Showing papers in "Annals of Mathematics in 1988"

Reconstructions from boundary measurements

Abstract: Soit Ω un domaine borne dans R n , n≥3 avec une frontiere C 1,1 . On considere l'operateur Lγ(u)=⊇•(γ⊇u) ou γ(x) est une fonction a valeur reelle dans C 1,1 (Ω) avec une borne superieure positive. On definit la forme quadratique Qγ sur H 1/2 (∂Ω) par Qγ(f)=∫ Ω γ(x)|⊇u(x)| 2 dx ou u∈H 1 (Ω) est la solution unique a Lγu=0 dans Ω, u| ∂Ω =f. On etudie la reconstruction de γ a partir de Qγ

Vector bundles of rank 2 and linear systems on algebraic surfaces

Abstract: Let S be a smooth complex algebraic surface and let L be a divisor on S. It is well-known that the linear system IL + KsI, where Ks is the canonical divisor, plays an important role in understanding the geometry of S. Several authors (see, for example, [8], [11]) studied various properties of this system. Our paper treats the same subject, but the point of view is different from the works mentioned above. The philosophy underlying our approach is almost classical: Points on S (more generally, effective 0-cycles) in special position with respect to IL + KsI (see definition below) contain information about the geometry of S. This point of view was recently revived in [5] (also [10]). In fact, this paper contains the technique of going from the effective 0-cycle Z in special position with respect to IL + KsI to the geometry of S. For the sake of completeness we give a definition and state a theorem which can be found in [5], [10].

A sharp inequality of J. Moser for higher order derivatives

Abstract: In fact go = nwl/(1), where wnis the area of the surface of the unit n-ball. Also, he notes that if /3 exceeds go then (1) can no longer hold with some co independent of the u, satisfying (2). The fact that this result holds for some /3 > 0 independent of u has been observed now by several authors with a variety of proofs; see [24], [20], [23], [22], [10], [4], [1], [2], [13], [14], for example. The earliest of these arguments expands the exponential function in a power series. Then observing that the norm in the imbedding

Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

Abstract: We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of small cardinality. Part II is logically independent of Part I.

Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes

Abstract: 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

On the Ramanujan conjecture and finiteness of poles for certain L-functions

Abstract: In this paper we prove a number of general results about automorphic L-functions which appear in the constant terms of Eisenstein series for an arbitrary quasi-split connected reductive algebraic group over a number field. These L-functions were first introduced by Langlands for Chevalley groups [20] and through them he was led to make some of his deep conjectures [21]. One significance of these L-functions is that all the automorphic L-functions studied so far are among them. There are three general results. First, we obtain a uniform line of absolute convergence for all of these L-functions (Theorem 5.1). A uniform estimate for the Hecke eigenvalues of generic cusp forms on many absolutely simple quasi-split connected reductive algebraic groups over number fields (Corollary 5.4) and an improvement on the best available estimate for the Fourier coefficients of Maass wave forms (Corollary 5.5) also follow. Next, we establish a meromorphic continuation and functional equation for each of these L-functions (Theorem 6.1). Finally, in Theorem 6.2, we prove finiteness of poles on the whole complex plane for an important class of these L-functions (Corollaries 6.6 through 6.10). More precisely, let G be a quasi-split connected reductive algebraic group over a number field F. Set G = G(A F)' where A F is the ring of adeles of F. Fix a Borel subgroup B of G over F and let U be its unipotent radical. Let P be a maximal F-parabolic subgroup of G with P D B. In the context of the problems studied here, nothing new will be obtained if one drops the maximality condition on P. Write P = MN, a Levi decomposition, N C U. and let B, U, P, M, and N be the corresponding groups of adelic points. For every place v of F. let GV = G(FJ). Similarly we have Bv, Us, Pv, Me, and Nv. Let X = ?~ Xv be a generic character of U(F) \ U (cf. Section 3). Then each Xv is generic. Let r = Ov 7Tv be a cusp form on M. We shall say 7T is

