on properly embedded minimal surfaces with three ends we classify all complete embedded minimal surfaces in r with three ends of genus g and at least 2g 2 symmetries the surfaces in this class are the costa hoffman meeks surfaces which have 4g 4 symmetries in the case of a flat middle end the proof consists of using the symmetry assumptions to deduce the possible weierstras data and then to study the period problems in all cases to handle the 1 dimensional period problems we develop a new general method to prove convexity results for period quotients the 2 dimensional period problems are reduced to the 1 dimensional case by an extremal length argument 1991 mathematics subject classification primary 53a10 secondary 53c42

embedded minimal surfaces of finite topology in this paper we prove that any complete embedded minimal surface m in r 3 with finite topology and compact boundary possibly empty is conformally a compact riemann surface m with boundary punctured in a finite number of interior points and that m can be represented in terms of meromorphic data on its conformal completion m in particular we demonstrate that m is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofm

embedded minimal surfaces with an infinite number of ends we prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends 25 refs 29 figs

the global smooth symmetric solution to 2 d full compressible euler system of chaplygin gases abstract for one dimensional or multidimensional compressible euler system of polytropic gases it is well known that the smooth solution will generally develop singularities in finite time however for three dimensional chaplygin gases due to the crucial role of null condition in the potential equation which is derived by the irrotational and isentropic flow p godin in 9 has proved the global existence of a smooth 3 d spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state it is noted that there are some clear differences for the global solution or blowup problems between 2 d and 3 d hyperbolic equations or systems in this paper we will focus on the global symmetric solution problem of 2 d full compressible euler system of chaplygin gases through carrying out involved analysis and finding an appropriate weight we can derive some uniform weighted energy estimates on the small symmetric solution to 2 d compressible euler system of chaplygin gases and further establish the global existence of the smooth solution by the continuous induction method

temps de vie des solutions regulieres des equations d euler compressibles axisymetriques en dimension deux nous considerons le systeme des equations d euler isentropiques en dimension deux pour des donnees initiales invariantes par rotation et perturbations de taille e d un etat de repos on etablit un equivalent du temps de viet e de la solution classique lim 2 t 2 de plus on donne pour t leqq frac a 2 varepsilon 2 a tau une estimation de la vraie solution en calculant la taille de son ecart a une solution approchee construite dans un precedent travail

stabilizing effect of the power law inflation on isentropic relativistic fluids abstract this paper is concerned with the stabilizing effect of the power law inflation on relativistic eulerian fluids we prove the global stability of the background solutions to the relativistic fluids including the isothermal gases and generalized chaplygin gases by the method of conformal transformation when the initial data is a small perturbation to the background solution we also prove the blowup phenomena of the relativistic euler fluids including the isothermal gases and polytropic gases when the initial data satisfies suitable assumptions

an embedded minimal surface with no symmetries we construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes we use weierstrass representation and we solve the period problem using the implicit function theorem as a corollary we obtain the existence of minimal surfaces with no symmetries

delayed singularity formation in 2d compressible flow the initial value problem for the compressible euler equations in two space dimensions is studied of interest is the lifespan of classical solutions with initial data that is a small perturbation from a constant state the approach taken is to regard the compressible solution as a nonlinear superposition of an underlying incompressible flow and an irrotational compressible flow this viewpoint yields an improvement for the lifespan over that given by standard existence theory the estimate for the lifespan is further improved when the initial data possesses certain symmetry in the case of rotational symmetry a result of s alinhac is reconsidered the approach is also applied to the study of the incompressible limit the analysis combines energy and decay estimates based on vector fields related to the natural invariance of the equations 0 introduction the compressible euler equations comprise a nonlinear symmetric hyperbolic system of pde s that model ideal fluid flow the charac teristic wave speeds of this system are given by the fluid velocity and the local sound speed the two wave families associated to these speeds may be con sidered separately hydrodynamical waves are described by the incompressible euler equations while acoustical waves are approximated by the linear wave equation or more precisely by the compressible euler equations with irrotational fluid velocity this paper attempts to use the nonlinear superposition of these two flows to study the long time behavior of small amplitude disturbances in two dimensional compressible fluid flow consider the initial value problem for the compressible euler equations in two or three space dimensions with smooth initial data which is a small perturbation of amplitude e from a constant state the life span is defined as the largest time interval on which there exists a classical solution to the initial value problem from the theory of symmetric hyperbolic systems 6 8 the life span is at least 0 e results on formation of singularities show that the life span of a classical solution is no better than 0 e2 in 2d 16 and o exp l 2 in 3d 17 in this paper we extend the length of the life span under various assumptions on the initial data in two space dimensions the simplest case is the one in which the initial fluid velocity is irrotational that is its curl is zero the compressible euler equations then behave much the same as scalar nonlinear wave equations this can readily be guessed by writing the velocity as the gradient of a potential although we will not use this approach here we shall see in theorem 2 that the life span of 2d irrotational

