Showing papers in "Annals of Mathematics in 1993"

A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation

Abstract: In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation

Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality

Abstract: where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see [8]). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is "What is the situation in dimension 2?" Two important arguments ([25], [26], [27]) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in part

Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space

Abstract: In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant [Br] showed that an analogue of the Weierstrass-representation formula holds for surfaces of constant mean curvature-i in the hyperbolic 3-space X3. In this article we abbreviate the term "constant mean curvature-i" as CMC-1. Like minimal surfaces in the euclidean space, the hyperbolic Gauss map of CMC-1 surfaces is defined as a holomorphic map to C U {oo}. However, in contrast to the euclidean case, the hyperbolic Gauss map of a CMC-1 surface may not be extended across the ends even if the total Gaussian curvature is finite. We call a complete CMC-1 surface, whose Gauss map can be extended across all of its ends, a CMC-1 surface of regular ends. In this article we produce an explicit tool to construct CMC-1 surfaces of regular ends. In Section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity. In our CMC-1 category, Ossermann's inequality is not expected and the Cohn-Vossen inequality is the best possible one. We show in Section 4 that the equality of the Cohn-Vossen inequality never holds for complete CMC-1 surfaces in XH3. In Section 5 we give a necessary and sufficient condition that a regular end of a CMC-1 surface be embedded. In Section 6 we classify complete CMC-1 surfaces of genus 0 with two regular ends. Our classification contains new examples. Furthermore, in Section 7, we construct several new CMC-1 surfaces with regular embedded ends. Each of these examples has a nontrivial deformation, which is mentioned in Section 3. It should be remarked that our construction does not work for surfaces with irregular ends. But there is another construction: By perturbing minimal surfaces in the euclidean 3-space, the authors constructed CMC-1 surfaces, all of whose ends are irregular (see [UY1]).

Fixed points, Koebe uniformization and circle packings

Abstract: A domain in the Riemann sphere \(\hat{\mathbb{C}}\) is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following conjecture, known as Koebe’s Kreisnormierungsproblem: Any plane domain is conformally homeomorphic to a circle domain in \(\hat{\mathbb{C}}\). When the domain is simply connected, this is the content of the Riemann mapping theorem.

Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras

Abstract: Much of the qualitative behaviour of the problem can be seen in embryo form in the simplest possible case: that of harmonic maps of a compact Riemann surface M into S. If M is the Riemann sphere, all harmonic maps into S are ±-holomorphic and so are given by rational functions. A similar picture obtains when the target is a higher dimensional symmetric space [6,7,11,34,35]: in this setting, harmonic maps are obtained from holomorphic curves in some auxiliary complex manifold (a twistor space) such as a flag manifold [6,11] or a loop group [34].

Asymptotic completeness of long-range N-body quantum systems

Abstract: where X is a euclidean space and {lra: a E A} is a finite family of projections onto certain subspaces xa of X. We will always assume that the family {Xa: a E A} is closed with respect to the algebraic sum and contains Xamin : = {0}. The class of operators of the form (0.2) is a generalization of (0.1) up to a linear change of coordinates in the configuration space X. They usually go under the name of generalized N-body Schrbdinger operators and were first considered in [A]. (In what follows we will drop the word "generalized".) We refer the reader to [RSi], vol. III, for a general introduction to N-body Schr6dinger operators defined as in equation (0.1). We will consistently use the definition (0.2) and assume that the reader is familiar with the physical meaning of various objects under study.

Asymptotics of the ground state energies of large Coulomb systems

Abstract: We prove the Scott conjecture for molecules.

Newton polygons of zeta functions and L functions

Abstract: In this article we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, is false in its full form, but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form

Uniqueness and regularity of proper harmonic maps

Abstract: Introduction The purpose of this paper is to study some uniqueness and regularity properties of the Dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces. In general, if the metrics of two complete noncompact manifolds Mm and N' are given locally by ds2 = iMj gij dxzdxj and dsy = -En h:,3 du'du$, respectively, then the energy-density function of a C1 map u: M -N is defined by

