# Showing papers in "Annals of Mathematics in 1992"

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TL;DR: In this article, the Riemann mapping theorem was generalized to higher dimensions and it was shown that a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways: it has zero scalar curvature and its boundary has constant mean curvature.

Abstract: One of the most celebrated theorems in mathematics is the Riemann mapping theorem. It says that an open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. In higher dimensions, very few regions are conformally diffeomorphic to the ball. However we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways, namely, it has zero scalar curvature and its boundary has constant mean curvature. In this paper we generalize the Riemann mapping theorem to higher dimensions in that sense.

401 citations

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267 citations

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257 citations

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195 citations

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190 citations

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TL;DR: The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia.

Abstract: *The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He would like to thank the Physics Department and others at the Universidad de Chile for their hospitality during his visit in April, 1990, when much of this research was completed. The second author was supported in part by FONDECYT (Chile), projects 0132-88 and 1238-90. Both authors also thank Fritz Gesztesy for general comments and encouragement.

188 citations

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TL;DR: In this article, the existence of noncollision singularities in the 5-body problem was shown to be true in the case of point-masses moving in a Euclidean space.

Abstract: In this paper we solve a long-standing problem in celestial mechanics proposed by Painleve and Poincare in the last century. The problem, which concerns the nature of the singularities in the n-body problem, asks whether there exists a noncollision singularity in the newtonian n-body problem? Here we give an affirmative answer to this problem by proving the existence of noncollision singularities in the 5-body problem. We consider n point-masses moving in a euclidean space W3. Let the mass of the ith particle be mi > 0, let its position be qj E R3 and let 4i E ]3 be its velocity. According to Newton's law,

171 citations

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TL;DR: In this paper, it was shown that the higher limits of certain functors n 1H 2: f p(G) Z(p-Wall vanish and that H 0(Mp(G); f i) has the expected value.

Abstract: To complete the classification of self-maps of BG (Theorem 4.2), it remains to prove Theorem 4.1, i.e., to show that the higher limits of certain functors n 1H 2: f p(G) Z(p-Wall vanish and that H0(Mp(G); f i) has the expected value. In this section, these limits H*(Mp(G); Hi) will be calculated in the "easy" cases, when G is a simply connected classical group (i.e., one of the groups SU(n), Sp(n), or Spin(n)), or when i = 2. The computation of H*(Mp(G); Hi) for the simply connected exceptional Lie groups will be put off until Section 6. We begin with some general properties of higher limits. Let 6 be any small discrete category. It is convenient, when dealing with higher limits of functors on -, to let 4-moldenote the category of contravariant functors from v to Ad. Then for any F e W-muok, we can identify

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TL;DR: In this paper, a simple proof of Bourgain's circular maximal theorem for the wave equation was given using easy wave front analysis along with techniques previously used in proofs of Carleson-Sj6lin theorem and in the proof of sharp regularity properties of Fourier integral operators.

Abstract: The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgain's circular maximal theorem [1]. We use easy wave front analysis along with techniques previously used in proofs of the Carleson-Sj6lin theorem (see [3],[5],[7]) and in the proof of sharp regularity properties of Fourier integral operators [13]. The circular means operators are defined by

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Abstract: Introduction 1. Excision and the homology of affine groups 2. Volodin's spaces 3. Excision for rings with the Triple Factorization Property 4. Lie analogue of Volodin's construction 5. The homology of nilpotent groups and nilpotent Lie algebras 6. A proof of the main theorems 7. The tensor product of H-unital algebras 8. The matrix ring of a small category 9. The acyclicity of triangular complexes 10. A proof of Karoubi's Conjecture

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TL;DR: In this paper, the authors introduce the notion of algebraic cocycles as the algebraic analogue of a map to an Eilenberg-MacLane space and develop a cohomology theory for complex algebraic varieties.

Abstract: We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational (p, p)-cohomology class. In this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle. Details will appear in [FL]. Our cohomology is a companion to the L-homology theory recently studied in [L, F, L-F1, L-F2, FM]. This homology is a bigraded theory based directly on the structure of the space of algebraic cycles. It admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class. Our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles. It carries a ring structure coming from the complex join (an elementary construction of projective geometry), and it admits a natural transformation Φ to integral cohomology. Chern classes are defined in the theory and transform under Φ to the usual ones. Our definition of cohomology is very far from a duality construction on L-homology. Nonetheless, there is a natural and geometrically defined Kronecker pairing between our “morphic cohomology” and L-homology. The foundation stone of our theory is the notion of an effective algebraic cocycle, which is of some independent interest. Roughly speaking, such a cocycle on a variety X , with values in a projective variety Y , is a morphism from X to the space of cycles on Y . When X is normal, this is equivalent (by “graphing”) to a cycle on X × Y with equidimensional fibres over X . Such cocycles abound in algebraic geometry and arise naturally in many circumstances. The simplest perhaps is that of a flat morphism f : X → Y whose corresponding cocycle associates to x ∈ X , the pullback cycle f({x}). Many more arise naturally from synthetic constructions in projective geometry. We show that every variety is rich in cocycles. Indeed if 1991 Mathematics Subject Classification. Primary 14F99, 14C05.

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TL;DR: In this article, the authors studied the stability properties of an analytic "standard" action of lattices in semisimple Lie groups on locally homogeneous spaces and showed that it can be shown by studying the behavior of the periodic orbits for the action.

