Showing papers in "Annals of Mathematics in 1986"

Ergodicity of billiard flows and quadratic differentials

Abstract: On considere un systeme billard de 2 objets de masses m 1 et m 2 . On montre que pour un ensemble dense de paires (m 1 ,m 2 ) ce systeme est ergodique

Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds

Abstract: This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an approximation to the published version, which is the definitive form for part I, and is provided for convenience only. All references and quotations should be taken from the published version, since the theorem numbering is different and not all corrections have been incorporated into the present version. Parts II and III will be made available as eprints shortly.

A numerical criterion for uniruledness

Abstract: An n-dimensional variety X over an algebraically closed field k is said to be uniruled if there exist an (n 1)-dimensional k-variety W and a dominant rational map f: P' x W -* X. X is called separably uniruled if f is a separable map (i.e. k(P' x W) is a finite separable extension of k(X) via f). If X is uniniled, then there exists a rational curve passing through a general k-valued point of X, and the converse is true (by the countability of the components of the Hilbert scheme) provided that the ground field k is uncountable. Uniruled varieties are of importance in the classification of varieties; it is conjectured that in characteristic 0 uniruledness is equivalent to K = Xc. This article gives a numerical criterion for a projective variety to be uniruled:

Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature

Abstract: Sur une variete de Riemann complete M, on considere le noyau de la chaleur H(x,y,t) qui resout l'equation de la chaleur (Δ−∂/∂t) F(x,t)=0 sur MX(0,∞) avec la donnee initiale F(x,0)=f(x), en posant F(x,t)=∫ M H (x, y,t)•f(y)dy

Fundamental solutions for second order subelliptic operators

Abstract: Estimation de la solution fondamentale d'un operateur aux derivees partielles, lineaire, du second ordre sur une variete compacte avec une mesure reguliere

Profinite braid groups, Galois representations and complex multiplications

Abstract: In this paper, we consider two closely correlated subjects. One is a pro-i analogue of the braid group, and the other is a construction of the universal 1-adic power series for complex multiplications of Fermat type, or equivalently, for Jacobi sums. Both arise from, and constitute, a first step in the study of the canonical representation of the absolute Galois group GQ = Gal(Q/Q) in the outer automorphism group of the profinite fundamental group of PQ \{O,1,oo}.

Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator

Abstract: : Second order subelliptic operators have been the subject of a considerable amount of research in recent years. Starting with the paper Rothschild and Stein, in which the sharp form of Hormander's famous subellipticity theorem is proved, and continuing through the work of Fefferman and Phong and Sanchez-Calle, it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration. The main purpose of this article is to obtain bounds, from above and below, on p(t,x,y), t epsilon (1, infinity), in terms of standard heat kernels.

K-theory and dynamics. I

Abstract: Un theoreme de H-cobordisme avec commande feuillettee. La variete associee. Le transfert asymptotique. Dynamique du flot geodesique. Completion de la demonstration du theoreme principal. Revue de la theorie de la commande. Structures de longues cellules minces

On the critical values of Hecke $L$-series

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High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials

Abstract: On considere la dynamique d'un systeme quantique constitue de particules de masse elevee et de particules de masse faible, qui interagissent par des potentiels reguliers. On demontre que si les grandes masses sont proportionnelles a e −4 , alors certaines solutions de l'equation de Schrodinger dependante du temps ont des developpements asymptotiques a des ordres arbitrairement eleves en puissances de e, et quand e→0

On a class of nearly holomorphic automorphic forms

Abstract: with a, b,1 e C and 0 < N e Z. The point that attracts our attention is its nonholomorphic nature. It is noteworthy that f is holomorphic if k # 2, and also such nonholomorphy doesn't appear in the Hilbert modular case. Therefore one may naturally ask whether this is an isolated oddity. In our previous papers [10] and [13], we showed that it was not so. In fact, a similar Eisenstein series on Sp(n, Q) or on SU(n, n) of a particular weight can have the form

Smooth maps, null-sets for integralgeometric measure and analytic capacity

Abstract: Here IFI is the one-dimensional Lebesgue measure of a set F lying on a line, and p0 denotes the orthogonal projection onto the line Lo through the origin having normal (cos 0, sin 0). We call a map segmental if it maps every line segment onto a line segment. Note that there are nonaffine segmental diffeomorphisms; for example (x, y) -* (1/x, y/x). The converse of the above theorem also holds: If f is a segmental diffeomorphism, then I p0E I = 0 for almost all O e [0, ST) implies Ijp(JE) I = 0 for almost all 0 E [0, ST). In fact, an application of Fubini's theorem shows that IpoE I = 0 for almost all 0 if and only if almost all lines through almost any point of A do not meet E. This latter condition is easily seen to be preserved under segmental diffeomorphisms. We can restate the above remarks by saying that a C2 diffeomorphism preserves the null-sets for the integralgeometric (Favard) measure if and only if it is segmental. For a Borel set E C R2 the integralgeometric measure is given by

