Showing papers in "Annals of Mathematics in 2000"

A remarkable periodic solution of the three-body problem in the case of equal masses

Abstract: Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a flxed eight-shaped curve. Setting aside collinear motions, the only other known motion along a flxed curve in the inertial plane is the \Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every \Euler conflguration" in which one of the bodies sits at the midpoint of the segment deflned by the other two (Figure 1). Numerical computations

Gauge theory and calibrated geometry, I

Abstract: The geometry of submanifolds is intimately related to the theory of functions and vector bundles It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles There are two more very recent examples of such a relation The mirror symmetry in string theory has revealed a deeper phenomenon involving special Lagrangian cycles (cf [SYZ]) On the other hand, C Taubes has shown that the Seiberg-Witten invariant coincides with the Gromov-Witten invariant on any symplectic 4-manifolds In this paper, we will show another natural interaction between Yang-Mills connections, which are critical points of a Yang-Mills action associated to a vector bundle, and minimal submanifolds, which have been studied extensively for years in classical differential geometry and the calculus of variations

Invariant measures for Burgers equation with stochastic forcing

Abstract: In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.

The Poisson formula for groups with hyperbolic properties

Abstract: The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)

Abstract: Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half-plane H of levels N,M respectively, with (unitarily normalized) L-functions L(s, f) = [equation] and L(s, g) = [equation]. When p does not divide N (resp. M), the inverse roots αp, βp (resp. α′p, β′p ) are nonzero with sum ap (resp. bp). For every p prime to NM, set Lp(s, f × g) = [(1 − αpα′pp−s)(1 − αpβ′pp−s)(1 − βpα′pp−s)(1 − βpβ′pp−s)]^−1. Let L∗(s, f × g) denote the (incomplete Euler) product of Lp(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series [sum over n≥1] a[sub]n b[sub] n n^−s, whose miraculous properties were first studied by Rankin and Selberg.

Curvature and symmetry of Milnor spheres

Abstract: Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

Homoclinic tangencies and hyperbolicity for surface diffeomorphisms

Abstract: We prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C 1 dense. This represents a step towards the global understanding of dynamics of surface diffeomorphisms.

On nonperturbative localization with quasi-periodic potential

Abstract: The two main results of the article are concerned with Anderson Localization for one-dimensional lattice Schroedinger operators with quasi-periodic potentials with d frequencies. First, in the case d = 1 or 2, it is proved that the spectrum is pure-point with exponentially decaying eigenfunctions for all potentials (defined in terms of a trigonometric polynomial on the d-dimensional torus) for which the Lyapounov exponents are strictly positive for all frequencies and all energies. Second, for every non-constant real-analytic potential and with a Diophantine set of d frequencies, a lower bound is given for the Lyapounov exponents for the same potential rescaled by a sufficiently large constant.

Real algebraic curves, the moment map and amoebas

Abstract: In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexiflcations.

Distribution of the partition function modulo $m$

Abstract: Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).

The monodromy groups of Schwarzian equations on closed Riemann surfaces

Abstract: Let θ : π1(R) → PSL(2, ℂ) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one We will establish the following theorem THEOREM Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary A branch point is required if and only if the representation θ does not lift to SL(2, ℂ)

Nonresonance and global existence of prestressed nonlinear elastic waves

Abstract: This article considers the existence of global classical solutions to the Cauchy problem in nonlinear elastodynamics. The unbounded elastic medium is assumed to be homogeneous, isotropic, and hyperelastic. As in the theory of 3D nonlinear wave equations in three space dimensions, global existence hinges on two basic assumptions. First, the initial deformation must be a small displacement from equilibrium, in this case a prestressed homogeneous dilation of the reference conflguration, and equally important, the nonlinear terms must obey a type of nonresonance or null condition. The omission of either of these assumptions can lead to the breakdown of solutions in flnite time. In particular, nonresonance complements the genuine nonlinearity condition of F. John, under which arbitrarily small spherically symmetric displacements develop singularities (although one expects this to carry over to the nonsymmetric case, as well), [4]. John also showed that small solutions exist almost globally [5] (see also [10]). Formation of singularities for large displacements was illustrated by Tahvildar-Zadeh [16]. The nonresonance condition introduced here represents a substantial improvement over our previous work on this topic [13]. To explain the difierence roughly, our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones. Here, only the cancellation of nonlinear wave interactions among individual wave families is required. The di‐culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal Helmholtz projection, which is ill-suited to nonlinear analysis. However, our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous, and this permits the usage of approximate local decompositions.

Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group

Abstract: The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.

A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

Abstract: We continue the study of the A-amplitude associated to a half-line Schrodinger operator, - d^2/dx^2 + q in L^2((0,b)), b ≤ ∞ A is related to the Weyl-Titchmarsh m-function via m(-k^2) = -k- ʃ^a_0 A(α)e^(-2αk) dα+O(e^(-(2α-Є)k)) for all Є > 0. We discuss five issues here. First, we extend the theory to general q in L^1((0,α)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure p: A(α) = -2 ^ʃ∞_(-∞)λ^(-1/2) sin (2α√λ) dp(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b < ∞. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.

