Showing papers in "Annals of Mathematics in 1989"

On the Cauchy problem for Boltzmann equations: global existence and weak stability

Abstract: We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.

Métriques de Carnot-Carthéodory et quasiisométries des espaces symétriques de rang un

Abstract: We exhibit a rigidity property of the simple groups Sp(n, 1) and F7-20 which implies Mostow rigidity. This property does not extend to O(n, 1) and U(n, 1). The proof relies on quasiconformal theory applied in the CR setting. Extensions are given to a class of solvable Lie groups. As a byproduct, a result on quasiisometries of infinite nilpotent groups is obtained. Dans cet article, on etablit une propriete de rigidite' des groupes simples de rang un Sp(n, 1), n 2 2 et Fqui entraine la rigidite de Mostow: 1. THEOREME. Toute quasiisornwtrie de 1'espace hyperbolique quaternionien HHn, n > 2, (resp. du plan hyperbolique de Cayley CaH2) est d distance bornee d'une isometrie, i.e., differe d'une isomnetrie par une application qui diplace les points d 'une distance bornee. Une application f entre espaces metriques est une quasiisometrie s'il existe des constantes L et C telles que l'image de f soit C-dense et que, pour tous X y, C + -d(x, y) < d(fx, fy) < Ld(x, y) + C. L Une quasiisometrie entre des G et G' est une sorte "d'isomorphisme virtuel" dans la categorie topologique (en effet, cela correspond 'a une action de G sur un fibre principal C' de groupe G' sur une base compacte; cf. [Ra]). Un isomorphisme entre sous-groupes cocompacts (covolume fini suffit pour les groupes simples de rang un) de groupes de Lie s'etend en une quasiisometrie des espaces symetriques ou des groupes de Lie. Si celle-ci est proche d'une isometrie des espaces symetriques (resp. un isomorphisme des groupes de Lie), les sous-groupes sont conjugues, c'est la rigidite de Mostow [M2]. La propriete ci-dessus ne s'etend pas aux groupes O(n, 1) et U(n, 1). En effet, (paragraphe 11.7) ceux-ci ont beaucoup de quasiisometries, au moins This content downloaded from 207.46.13.92 on Sun, 20 Nov 2016 04:26:46 UTC All use subject to http://about.jstor.org/terms

Interior a priori estimates for solutions of fully non-linear equations

Abstract: On etend une theorie de perturbations aux solutions d'equations uniformement elliptiques d'ordre 2 totalement non lineaires

Continuity properties of entropy

Abstract: On donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d'une auto-application C r d'une variete C r , r>1

Estimates for the Bergman and Szegö kernels in $\mathbf{C}^2$

Abstract: The purpose of this paper (some of whose conclusions were announced in [NRSW]) is to study the Bergman and Szegd projection operators on pseudoconvex domains Q of finite type in C2. The results we obtain are of three kinds: (i) The "size" estimates of the Bergman and Szeg6 kernels, and their derivatives. (ii) The "cancellation" properties of those kernels, expressed in terms of the actions of these operators on suitable "bump" functions; these properties have an interest in their own right but are also crucial in (iii) below. (iii) The sharp mapping properties of these operators on functions spaces, such as LP, the nonisotropic Sobolev spaces, and the nonisotropic Lipschitz spaces, which are naturally attached to the geometry of the boundary. This also leads ultimately to sharp results in the regularity properties of solutions of the equation du = f in Q, and 9bU = f on the boundary. Let us describe these matters in more detail. Our starting points are certain basic geometric constructs associated to Q: a metric p defined in terms of the vector fields X1 and X2 which are the real and imaginary parts of the (tangential) Levi vector field on d 2; and a function A(p, 8), (a polynomial in 8 for p E d 2), which represents the "higher" Levi-invariant attached to d 9 . The significance of these to the geometry of d Q can be understood from the following facts: the ball B(p, 8) C d Q (centered at p, of radius 8 in the metric p) when viewed in the appropriate coordinates has width - 8 in the complex tangential directions, but its width is - A(p, 8) in the complementary direction. Thus the volume of the ball is - 82A(p, 8). Among other crucial properties

