the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

classification of rotational special weingarten surfaces of minimal type in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r in this paper we classify the complete rotational special weingarten surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r i e rotational surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r whose mean curvature h and extrinsic curvature ke satisfy h f h2 ke for some function f in mathcal c 1 0 infty such that f 0 0 and 4x f x 2 1 for any x 0 furthermore we show the existence of non complete examples of such surfaces

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

a new method for the calculation of the conductivity of inhomogeneous systems a new method for computing the conductivity of random irregular resistor networks is developed this method is a generalization of the transfer matrix technique proposed by derrida and vannimenus for regular 2d and 3d lattices at the same time for large systems the method presented in this paper is more efficient than the transfer matrix technique to demonstrate the method it is applied to a cubic lattice at the percolation threshold and away from it the conductivity has been found for lattices with size up to 324 3 the ratio between the conductivity exponent t and the correlation length exponent was estimated to be t 2 315 in good agreement with the literature data

index one minimal surfaces in spherical space forms abstract we prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two this confirms a conjecture of r schoen for an infinite class of 3 manifolds

on the growth of supersolutions of nonlinear pde s on exterior domains abstract we obtain a comparison principle on annuli with catenoid like functions for supersolutions of non linear elliptic pdes over exterior domains in a non positively curved manifold with a pole this result is applied to get an upper estimate on the growth of such supersolutions and in particular of exterior graphs of non negative mean curvature

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

complete embedded minimal n dimensional submanifolds in ℂn a classical problem in the theory of minimal submanifolds of euclidean spaces is to understand whether a minimal submanifold exists with a prescribed behavior at infinity or to determine from the asymptotes the geometry of the whole submanifold beyond the intrinsic interest of these questions they are also of crucial importance when studying the possible singularities of minimal submanifolds in general riemannian manifolds for minimal surfaces i e two dimensional submanifolds the standard tool to solve these problems is given by the weierstrass representation formula which relates the geometry of the minimal surface to complex analytic properties of holomorphic 1 forms on riemann surfaces recently gluing techniques have been developed that provide an abundant number of new examples of minimal hypersurfaces in euclidean space for higher dimensional minimal submanifolds such a link clearly disappears and no complex analysis can be put into play although gluing techniques have been extensively used in the study of minimal hypersurfaces they have not been adapted to handle higher codimensional submanifolds the aim of this paper is to use a gluing technique for minimal submanifolds to make a step toward understanding these questions in higher codimension more precisely we will restrict ourselves to the case of real n dimensional submanifolds of cn there are two main reasons for doing so the first one comes from the fact that when trying to desingularize the intersection for example of a pair of n planes that give the desired asymptotic behavior one needs a model of a minimal submanifold with this behavior at infinity to rescale and to glue into the pair of planes where a neighborhood of the intersection is removed this local model needs to be sufficiently simple to allow a detailed study of the linearized mean curvature operator in our situation this is provided by a generalization of an area minimizing submanifold found by lawlor in 12 while in more general cases such an example is not known the second reason is that among minimal n submanifolds ofcn there is a

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

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

constant mean curvature graphs on exterior domains of the hyperbolic plane we prove an existence result for non rotational constant mean curvature ends in mathbb h 2 times mathbb r where mathbb h 2 is the hyperbolic real plane the value of the curvature is h in big 0 frac 1 2 big we use schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of mathbb h 2 we also prove a fine property of the asymptotic behavior of the rotational ends introduced by sa earp and toubiana

asymptotic expansion of the tricomi carlitz polynomials and their zeros the tricomi carlitz polynomials f n a x are non classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are d j a x k a k 1e kk atx x k k a 1 2 k 0 1 2 in this paper we derive an asymptotic expansion for f n a t n as n valid uniformly for bounded real t where n n 2 a 1 2 and a is a positive parameter the validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by olde daalhuis and temme the expansion involves the airy functions and their derivatives error bounds are given for one term and two term approximations asymptotic formulas are also presented for the zeros of f n a t n

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

calibrations modulo v abstract the theory of calibrations is extended to show that certain surfaces are area minimizing modulo v for example a complex algebraic variety in c n of degree d is area minimizing modulo v for all v 2 d