Generalizations of the Poincaré-Birkhoff Theorem

Abstract: In this article we prove some generalizations of the classical Poincar&Birkhoff theorem on area-preserving homeomorphisms of the annulus which satisfy a boundary twist condition. The work of G. D. Birkhoff on this theorem and its applications can be found in [B1], [B2], and Chapter V of [B3]. A more modern treatment can be found in [B-N]. We prove a theorem for the open annulus A = S1 x (0, 1), since, as we will see, the theorem for the closed annulus can easily be obtained from this result. For the open annulus, however, it is not immediately obvious what should be the analogue of the twist condition. It turns out that the most general hypothesis, and the most natural from the point of view of our proof, involves the notion of positively and negatively returning disks for some lift of f to the covering space A = R x (0, 1). More precisely, if f: A -, A is a lift of f: A -, A we will say that the e is a positively returning disk for f if there is an open disk U c A such that f(U) n U = 0 and fP(U) n (U + k) # 0 forsome n, k > 0(here U + k denotes the set {(x + k, t) I(x, t) e U }). Thus U is disjoint from its image but under iteration by f returns to a positive translate of itself. A negatively returning disk is defined similarly but with k 0 such that fl(U) n U # 0. In Section 2 we prove the following.

The Existence of Kahler-Einstein Metrics on Manifolds with Positive Anticanonical Line Bundle and a Suitable Finite Symmetry Group

Abstract: On utilise la methode de continuite pour demontrer l'existence de metriques de Kahler-Einstein pour des varietes de Kahler de fibre linge positif anticanonique sous l'hypothese additionnelle de l'existence d'un bon groupe de symetries fini ou compact

The image of a derivation is contained in the radical

Abstract: In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conjectured that this result holds even if the derivation is unbounded. We give a proof of this conjecture. The central idea in the proof is the introduction of the concept of a recalcitrant system of elements in a commutative radical Banach algebra. Such systems put algebraic constraints upon a derivation which prevent the derivation from mapping outside of the radical.

Convex hypersurfaces and Fourier transforms

Abstract: Hilbert transforms and maximal functions along curves and surfaces, spectral synthesis problems, and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators. The problem of estimating such Fourier transforms has a long history. See for example Hlawka [3], Herz [2], Littman [4], Randol [9], [10], Svensson [15], Sogge and Stein [12], [13], Stein [14], Marshall [5], etc. The results of these investigations suggest a strong relation between the local curvature properties of the surface S and the decay of the function a'. We are mainly interested in how the geometry of S near a fixed point xO affects the decay of a6, for by using a partition of unity on S, it is clearly enough to study integrals of the type

Covering minima and lattice-point-free convex bodies

Abstract: The covering radius of a convex body K (with respect to a lattice L) is the least factor by which the body needs to be blown up so that its translates by lattice vectors cover the whole space. The covering radius and related quantities have been studied extensively in the geometry of numbers (mainly for convex bodies symmetric about the origin). In this paper, we define and study the "covering minima" of a general convex body. The covering radius will be one of these minima; the "lattice width" of the body will be the reciprocal of another. We derive various inequalities relating these minima. These imply bounds on the width of lattice-point-free convex bodies. We prove that every lattice-point-free body has a projection whose volume is not much larger than the determinant of the projected lattice.

Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds

Abstract: That paper concerned the general theory of groups acting on R-trees and the relationship of these actions to representations into SL2(C). The purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3-manifolds. In the third paper [8] the results of this paper will be applied to study actions of 3-manifold groups on trees. (The starting point of [8] is the idea that "dual" to an action of qg,(M) on an R-tree is a codimension-1 measured lamination in M.) Combining the latter results with [7] gives information about how hyperbolic structures on 3-manifolds can degenerate. We view measured laminations in 3-manifolds as generalizations of both geodesic laminations on surfaces and surfaces in 3-manifolds. Most of this paper consists in extending Thurston's theory of geodesic laminations and the HakenStallings-Waldhausen theory of incompressible surfaces to our context. The paper is organized along the following lines. In Chapter I we develop the basic notions of codimension-1 measured laminations. We prove one new, and quite useful, result (Theorem I.3.2) concerning a decomposition of these objects. We also give a definition of the Euler characteristic of a measured lamination and show that it has all the usual properties. In Chapter II we introduce the branched surfaces of Williams [16] and Hatcher-Thurston. We establish analogues for measured laminations carried by branched surfaces of some results of Thurston's for train tracks and geodesic laminations. By imitating an argument of Plante's [10], we show that the leaves of a measured lamination have polynomial growth. This is used to show that if a branched surface N carries some lamination all of whose leaves have virtually abelian fundamental groups and carries no disks or spheres, then all laminations carried by N have zero Euler characteristic. We express the latter property by saying that N is flat. In Chapter III, generalizing Haken's notion of normal surfaces, we develop the theory of normal measured laminations and of normal branched surfaces in a

A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$-symmetry

Abstract: On donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds

Abstract: We study maps qp from domains Q in R3 to S2 which have prescribed boundary value functions 4 and which minimize Dirichlet's integral energy. Such (p's can have isolated singular points, and our principal concern is the number and arrangement of such singularities. Our main result is that the number of singular points is bounded above by a constant times Dirichlet's integral of 4 on d Q, and we show that this linear law is the best possible in two different senses. Furthermore, singularities of minimizing (q's sometimes have peculiar and counterintuitive properties: (i) We illustrate by example that the mapping area (Jacobian integral) of 4 can be zero although qp has many singular points. (ii) We also show how to construct uniquely minimizing qp's having many singular points stacked up vertically near d Q (like bubbles in a pan of water about to boil). (iii) Finally, we show by example that symmetries of Q and 4 do not insure corresponding symmetries of qp and its singularities; in particular, singularities in (p can be unstable under small perturbations of 4.

Coordinates for triangular operator algebras

Abstract: On demontre un theoreme de structure pour des algebres sous diagonales contenant une sous algebre de Cartan. On etudie les isomorphismes pour ces algebres

On real algebraic morphisms into even-dimensional spheres

Abstract: Given affine real algebraic varieties X and Y let us denote by R( X, Y) the set of regular mappings (real algebraic morphisms) from X into Y (for definitions and notions of real algebraic geometry see [2], where the theory of real algebraic varieties is treated systematically). Our aim in this paper is to study the set M(X,S') of regular mappings from affine nonsingular real algebraic varieties X into S ', the unit sphere EZn+?1x2 = 1. Earlier we obtained several results in this direction [3], [4]; cf. also [2], Chap. 13. In particular, it was shown in [4] that given a compact connected nonsingular orientable real algebraic subset X of RP, of odd dimension k, either each smooth mapping from X to Sk is homotopic to a regular mapping, or precisely those mappings of even topological degree have this property. Now we shall study the case of regular mappings into even-dimensional spheres. Strangely enough the situation then is radically different from that mentioned above for odd-dimensional spheres. For a very large class of compact smooth manifolds of even dimension 2k, "most" algebraic models X of these manifolds have the property that every regular mapping from X into S2k is null homotopic (the meaning of "most" will be made precise later; cf. Remark 1.6, Theorem 2.1, Example 2.3). This class of manifolds contains all compact Co hypersurfaces of R2k?1*

Factorization theorems and the structure of operators on Hilbert space

Abstract: Soit #7B-H un espace de Hilbert complexe, et soit #7B-L(#7B-H) l'algebre des operateurs lineaires bornes agissant sur #7B-H. On demontre des theoremes de factorisation. On etudie la structure des operateurs sur #7B-H

Solution of the Littlewood-Offord problem in high dimensions

Abstract: Consider the 2' partial sums of arbitrary n vectors of length at least one in d-dimensional Euclidean space. It is shown that as n goes to infinity no closed ball of diameter A contains more than ([A] + 1 + o(l))(ln2I) out of these sums and this is best possible. For A - [A] small an exact formula is given.