The topology of instanton moduli spaces, I: The Atiyah-Jones conjecture

Abstract: In this paper we study the global geometry and topology of the moduli spaces of based SU(2)-instantons over the 4-sphere S4 . These instanton moduli spaces have a rich history and have been analyzed from many points of view. Originally these spaces, which we denote by Mk, were defined as solution spaces (modulo gauge equivalence) to certain partial differential equations, namely the self-duality equations associated to the Yang-Mills functional in SU(2) gauge theory. They have been successfully studied from this point of view by Taubes ([T1], [T2]), Uhlenbeck [U] and others. An important alternative approach was initiated by Ward [W], who related instantons to certain holomorphic bundles on CP3, and was continued by Atiyah and Ward in [AW]. This allowed the classification of instantons on $4 in terms of quaternionic linear algebra by Atiyah, Drinfeld, Hitchin and Manin [ADHM]. This holomorphic approach was further extended by Donaldson [D], who showed that these bundles were determined by their restriction to a Cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in C2R2. Hurtubise [Hul] then exploited this fact to study the moduli spaces Mk, as did Atiyah [A] to show that Mk arise naturally in the theory of holomorphic maps into loop groups; this latter approach was continued by Gravesen [G]. Atiyah and Jones [AJ] obtained the first results and formulated the foundational questions on the global topology of these moduli spaces. Recall that an element of Mk is a based gauge-equivalence class of a connection on the principal SU(2) bundle over $4, denoted by Pk with second Chern class k, satisfying the self-duality equations. There is a natural forgetful map to the based equivalence classes of all connections in Pk. Atiyah and Jones [AJ] showed that this target space, which we denote by Bk, is homotopy equivalent

Enumerating finite groups of given order

Abstract: Let f(n) denote the number of (isomorphism classes of) groups of order n Let n = HW p9ig be the prime factorization of n and let A = A(n) = Ek and ,u = ii(n) = maxkL1 gi The function f(n) has been subject to considerable study G Higman [15] and CC Sims [26] showed that if n is a prime power, say, n = p', then f(n) = n(2/27+o(l))a2 for a -x oc PM Neumann [22] proved that if the number of (isomorphism classes of) simple groups of order n is bounded by ncA2 for some constant c, then f(n) < n(1+c)A2 holds in general Combining the methods of [22] with some consequences of the Classification Theorem of finite simple groups, A McIver and Neumann [21] obtained that in fact the estimates f(n) < nA2/2 and f(n) < n1,2+1,+2 hold Recently DF Holt [16] found a rather elegant proof for the bound

Finitely generated groups, $p$-adic analytic groups and Poincaré series

Abstract: This is equivalent to the coefficients apn(G) satisfing a linear recurrence relation with constant coefficients for sufficiently large n. These Poincare series were first studied in the case when G is a finitely generated, torsion-free nilpotent group in a paper by Grunewald, Segal and Smith [GSSm]. There the authors established that, for such a group, (G,p(s) is a rational function in p8s. In that setting the functions (G,p(S) are the local factors associated with the Dirichlet series

The classification of the simple modular Lie algebras: IV. Determining the associated graded algebra

Abstract: F (c1, . . . , cn) : F, c p i ∈ F ∀i. He could prove a Galois-like theorem by considering derivation algebras instead of automorphism groups. The set DerF F (c1, . . . , cn) of F -derivations of F (c1, . . . , cn) carries the structures of an F (c1, . . . , cn)-module in the obvious fashion, an associative p-power mapping D → D, a Lie algebra under the bracket operation [D1, D2] = D1 ◦D2 −D2 ◦D1.

Etale cohomological dimension and the topology of algebraic varieties

Abstract: The purpose of this paper is twofold: to develop a theory of etale cohomological dimension in the context of schemes of finite type over a separably closed field that would be analogous to the well-known theory of quasicoherent cohomological dimension, and to apply our theory to prove new results about the topology of algebraic varieties of small codimension in n-space. The cohomological dimension of a scheme X relative to a closed subscheme Y, denoted by cd(X, Y), is the largest integer r such that the local cohomology group H' (X, F) $& 0 for some abelian torsion sheaf F on Xet, where F consists only of torsion prime to all the residual characteristics of X. The gist of our theory is a technique for proving various upper bounds on cd(X, Y), especially in the case where Y has small codimension in X. Local bounds are the most important as well as the easiest to state. Accordingly, for the purposes of this introduction, let X = Spec A, where A is a strictly Henselian local ring of a finite-type scheme over a separably closed field. Just what kind of bounds on cd(X, Y) one should expect is indicated by the theory of quasicoherent cohomological dimension, which has been developed by several authors; see, for example, [Fa], [Grl], [Hal], [Ha2], [HaSp], [HuLy], [01] and [PesSz]. The quasicoherent cohomological dimension of X relative to Y, denoted by qccd(X, Y), is the largest integer r such that there exists a quasicoherent sheaf F on Xzar with H'(X, F) 54 0. In the above-stated local case, the main results of the theory of quasicoherent cohomological dimension are the following (where n = dim X): (i) qccd(X, Y) < n (see [Gri], 1.12). (ii) qccd(X, Y) < t if Y is set-theoretically defined by t equations (see [Gri], 2.3).