Essential laminations in 3-manifolds

Abstract: It is a well-established principle in manifold topology that by studying codimension-1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself. This is particularly true for 3-dimensional manifolds. Using incompressible surfaces a great many advances have been made during the last 30 years; the most spectacular results were obtained by Haken [Ha], Waldhausen [W] and Thurston [T1]. More recently, work [T2], [G14] using taut foliations in 3-manifolds has proven fruitful. Unfortunately (in some sense) most closed 3-manifolds do not contain incompressible surfaces and it is currently not known exactly which 3-manifolds have taut foliations. The purpose of this paper is to study another codimension-1 object, the essential lamination, which is a generalization of the incompressible surface as well as the taut foliation. We will show that the existence of an essential lamination in a 3-manifold M implies that M has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation. We will also show that the universal cover of a manifold containing an essential lamination is R3. Precise definitions of terms used in the following theorem will be given in Section 1. Here we give only a rough idea. We begin with two alternative approximate definitions of "essential lamination." (I) An essential lamination in a 3-manifold M is a lamination satisfying four conditions: The inclusion of leaves of the lamination into M should induce an injection on sTJ, the complement of the lamination should be irreducible, no leaf should be a sphere, and the lamination should be "end-incompressible." To say that a lamination is end-incompressible means, roughly, that a "folded" leaf can be straightened using an isotopy; there are no infinite folds. (II) Alternatively

Parabolic Equations for Curves on Surfaces. 2. Intersections, Blow Up and Generalized Solutions

Abstract: We describe a theory for parabolic equations for immersed curves on surfaces, which generalizes the curve shortening or flow-by-mean-curvature problem, as well as several models in the theory of phase transitions in two dimensions. A class of equations is described for which the initial value problem is well-posed for rough initial data, for which one can give a description of the way a smooth solution becomes singular, and for which one can define generalized solutions, i.e., solutions which are smooth, except at a discrete set of times. The methods which are used in this paper are more geometrical than those of Part I. By comparing arbitrary solutions with certain special solutions, and by considering the way they intersect, we derive estimates for the curvature and the tangent, which allow one to study the initial value problem, and the way solutions become singular.

Exponential sums and Newton polyhedra: Cohomology and estimates

Abstract: The basic objects of this study are exponential sums on a variety defined over a finite field Fq (q = pa, p = char Fq). As we have remarked in some earlier articles [1], [2], we find it more natural to begin with exponential sums on the torus (Gm)n, extend via the usual toric decomposition of An to exponential sums on affine n-space, and finally proceed via a standard character argument [4] to exponential sums on an affine variety defined over Fq. While this is the natural order of the work, what we do, in fact, in the first part of this article is to combine the first two steps and deal with exponential sums on varieties V of the form V = (Gm)r X As (r + s = n). Let f E Fq[xl,..., xn, (xl ... xr-'] be an arbitrary regular function on V. Then f is a sum of monomials and as such has a well-defined Newton polyhedron A( ff) at infinity. This is the convex closure in Rn of the lattice points which occur as exponents of the terms of f together with the origin. We have indicated in our previous work [1], [2] the description of some of the invariants of the associated L-function in terms of properties of this polyhedron. For example, in [1] we showed how bounds for the degree and total degree of the L-function associated with a general exponential sum on V can be expressed in terms of the volumes of A(f) and the intersections of A(f) and the various coordinate spaces. In the present article, assuming f is nondegenerate and commode with respect to Af f), we show these estimates are sharp. Our methods are p-adic and are based on the work of Dwork [11], [12]. Our main accomplishment, from which our other results follow, is the extension of Dwork's cohomology theory from smooth, projective hypersurfaces in characteristic p to a general class of exponential sums. Given f regular on V, we construct a complex of p-adic Banach spaces on which Frobenius acts. The alternating product of characteristic polynomials of Frobenius describes the associated L-function. In fact, when f is nondegenerate and commode with respect to A( f), the complex is acyclic in dimensions other than 0 and the characteristic