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

universality of 3d percolation exponents and first order corrections to scaling for conductivity exponents abstract by using a fast nested dissection algorithm we compare the results of finite size scaling at p c and of p scaling l const on large cubic random resistor networks up to 500 500 500 the p scaling for conductivity of both site and bond networks leads to an exponent t 2 00 1 the finite size scaling leads to the ratio of this conductivity exponent to the coherence length exponent t 2 283 3 combining these results we estimate 0 876 6 in excellent agreement with a value proposed by ballesteros et al the first order correctional exponent is found to be 1 0 2

corrections to scaling in random resistor networks and diluted continuous spin models near the percolation threshold we investigate corrections to scaling induced by irrelevant operators in randomly diluted systems near the percolation threshold the specific systems that we consider are the random resistor network and a class of continuous spin systems such as the x y model we focus on a family of least irrelevant operators and determine the corrections to scaling that originate from this family our field theoretic analysis carefully takes into account that irrelevant operators mix under renormalization it turns out that long standing results on corrections to scaling are respectively incorrect random resistor networks or incomplete continuous spin systems

uniform asymptotic expansion of j_ nu nu a via a difference equation there are two ways of deriving the asymptotic expansion of j nu nu a as nu to infty which holds uniformly for a geq 0 one way starts with the bessel equation and makes use of the turning point theory for second order differential equations and the other is based on a contour integral representation and applies the theory of two coalescing saddle points in this paper we shall derive the same result by using the three term recurrence relation j nu 1 x j nu 1 x 2 nu x j nu x our approach will lead to a satisfactory development of a turning point theory for second order linear difference equations

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 hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

fundamental groups of manifolds with positive isotropic curvature a central theme in riemannian geometry is understanding the relationships between the curvature and the topology of a riemannian manifold positive isotropic curvature pic is a natural and much studied curvature condition which includes manifolds with pointwise quarter pinched sectional curvatures and manifolds with positive curvature operator by the results of micallef and moore there is only one topological type of compact simply connected manifold with pic namely any such manifold must be homeomorphic to the sphere on the other hand there is a large class of nonsimply connected manifolds with pic an important open problem has been to understand the fundamental groups of manifolds with pic in this paper we prove a new result in this direction we show that the fundamental group of a compact manifold m n with pic n 5 does not contain a subgroup isomorphic to the techniques used involve minimal surfaces

asymptotics of racah polynomials with varying parameters abstract within the askey scheme of hypergeometric orthogonal polynomials racah polynomials stay on the top of the hierarchy and they generalize all of the discrete hypergeometric orthogonal polynomials in this paper we investigate asymptotic behaviors of racah polynomials with varying parameters when the polynomial degree tends to infinity using the difference equation technique developed in our earlier papers we obtain an asymptotic formula in the outer region via ratio asymptotics and then derive asymptotic formulas in the oscillatory region via a matching method our asymptotic formulas are explicitly given in terms of the polynomial degree variable and parameters using elementary functions such as logarithmic exponential and rational functions by taking limits our results also yield asymptotic formulas for orthogonal polynomials in the lower hierarchy of the askey scheme such as hahn polynomials and krawtchouk polynomials

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

asymptotic expansions for second order linear difference equations abstract formal series solutions are obtained for the difference equation y n 2 a n y n 1 b n y n 0 where a n and b n have asymptotic expansions of the form a n s 0 a s n s and b n s 0 b s n s for large values of n and b0 0 these solutions are characterized by the roots of the characteristic equation 2 a0 b0 0 our discussion is divided into three cases according to whether the roots are distinct or equal and do not satisfy the auxiliary equation a1 b1 0 or equal and do satisfy the auxiliary equation the last case is further divided into three subcases according to whether the roots of the indicial equation 1 2 a1 a2 b2 0 do not differ by a nonnegative integer or differ by a positive integer or are equal in all cases the formal series solutions will be shown to be asymptotic our approach is based on the method of successive approximations

a schoen theorem for minimal surfaces in h 2 r abstract in this paper we prove that a complete minimal surface immersed in h 2 r with finite total curvature and two ends each one asymptotic to a vertical geodesic plane must be a horizontal catenoid moreover we give a geometric description of minimal ends of finite total curvature in h 2 r we also prove that a minimal complete end e with finite total curvature is properly immersed and that the gaussian curvature of e is locally bounded in terms of the geodesic distance to its boundary

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

asymptotic expansions for second order linear difference equations with a turning point a turning point theory is developed for the second order difference equation where the coefficients a n and b n have asymptotic expansions of the form 0 being a real number in particular it is shown how the airy functions arise in the uniform asymptotic expansions of the solutions to this three term recurrence relation as an illustration of the main result a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the freud weight exp x 4 x

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

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

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