On the local Langlands conejcture $\mathrm{GL}(n)$: the cyclic case

Abstract: Let F be a locally compact nonarchimedean field, and p its residue characteristic. Let F be a separable algebraic closure of F, WF the Weil group of F over F, E a character of FX of order n, and E the corresponding character of WF. Then there exists a canonical map from the set of equivalence classes a of irreducible degree n representations of WF such that E ? a = a to the set of equivalence classes 7r of admissible irreducible supercuspidal representations of GL(n, F) verifying ( odet) ? X = 7T. This map is shown to be bijective and to preserve e-factors for pairs.

Symmetrization and the Poincare metric

Abstract: In the 1930's and 1940's Polya and Szegd developed an elegant theory of symmetrization which is summarized in [PS]. In this work they derived the effects of symmetrization on domain constants such as capacity, inner radius, and principal frequency. The monotonicity in the behavior of the inner radius under symmetrization yields the principle of symmetrization for simply connected regions [Ha2; p. 84] which gives bounds on the derivative of an analytic function mapping the unit disk onto a region D in terms of the conformal mapping of U onto the symmetrized region D*, with D* simply connected. Hayman later used the ideas of Polya and Szegd in the study of analytic functions [Hal] and raised the problem [Ha3; p. 32] of determining if this principle extends to general universal covering maps. The difficulty in obtaining such an extension stems from the fact that the methods rely heavily on the simple connectivity of D*. In the present work we develop a technique which does not depend on the connectivity of the region, and thus enables us to answer Hayman's question affirmatively. The proof relies on a method of symmetrization introduced by Baernstein in [Bi] and further developed in [B2]. I would like to thank Professor Baernstein for several improvements and simplifications in the proofs of this paper.

Examples and counterexamples to a conjecture in the theory of differentiation of integrals

Abstract: In this paper we present some positive and negative results related to a conjecture in the theory of differentiation of integrals to the effect that the differentiation properties of a basis of intervals in R' consisting of all those intervals whose side length in the j-th direction is given by a function 4j, increasing in each variable, would only depend on the number of parameters of these Pj's. First, we will introduce some notation. B [x1, x2, . . ., xn] will denote the collection of all intervals in Rn with side lengths xl, x2, . . . xn, where, for i = 1, 2, ... ., n, xi is a positive parameter, or a positive function defined on some parameters, to be specified at each time. Such a collection B is a particular example of a translation invariant basis in RW. B is said to be a differentiation basis if, moreover, given e > 0, B contains elements with side lengths less than e. Associated to B, we will consider the maximal operator M = MB defined on each f E L10c(Rn) by

Universal formulae and universal differential equations

Abstract: Etude des solutions des equations differentielles algebriques en relation avec l'approximation uniforme des fonctions

Analytic Functions Sharing Level Curves and Tracts

Abstract: A level curve for a nonconstant analytic function is a curve on which the function has constant modulus. Analytic functions should be symmetric about their level curves; any behavior on one side should be the reflection of behavior on the other. This symmetry may be realized locally-one straightens out a bit of the level curve with a local change of coordinates and appeals to the Schwarz reflection principle. Here we prove that, in fact, the only obstructions to global symmetry are presented by compositional factors. Removing these factors allows us to straighten out complete level curves and to exploit the perfect symmetry of the functions remaining. We can then establish, in a unified way, numerous old and new results about shared level curves and tracts-including work going back to Valiron and Cartwright and more recent work of Fuchs, Heins, and others. Given a Riemann surface O' and a nontrivial arc r in 4, we denote by Yr the nonconstant meromorphic functions having constant modulus one on F. When 4" is the Riemann sphere P, the plane C, or the unit disc D and F lies in the extended real line RX, we use the notation 3VR. The key theorem of our work is this:

Admissible convergence of Poisson integrals in symmetric spaces

Abstract: Let f be an LP function, p > 1, on the distinguished boundary of a Furstenberg-Satake compactification of a symmetric space. We prove that the Poisson integral Pf of f converges admissibly at almost all components of each boundary of the compactification. This was known previously only for large p. In particular, Pf converges admissibly to f almost everywhere at the distinguished boundary. A similar result is obtained for the normalized X-Poisson integral 9jf. The method of proof uses maximal functions, and it also gives a new proof of almost everywhere restricted convergence of Pf and 9'Xf for f E L'.