Excision in cyclic homology and in rational algebraic $K$-theory

Abstract: (for a precise definition, see ?1 below). By replacing everywhere K*( ) by K*( ) ? Q, one obtains the corresponding notion in rational algebraic K-theory. The above definition has an obvious counterpart for cyclic homology and algebras over a fixed field. In the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable (due to the well-known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring). An extension of k-algebras will be called pure if it is pure as an extension of k-modules (in the sense of P. M. Cohn [6]; cf. also Appendix A.3 below). The class of pure extensions contains, e.g., (i) extensions which admit a k-module splitting, (ii) extensions A >-e R -* S with S flat over k. In fact, one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k-modules is an inductive limit of split extensions (see Theorem A.4 of Appendix A below). Everywhere in this paper the word "excision" used in the context of cyclic homology will mean "excision with respect to the class of pure extensions." The second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology. Before stating the corresponding result we need the following definition. Let us

Endomorphisms of C*-algebras, cross products and duality for compact groups

Abstract: On definit une nouvelle sorte de produit croise d'une C*-algebre par des semigroupes d'endomorphismes avec des proprietes reliees a la symetrie de permutation et a l'existence de conjugues. Un groupe compact est alors defini intrinsequement par son action duale sur le produit croise. Cette construction donne une caracterisation des duaux de groupes compacts abstraits qui est hors d'atteinte de la theorie de dualite de Taunaka-Kein et est independante de cette theorie. Ceci resout le probleme de demontrer l'existence d'un groupe de jauge global compact en physique des particules etant seulement donnees les observables locales

Algebraic cycles and homotopy theory

Abstract: examine its homotopy groups which, after a certain idealization of the space, turn out to be astonishingly simple. In fact for complex projective space pn the structure can be understood completely. This yields new information about the topology of the classical Chow varieties, and establishes an explicit relationship with universal cohomology operations. In the general case complete computations are difficult, but we shall establish a "complex suspension" theorem and lay the foundations for a theory based on the homotopy groups of Chow varieties. To state the results we must give precise meaning to " the space of subvarieties". We begin with the fundamental case of subvarieties in complex projective n-space pln. For each pair of integers p and d with d ? 1 and 0 < p < n, consider the set Wp d(pn) of all finite formal sums

The trace class conjecture in the theory of automorphic forms

Abstract: Let G be the group of real points of a reductive algebraic \( \Bbb {Q} \)-group satisfying the assumptions made in [H, Chapter I] and let \( \Gamma \) be an arithmetic subgroup of G. Let \( R_{\Gamma} \) be the right regular representation of G on \( L^2(\Gamma \backslash G) \) and denote by \( R^d_\Gamma \) the restriction of \( R_\Gamma \) to the discrete subspace. In this paper we prove that for every integrable rapidly decreasing function f on G, the operator \( R^d_\Gamma (f) \) is of the trace class.

Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I

Abstract: We prove that for a simply connected domain Q C C whose boundary a Q is self-similar there is the following dichotomy, concerning the harmonic measure X on a Q viewed from 0: Either a Q is (piecewise) real-analytic or else o is singular with respect to the Hausdorff measure A,, (notation o I A,,,) using Makarov's function 4Dc(t) = t exp(c log 1/t log log log 1/t) for some c = c(X) > 0, and X is absolutely continuous with respect to A,:, (notation X c(@). So if Q has "fractal" boundary then the boundary compression and the "radial growth" of j logIR'l j for a Riemann mapping R: D -* are as strong, respectively as fast, as permitted by Makarov's theory. We prove that c(X) = ,2a 2/X for some asymptotic variance a 2 for a sequence of weakly dependent random variables and a Lyapunov characteristic exponent X. This includes the case where a Q is a mixing repeller (in Ruelle's sense) for a holomorphic map f defined on its neighbourhood, the case a Q is a quasi-circle, invariant under the action of a quasi-Fuchsian group (for a pair of isomorphic, compact surface, Fuchsian groups) and the cases of the boundary of the "snowflake" and, more generally of Carleson's "fractal" Jordan curves. This dichotomy is partially deduced from the dichotomy concerning Gibbs measures for Holder continuous functions on an arbitrary mixing repeller X C C for a holomorphic map. Either i I AK where K is the Hausdorff dimension of y

