustrated using the model of binary fluid convection between free slip boundaries further away from threshold the phase diagram of the spatial structure involving four distinct phases is calculated using the amplitude equation approach the validity of this method is judged by comparison with the free slip convection model the transition between a state where the disturbance fills the system and one where only a portion is filled filling to confined is analyzed in detail our theory is compared with experiments on the oscillatory instability in pure fluid convection

on the dissipative evolution equations associated with the zakharov shabat system with a quadratic spectral parameter in this paper we derive some results for the zakharov shabat system of the form dm dx z2 j m zq p m j is diagonal and skewhermitian 8 10 12 following the idea of r beals and r r coifman we estimate the wedge products of the columns of m by l2 norm of the potential q p 4 by this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data q x 0 p x 0 qo po is of schwartz class 1 direct and inverse scattering for the z s system with a quadratic parameter by a potential here we mean a pair of functions q p r m c set of n x n complex matrices q is off diagonal and the diagonal part of p equals the diagonal part of q adj f q and q qx p p e l we consider the following spectral problem given z 0 x z im z o find m z r 4m c with 1 1 am z z 0x z 2 j m x z zq p m x z where j diag id id2 idn d 0 q z im z 0 for a certain set of potentials called generic m has the following properties 8 10 1 3 for any z e x there is a unique matrix v z such that for all x m x z m x z exp xz2j v z exp xz2j where m x z limit of m on i from q received by the editors may 4 1988 1980 mathematics subject classification 1985 revision primary 35q20 35q15

the isoperimetric problem for lens spaces we solve the isoperimetric problem in the lens spaces with large fundamental group namely we prove that the isoperimetric surfaces are either geodesic spheres or quotients of clifford tori we also show that the isoperimetric problem in the lens spaces l 3 1 and l 3 2 follows from the proof of the willmore conjecture by marques and neves

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

curvature estimates for cmc surfaces in three dimensional manifolds in this paper we give curvature estimates for strongly stable cmc surfaces in complete three dimensional manifolds

bifurcation structure of dissipative solitons in this paper we analyze in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity we compare our results with the general picture described in p d woods a r champneys physica d 129 1999 147 p coullet c riera c tresser phys rev lett 84 2000 3069 and find that the stable and unstable manifolds of homogeneous and patterned solutions present a much higher level of complexity than predicted including the existence of additional localized solutions and fronts this extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low amplitude periodic solutions and to the fact that these periodic solutions together with the high amplitude ones constitute a one parameter family generating a closed line on the symmetry plane

the second variation of nonorientable minimal submanifolds suppose m is a complete nonorientable minimal submanifold of a riemannian manifold n we derive a second variation formula for the area of m with respect to certain perturbations giving a sufficient condition for the instability of m some simple applications are given we show that the totally geodesic irip2 is the only stable surface in irip3 and we show the non existence of stable nonorientable cones in r we reproduce and marginally extend some known results in the truly non compact setting

a general halfspace theorem for constant mean curvature surfaces in this paper we prove a general halfspace theorem for constant mean curvature surfaces under certain hypotheses we prove that in an ambient space m 3 any constant mean curvature h 0 surface on one side of a constant mean curvature h 0 surface sigma 0 is an equidistant surface to sigma 0 the main hypotheses of the theorem are that sigma 0 is parabolic and the mean curvature of the equidistant surfaces to sigma 0 evolves in a certain way