global smooth axisymmetric solutions to 2d compressible euler equations of chaplygin gases with non zero vorticity abstract for 2d compressible isentropic euler equations of polytropic gases s alinhac establishes that the axisymmetric smooth solution blows up in the present paper for 2d compressible isentropic euler equations of chaplygin gases we show that the small perturbed smooth solution exists globally when the rotationally invariant initial data are a perturbation of size e 0 of a rest state near the light cone 2d euler equations of chaplygin gases can be transformed into a second order quasilinear wave equation which satisfies both the null conditions this will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone however away from the light cone the hydrodynamical waves have no decay in time and strongly affect the related acoustical waves by introducing a nonlinear ode we can distinguish the fast decay part and non decay part so that the global energy estimates are derived

completely linear degeneracy for quasilinear hyperbolic systems abstract in this paper we introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition for applications we give some examples arising from mathematics and physics at last

rationality results for the exterior and the symmetric square l function with an appendix by nadir matringe let g mathrm gl 2n over a totally real number field f and n ge 2 let pi be a cuspidal automorphic representation of g mathbb a which is cohomological and a functorial lift from so 2n 1 the latter condition can be equivalently reformulated that the exterior square l function of pi has a pole at s 1 in this paper we prove a rationality result for the residue of the exterior square l function at s 1 and also for the holomorphic value of the symmetric square l function at s 1 attached to pi as an application of the latter we also obtain a period free relation between certain quotients of twisted symmetric square l functions and a product of gaus sums of hecke characters

the lifespan of a class of smooth spherically symmetric solutions of the compressible euler equations with variable entropy in three space dimensions we consider smooth three dimensional spherically symmetric eulerian flows of ideal polytropic gases with variable entropy whose initial data are obtained by adding a small smooth perturbation with compact support to a constant state under a natural assumption we obtain precise information on the asymptotic behavior of their lifespan when the size of the initial perturbation tends to 0 this is achieved by the construction and estimate of a suitable approximate flow

minimal surfaces from circle patterns geometry from combinatorics the theory of polyhedral surfaces and more generally the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry whereas classical differential geometry investigates smooth geometric shapes such as surfaces and discrete geometry studies geometric shapes with a finite number of elements polyhedra the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory the latter appears then as a limit of the refinement of the discretization current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization one of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces such as minimal con stant mean curvature isothermic surfaces etc usually one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties the goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible our discretizations are based on quadrilateral meshes i e we discretize parametrized surfaces for the discretization of a special class of surfaces it is natural to choose an adapted parametrization in this paper we investigate conformai discretizations of surfaces i e discretizations in terms of circles and spheres and introduce a new discrete model for minimal surfaces see figures 1 and 2 in comparison with direct methods see in particular 23 leading

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

minimal surfaces with the area growth of two planes the case of infinite symmetry we prove that a connected properly immersed minimal surface in with infinite symmetry group and area growth constant less than is a plane a catenoid or a scherk singly periodic minimal surface as a consequence the scherk minimal surfaces are the only connected periodic minimal desingularizations of the intersections of two planes

remarks on global solutions for nonlinear wave equations under the standard null conditions abstract a combination of some weighted energy estimates is applied for the cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions alternative proofs for global solutions are shown including the exterior domain problems

the small data solutions of general 3 d quasilinear wave equations ii abstract this paper is a continuation of the work in 8 where the authors established the global existence of smooth small data solutions to the general 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0 when the weak null condition holds in the present paper we show that the smooth small data solutions of equation i j 0 3 g i j u u i j 2 u 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied moreover a precise blowup time is completely determined therefore collecting the main results in this paper and 8 we have given a basically complete study on the blowup or global existence of small data solutions to the 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

intersection numbers of special cycles let g q be a connected semisimple algebraic group defined over q and let a r be two q rational automorphisms of finite order of g which commute with each other we assume that the group g r of real points of g is noncompact let i be a torsionfree arithmetic subgroup of g q which is stable under the group a r generated by a and r then a and r act on the space x of maximal compact subgroups of g r and on the locally symmetric space x i in a natural way if g f l denotes the reductive algebraic q group of fixed points of f l a r acting on g then the corresponding symmetric space x f l is acted upon by the arithmetic group 1 f l of fixed points of f l acting on i these give rise to nonempty closed immersed submanifolds c a x a 1 a and c r x r 1 r of x i they are called special cycles we assume that x i c a and c r are oriented and that the intersection c a n c r is compact then if dim c a dim c r dim x the intersection number

cohomologically induced distinguished representations and cohomological test vectors let g be a real reductive group and let be a character of a reductive subgroup h of g we construct invariant linear functionals on certain cohomologically induced representations of g and we show that these linear functionals do not vanish on the bottom layers applying this construction we prove two archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain l functions via modular symbols