Limitations to the equi-distribution of primes I

Abstract: In an earlier paper FG] we showed that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept xed. However, by a new construction, we show herein that this fails in the same ranges, for a xed and, indeed, for almost all a satisfying 0 < jaj < x= log N x.

A structure theorem in one dimensional dynamics

Abstract: On considere la classe #7B-A des applications C ∞ f:[0,1]→[0,1] telles que f(0)=f(1)=0 et f a un unique point critique C∈(0,1). Si le point critique de f∈#7B-A est non plat alors f n'a pas d'intervalle errant

Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen

Abstract: Das Ziel dieser Arbeit ist es, ein allgemeines Resultat uiber arithmetische Eigenschaften von analytischen Homomorphismen zwischen kommutativen algebraischen Gruppen zu beweisen. Viele Ergebnisse und Probleme in der Transzendenztheorie lassen sich auf diese Frage zuriuckfiuhren, und unser Resultat gibt auf eine ganze Reihe von offenen Fragen eine Antwort. Es hat sich herausgestellt, dal3 das Studium von analytischen Homomorphismen zwischen kommutativen algebraischen Gruppen in der Transzendenztheorie sehr nutzbringend ist und zu sehr schonen Resultaten gefuihrt hat. Dies wurde von S. Lang vor etwa zwanzig Jahren bemerkt, und er bewies in [LI], dal3 fir kommutative algebraischen Gruppen G, welche uiber Q definiert sind und fur Elemente a / 0 aus T(G)(Q) das Bild expG(a) unter der Exponentialabbildung im allgemeinen nicht in G(Q) liegt. Hier bedeuten T(G) der Tangentialraum von G im neutralen Element, T(G)(Q) die Menge der Q-rationalen Punkte von T(G) und G(Q) die Untergruppen der algebraischen Punkte von G. Mit anderen Worten ist G(Q) die Gruppe der Q-wertigen Punkte des Gruppenschemas G. Aus diesem Ergebnis von Lang konnen eine ganze Reihe von Transzendenzresultaten gewonnen werden. Unter anderem gewinnt man hieraus die Transzendenz von ea fur algebraisches a / 0, wenn man G = Gm setzt, wobei Gm das multiplikative Gruppenschema ist. Dies ist der beriihmte Satz von Lindemann. Dieses Ergebnis entspricht einem Ergebnis uiber I-Parameter-Untergruppen von algebraischen Gruppen. Es wurde kurze Zeit darauf von Lang [L2], [L3] in verschiedenen Richtungen auf d-Parameter-Untergruppen von algebraischen Gruppen erweitert. All diesen Arbeiten lag eine von Schneider [Schl], [Sch2] entwickelte Methode zugrunde. Eine zweite grundlegende Methode wurde von A. Baker [Bal], [Ba2] im Zusammenhang mit dem Studium von Linearformen in Logarithmen von algebraischen Zahlen entwickelt. Sie wird in unseren Untersuchungen eine zentrale Rolle spielen, zusammen mit den sogenannten Nullstellenabschatzungen auf algebraischen Gruppen. Diese wurden in den letzten Jahren von D. W. Masser