The moduli space of Riemann surfaces is Kähler hyperbolic

Abstract: Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant curvature -1 in the Teichmuller (=Kobayashi) metric. When $r=\dim_{\cx} \cM_{g,n}$ is greater than one, however, $\cM_{g,n}$ carries no complete metric of bounded negative curvature. Instead, Dehn twists give chains of subgroups $\zed^r \subset \pi_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$. In this paper we introduce a new Kahler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank.

Analytic properties of zeta functions and subgroup growth

Abstract: It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in [36]. The story so far has been well-documented in Lubotzky’s subsequent survey paper in [30]. In [24] the second author of this article, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let an(G) be the number of subgroups of index n in the finitely generated group G and sN (G) = a1(G) + · · ·+ aN (G) be the number of subgroups of index N or less. The zeta function is defined as the Dirichlet series with coefficients an(G) and has a natural interpretation as a noncommutative generalization of the Dedekind zeta function of a number field:

The symplectic Thom conjecture

Abstract: In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds.

Regularity of a free boundary with application to the Pompeiu problem

Abstract: In the unit ball B(0; 1), let u and › (a domain in N ) solve the following overdetermined problem: ¢u = ´› in B(0; 1); 02 @› ;u =jruj =0 inB(0; 1)n ›; where ´› denotes the characteristic function, and the equation is satisfled in the sense of distributions. If the complement of › does not develop cusp singularities at the origin then we prove @› is analytic in some small neighborhood of the origin. The result can be modifled to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

The quantization conjecture revisited

Abstract: A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X//G equal the invariant part of the cohomologies over X. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [li]) to RiemannRoch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodgeto-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Ti] for the moduli stack of C-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.

The bilinear maximal functions map into L^p for 2/3 < p <= 1

Abstract: The bilinear maximal operator defined below maps LP x Lq into Lr provided 1 0 2t t In particular Mfg is integrableif f and g are square integrable, answering a conjecture posed by Alberto Calder6n.

Proof of the gradient conjecture of R. Thom

Abstract: Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t )a tx0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x0 onto the unit sphere has flnite length.

An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$

Abstract: A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . In this paper we show that the Minkowski dimension must in fact be greater than 5= 2+ " for some absolute constant "> 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call \stickiness," \planiness," and \graininess."

Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions

Abstract: A method of \algebraic estimates" is developed, and used to study the stability properties of integrals of the form R B jf(z)j i‐ dV , under small deformations of the function f. The estimates are described in terms of a stratiflcation of the space of functionsfR(z )= jP (z)j " =jQ(z)j ‐ g by algebraic varieties, on each of which the size of the integral of R(z) is given by an explicit algebraic expression. The method gives an independent proof of a result on stability of Tian in 2 dimensions, as well as a partial extension of this result to 3 dimensions. In arbitrary dimensions, combined with a key lemma of Siu, it establishes the continuity of the mapping c! R B jf(z;c)j i‐ dV1¢¢¢dVn when f(z;c) is a holomorphic function of (z;c). In particular the leading pole is semicontinuous in f, strengthening also an earlier result of Lichtin.

Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension > 3

Abstract: In this paper we prove the following theorem. Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat real hypersurface M in P_n. The condition that M is Levi-flat means that when M is locally defined by the vanishing of a C^m real-valued function f, at every point of M the restriction of d d-bar f to the complex tangent space of M is identically zero. The case of the nonexistence of C^\infty Levi-flat real hypersurface in P_2 is motivated by problems in dynamical systems in P_2.

Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité

Abstract: Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g. generalized hypergeometric series with rational parameters, enjoy arithmetic counterparts of the Archimedean Gevrey condition; 2) the differential operators which occur in classical treatises on special functions have a rather simple structure: they are either Fuchsian, or have only two singularities, 0 and infinity, one of them regular, the other irregular with a single slope... The main idea of the paper is that the arithmetic property 1) accounts for the global analytic property 2): the existence of an injective arithmetic Gevrey solution at one point determines to a large extent the global behaviour of a differential operator with polynomial coefficients. Proofs use both p-adic and complex analysis, and a detailed arithmetic study of the Laplace transform.

Entropy and mixing for amenable group actions.

Abstract: For i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space, written asfT∞g∞2i acting on (X;F;„). We sayfT∞g∞2i has completely positive entropy (or simply cpe for short) if for any flnite and nontrivial partition P of X the entropy h(T;P) is not zero. Our goal is to demonstrate what is well known for actions of and even d , that actions of completely positive entropy have very strong mixing properties. Let Si be a list of flnite subsets of i. We say the Si spread if any particular ∞6 id belongs to at most flnitely many of the sets SiS i1 i . Theorem 0.1. For fT∞g∞2i an action of i of completely positive entropy and P any flnite partition, for any sequence of flnite sets Siµ i which spread we have

Getting rid of the negative Schwarzian derivative condition

Abstract: In this paper we will show that the assumption on the negative Schwarzian derivative is redundant in the case of C 3 unimodal maps with a non∞at critical point. The following theorem will be proved: For any C 3 unimodal map of an interval with a non∞at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative Schwarzian derivative. Another theorem proved in the paper provides useful cross-ratio estimates. Thus, all theorems proved only for unimodal maps with negative Schwarzian derivative can be easily generalized.