Abstract: This is the simplest example of a large class of analytic "standard" actions of lattices in semisimple Lie groups on locally homogeneous spaces. A basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics (cf. [13], [50], [51]). A Cr-action So: IF > X -x X of a group F on a compact manifold X is said to be Anosov if there exists at least one element, Yh E F, such that 'p(Yh) is an Anosov diffeomorphism of X. We begin in this paper to study the Anosov differentiable actions of lattices, including many standard algebraic examples, and especially to study their stability properties. Our main theme is that either the Cr-rigidity or the Cr-deformation rigidity of an Anosov action (for 1 < r < cr. or even for the real analytic case) can be shown just by studying the behavior of the periodic orbits for the action. There are two notions of "structural stability" that appear in this paper, rigidity and deformation rigidity. A Cl-perturbation of a Cr-action So is simply another Cr-action Spl such that for a finite set of generators {81, . . , ad of F, the Cr-diffeomorphisms 'P(8i) and 'Pl(8i) are Cl-close for all i. An action So is said to be Cr-rigid (or topologically rigid if r = 0) if every sufficiently small C'perturbation of So is Cr-conjugate to So, for 0 < r < oo, or r = w in the case of real analytic actions. A Cl-deformation of an action So is a continuous path of Cr-actions (Pt defined for some 0 < t < e with Spo = SD* An action SD is said to be Cr-deform-

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TL;DR: Antimonotonicity of one-parameter families of C area contracting diffeomorphisms of the Euclidean plane has been studied in this paper, where it has been shown that in each neighborhood of a parameter value at which a homoclinic tangency occurs, there are either infinitely many values at which periodic orbits are created or infinitely many at which they are annihilated.

Abstract: One-parameter families fx of diffeomorphisms of the Euclidean plane are known to have a complicated bifurcation pattern as X varies near certain values, namely where homoclinic tangencies are created. We argue that the bifurcation pattern is much more irregular than previously reported. Our results contrast with the monotonicity result for the wellunderstood one-dimensional family gx(x) = Àx (I x ) , where it is known that periodic orbits are created and never annihilated as A increases. We show that this monotonicity in the creation of periodic orbits never occurs for any one-parameter family of C area contracting diffeomorphisms of the Euclidean plane, excluding certain technical degenerate cases where our analysis breaks down. It has been shown that in each neighborhood of a parameter value at which a homoclinic tangency occurs, there are either infinitely many parameter values at which periodic orbits are created or infinitely many at which periodic orbits are annihilated. We show that there are both infinitely many values at which periodic orbits are created and infinitely many at which periodic orbits are annihilated. We call this phenomenon antimonotonicity.

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TL;DR: In this article, the absolute Galois group of a countable Hilbertian P(seudo)-A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A).

Abstract: We show that the absolute Galois group of a countable Hilbertian P(seudo)- A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., � ∞=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RG-Hilbertian property

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TL;DR: In this paper, it was shown that real-analytic hypoellipticity is false if P(z) = [Rz]m, m an even integer > 4.

Abstract: Consider the hypersurface {(Z1, Z2): Q(Z2) = P(zi)} cC C2, where P : R is a subharmonic, nonharmonic polynomial. Such a surface is pseudoconvex (more precisely, is the boundary of a pseudoconvex domain) and of finite type. It is the boundary of a "model domain" {f(Z2) > P(zi)}, so called because such domains provide good approximations to general pseudoconvex domains of finite type in C2 (see [37],[11]). A nonvanishing, antiholomorphic, tangent vector field is a/Oaz-1 -2i(aP/Oaz-i)O/a29z-. As coordinates for the surface we use C x JR 3 (z, t) -(z, t + iP(z)); the vector field pulls back to L = O/OZ i(OP/9Z-)a3/t. Equip C x JR with Lebesgue measure and consider the orthogonal projection from L2 onto its intersection with the kernel of L. The intersection of this kernel with the Schwartz space may be shown to have infinite dimension, and it is well known that L fails to be C? hypoelliptic. However C? hypoellipticity does hold in a modified form (see [30]): If Lu E C? in an open set, and if in addition u = Lv for some v E L2, where L is the formal adjoint of L, then u E C? there. Our main purpose is to point out that the analogous assertion for real-analytic hypoellipticity is false if P(z) = [Rz]m, m an even integer > 4. Specifically, for P a general, homogeneous, subharmonic, nonharmonic polynomial, let S((z, t); (w, s)) be the Szeg6 kernel; that is, the distribution kernel associated to the operator defined by the orthogonal projection of L2(C x JR), with respect to Lebesgue measure, onto the kernel of L. We shall see in Section 5 that this projection maps Co to C?. Further, S is smooth off the diagonal by [37]. Define the distribution

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TL;DR: In this paper, the G-orbit structure of X is well understood and a Cartan subalgebra of g is defined for the complex conjugation of gc over g.

Abstract: The G-orbit structure of X is well understood (see [13]). There are only finitely many orbits, in particular there are open orbits. If x E X, let Px be the corresponding parabolic subalgebra of g; that is, if x = gP, then Px = Ad(g)p. Let ( " denote the complex conjugation of gc over g. Then Px n px contains a Cartan subalgebra of g of the form [ = io OR C, where ho is a Cartan subalgebra of go. Let A = A(g, [) denote the root system. Fix

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