Moduli space, heights and isospectral sets of plane domains*

Abstract: Let Q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type. Then there exists a flat metric g on 2 such that Q is isometric with 29g. In what follows we do not distinguish between isometric domains. By the spectrum of 29 we mean the spectrum of the Laplace-Beltrami operator Ag on 29 with Dirichlet boundary conditions. The height h(Eg) = - log det Ag is a spectral invariant and plays a central role in this paper. Among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum. This metric is characterized by the fact that d 2 has constant geodesic curvature; we call such a metric uniform and denote it by u. The set of all such metrics is denoted by u(2). We can therefore identify ( with the moduli space #(2) of conformal structures on E. For n ? 3 we introduce a special parametrization for Mu(2) by means of which we show that h(u) - oo as u approaches the boundary of fu(2). Using this along with the heat invariants for the Laplacian we then show that any isospectral set of plane domains is compact in the C' topology. Similar results hold for n = 1 and 2.

Ergodic Actions of Compact Groups on Operator Algebras: I. General Theory

Abstract: We introduce some general machinery for studying ergodic actions of compact groups on operator algebras, the most important tool being that of the multiplicity map. This theory is used subsequently in [42] to prove that a von Neumann algebra on which SU(2) acts ergodically is necessarily of Type I.

On the zeros of the derivatives of real entire functions and Wiman’s conjecture

Abstract: Let f be an entire function of finite order, real on the real axis, and possessing only real zeros. A classical problem, proposed by G. P6lya [9], [10] and A. Wiman [1], is to determine, from the Hadamard canonical representation of the function, the number of nonreal zeros of the higher derivatives of f. In this paper we shall prove the following conjecture of Wiman, which has been open since 1915:

Multiplicity estimates on group varieties

Abstract: tions the field K will be the field Q of algebraic numbers. We remark at this point that the results remain true if we take instead of the field of complex numbers C its p-adic analogue C for some fixed prime p and for K a corresponding subfield Kp of CP. We restrict ourselves to the complex case in order to avoid some minor complications appearing in the p-adic domain. These complications always arise when analytic functions come in. Our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p-adic domain. These difficulties can be avoided by a purely algebraic approach. If we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero. But we prefer to avoid such an approach in order to keep the text understandable, also for those who are mainly interested in the applications in transcendence.

The nonexistence of certain level structures on abelian varieties over complex function fields

Abstract: Let A be a principally polarized abelian variety of dimension g over a number field k. The Mordell-Weil theorem tells us that the group A(k) of k-rational points of A is finitely generated; in particular, the torsion subgroup of A(k) is finite. For the important special case of an elliptic curve E/Q, Mazur [Mal,2] has proved that the torsion subgroup of E(Q) has order < 16. In general one would expect the torsion subgroup of A(k) to have order < C(g, k), a constant depending only on the number field k and the dimension g but not

Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature

Abstract: Siu-Yau [16] studied the compactification of complete Kaihler manifolds of finite volume and of Riemannian sectional curvature pinched between two negative constants. In [12] the first author of the present article started a systematic study of the more general problem of compactifying complete Kwhler manifolds of finite volume and of bounded curvature. A number of conjectures were formulated, which can all be regarded as conjectural generalizations of the compactification of arithmetic varieties (i.e. quotients of bounded symmetric domains Q by torsion-free arithmetic subgroups of Aut(Q)) in a differentialgeometric setting. Here and henceforth all discrete groups of automorphisms will be assumed torsion-free. In the same article we treated the case of Kaihler surfaces and proved:

Orbital integrals on p-adic groups: A proof of the Howe conjecture

Abstract: Let G be a connected reductive group over a p-adic field F of characteristic 0. Let G = G(F). Let o be a compact subset of G, oG the invariant subset { gxg1: g E G, x E co} of G. We say that a closed invariant subset of G, Q, is compact modulo conjugation if Q c wG for some compact co. (It is equivalent to assume that the traces of Q on all Cartan subgroups of G are compact.) The purpose of this article is to prove the following theorem concerning invariant distributions on G, which had been conjectured by Howe [10]. Let Of(Q) denote the set of invariant distributions supported in U. If K is a compact-open subgroup of G, let XK be the Hecke algebra of compactly supported functions on G with complex values, invariant on the left and right by K.