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

archimedean non vanishing cohomological test vectors and standard l functions of gl2n complex case abstract the purpose of this paper is to study the local zeta integrals of friedberg jacquet at complex place and to establish similar results to the recent work 4 joint with c chen and d jiang in this paper we will 1 give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation of gl 2 n c with a non zero shalika model 2 construct a new twisted linear period s and give a different expression of the linear model for 3 give a necessary and sufficient condition on the character such that there exists a uniform cohomological test vector v v which we construct explicitly for s as a consequence we obtain the non vanishing of local friedberg jacquet integral at complex place all of the above are essential preparations for attacking a global period relation problem in the forthcoming paper 11

constant mean curvature surfaces in m 2 times mathbf r the subject of this paper is properly embedded h surfaces in riemannian three manifolds of the form m 2 x r where m 2 is a complete riemannian surface when m 2 r 2 we are in the classical domain of h surfaces in r 3 in general we will make some assumptions about m 2 in order to prove stronger results or to show the effects of curvature bounds in m 2 on the behavior of h surfaces in m 2 x r

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

global radial solutions to 3d relativistic euler equations for non isentropic chaplygin gases in this paper we investigate the smooth spherically symmetric solutions to 3d relativistic chaplygin gases with variable entropy whose initial data is assumed to be a small smooth perturbation to a constant state due to the structure of the equation we can still take advantage of the null condition which is satisfied by the potential equation for isentropic and spacetime irrotational relativistic chaplygin gases and obtain the global existence of smooth solution by continuity method this generalizes the result for classical chaplygin gases of godin j math pures appl 87 9 91 117 2007 to the relativistic case

the classification of complete minimal surfaces with total curvature greater than 12 we classify complete orientable minimal surfaces with finite total curvature 87r introduction the classification of complete minimal surfaces with finite total curvature in r3 has been an important problem in the classical differential geometry some basic properties of these surfaces were studied by r osserman see 8 9 who showed the first nontrivial result about this subject concretely he characterized the catenoid and enneper surface as the unique complete orientable minimal surfaces of total curvature 47 however until recent years no more relevant results have been obtained w h meeks 6 gave the classification of nonorientable complete minimal surfaces with total curvature greater than 87 this paper is concerned with the total classification of orientable complete minimal surfaces with total curvature 87 chen and gackstatter 1 discovered the first example of a complete minimal surface properly of genus 1 see theorem 1 the picture of chen gackstatter surface is obtained by joining a handle on enneper s surface this genus one minimal surface has total curvature 87 and no other examples of such surfaces were found so it is expected that no other genus one orientable minimal surface of total curvature 87 does exist in this paper we give a proof of this fact more precisely we prove that chen gackstatter surface is the only genus one orientable complete minimal surface with finite total curvature 87 of course it is not difficult to find genus zero minimal surfaces with total curvature 8i a geometrically interesting example described by jorge and meeks 5 is the trinoid this surface has three embedded catenoid ends moreover its normal vectors at these ends are placed symmetrically in an equator of s2 received by the editors april 27 1990 and in revised form july 20 1990 1980 mathematics subject classification 1985 revision primary 53a10 research partially supported by dgicyt grant ps87 01 15 c03 02

archimedean non vanishing cohomological test vectors and standard l functions of mathrm gl _ 2n gl2n real case the standard l functions of mathrm gl 2n expressed in terms of the friedberg jacquet global zeta integrals have better structure for arithmetic applications due to the relation of the linear periods with the modular symbols the most technical obstacles towards such arithmetic applications are 1 non vanishing of modular symbols at infinity and 2 the existence or construction of uniform cohomological test vectors problem 1 is also called the non vanishing hypothesis at infinity which was proved by sun duke math j 168 1 85 126 2019 theorem 5 1 by establishing the existence of certain cohomological test vectors in this paper we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional lambda s chi which when evaluated with our explicitly constructed cohomological vector is equal to the local twisted standard l function l s pi otimes chi for all complex values s with the relations between linear models and shalika models we establish 1 with an explicitly constructed cohomological vector using classical invariant theory and hence proves the non vanishing results of sun 24 theorem 5 1 via a completely different method

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

siegel cusp forms of degree 2 are determined by their fundamental fourier coefficients we prove that a siegel cusp form of degree 2 for the full modular group is determined by its set of fourier coefficients a s with 4 det s ranging over odd squarefree integers as a key step to our result we also prove that a classical cusp form of half integral weight and level 4n with n odd and squarefree is determined by its set of fourier coefficients a d with d ranging over odd squarefree integers a result that was previously known only for hecke eigenforms