Showing papers in "Inventiones Mathematicae in 2004"

Time decay for solutions of Schrödinger equations with rough and time-dependent potentials

Abstract: In this paper we establish dispersive estimates for solutions to the linear Schrodinger equation in three dimensions 0.1 $$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$ where V(t,x) is a time-dependent potential that satisfies the conditions $$\sup_{t}\|V(t,\cdot)\|_{L^{\frac{3}{2}}(\mathbb{R}^3)} + \sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \int_{-\infty}^\infty\frac{|V(\hat{\tau},x)|}{|x-y|}\,d\tau\,dy < c_0.$$ Here c 0 is some small constant and $V(\hat{\tau},x$) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·)∈L ∞ t (L 2 x (ℝ3))∩L 2 t (L 6 x (ℝ3)) for any f∈L 2(ℝ3) satisfying the dispersive inequality 0.2 $$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$ For the case of time independent potentials V(x), (0.2) remains true if $$\int_{\mathbb{R}^6} \frac{|V(x)|\;|V(y)|}{|x-y|^2} \, dxdy <(4\pi)^2\text{\ \ \ and\ \ \ }\|V\|_{\mathcal{K}}:=\sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|V(y)|}{|x-y|}\,dy<4\pi.$$ We also establish the dispersive estimate with an e-loss for large energies provided $\|V\|_{\mathcal{K}}+\|V\|_2<\infty$ . Finally, we prove Strichartz estimates for the Schrodinger equations with potentials that decay like |x|-2-e in dimensions n≥3, thus solving an open problem posed by Journe, Soffer, and Sogge.

On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation

Abstract: We consider finite time blow-up solutions to the critical nonlinear Schrodinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

Abstract: The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R 3. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L 1 topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.

Récurrence et généricité

Abstract: Nous montrons un lemme de connexion C 1 pour les pseudo-orbites des diffeomorphismes des varietes compactes. Nous explorons alors les consequences pour les diffeomorphismes C 1-generiques. Par exemple, les diffeomorphismes conservatifs C 1-generiques (d’une variete connexe) sont transitifs.

On the existence of collisionless equivariant minimizers for the classical n -body problem

Abstract: We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree -α, with α>0) which ensures that the restriction of the Lagrangian action $\mathcal{A}$ to the space Λ G of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of Λ G are free of collisions we develop an averaging technique based on Marchal’s idea of replacing some of the point masses with suitable shapes (see [10]). As an application, several new orbits can be found with some appropriate choice of G. Furthermore, the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits.

Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation

Abstract: Let M be a C∞ second countable manifold without boundary. We denote by TM the tangent bundle and by π : TM → M the canonical projection. A point in TM will be denoted by (x, v)with x ∈ M and v ∈ Tx M = π−1(x). In the same way a point of the cotangent space T ∗M will be denoted by (x, p) with x ∈ M and p ∈ T ∗ x M, a linear form on the vector space Tx M. We will suppose that g is a complete Riemannian metric on M. For v ∈ Tx M, the norm ‖v‖ is g(v, v)1/2. We will denote by ‖ · ‖ the dual norm on T ∗ x M. We will also use the notations ‖v‖x , for v ∈ Tx M, and ‖p‖x , for v ∈ T ∗ x M. We will assume in the whole paper that H : T ∗M → R is a function of class at least C2, which satisfies the following three conditions (1) (Uniform superlinearity) for every K ≥ 0, there exists C∗(K ) ∈ R such that

On differentiability of SRB states for partially hyperbolic systems

Abstract: Consider a one parameter family of diffeomorphisms f e such that f 0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties Let νe be any u-Gibbs state for f e We prove (Theorem 1) that for any C ∞ function A the map e→νe(A) is differentiable at e=0 This implies (Corollary 22) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to e We apply this result (Corollary 33) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties

Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem

Abstract: This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts.

A stable cohomotopy refinement of Seiberg-Witten invariants: II

Abstract: The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. Part I discusses the definition of this stable homotopy invariant and its relation to the integer valued Seiberg-Witten invariants.

Characterization of a free arrangement and conjecture of Edelman and Reiner

Abstract: We consider a hyperplane arrangement in a vector space of dimension four or higher. In this case, the freeness of the arrangement is characterized by properties around a fixed hyperplane. As an application, we prove the freeness of cones over certain truncated affine Weyl arrangements which was conjectured by Edelman and Reiner.

The Cauchy problem for quasi-linear Schrödinger equations

Abstract: with x ∈ Rn, t ∈ R with ∇ = (∂x1, .., ∂xn) and summation convention. One may think of the equation in (1.1) as a non-linear Schrodinger equation where the operator modeling the dispersion relation is non-isotropic and depends also on the unknown function, its conjugate and their gradients in the space variables. Under appropriate assumptions on the function F the method presented here extends to fully non-linear Schrodinger equations of the form

Some prime factorization results for type II 1 factors

Abstract: We prove several unique prime factorization results for tensor products of type II1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if \(\mathcal{R}\overline{\otimes}\mathcal{L}\mathbb{F}_{r_1}\overline{\otimes}\cdot\cdot\cdot\overline{\otimes}\mathcal{L}\mathbb{F}_{r_m}\) is isomorphic to a subfactor in \(\mathcal{R}\overline{\otimes}\mathcal{L}\mathbb{F}_{s_1}\overline{\otimes}\cdot\cdot\cdot\overline{\otimes}\mathcal{L}\mathbb{F}_{s_n}\), for some 2≤ri,sj≤∞, then m≤n.

The entropy theory of symbolic extensions

Abstract: Fix a topological system (X,T), with its space K(X,T) of T-invariant Borel probabilities. If (Y,S) is a symbolic system (subshift) and ϕ:(Y,S)→(X,T) is a topological extension (factor map), then the function h ϕ ext on K(X,T) which assigns to each μ the maximal entropy of a measure ν on Y mapping to μ is called the extension entropy function of ϕ. The infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by h sex. In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X,T). The entropy structure ℋ is a sequence of entropy functions h k defined with respect to a refining sequence of partitions of X (or of X×Z, for some auxiliary system (Z,R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities. We develop the functional analysis and computational techniques to produce many dynamical examples; for instance, we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of (X,T) must always be attained, and we show that the maximum value of h sex need not be achieved at an ergodic measure. We exhibit several characterizations of the asymptotically h-expansive systems of Misiurewicz, which emerge as a fundamental natural class in the context of the entropy structure. The results of this paper are required for the Downarowicz-Newhouse results [DN] on smooth dynamical systems.

Einstein metrics and complex singularities

Abstract: This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkahler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kahler metric (which is hyperkahler if and only if K X is trivial), and that if K X is strictly nef, then X also admits a complete (non-Kahler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Harmonic analysis on the infinite symmetric group

Abstract: The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space l2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification \(\mathfrak{S}\supset{S(\infty)}\), which we call the space of virtual permutations. Although \(\mathfrak{S}\) is no longer a group, it still admits a natural two–sided action of S(∞). Thus, \(\mathfrak{S}\) is a G–space, where G stands for the product of two copies of S(∞). On \(\mathfrak{S}\), there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μt: t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz: z∈ℂ} of unitary representations of G, called generalized regular representations (each representation Tz with z≠=0 can be realized in the Hilbert space \(L^2(\mathfrak{S}, \mu_t)\), where t=|z|2). As |z|→∞, the generalized regular representations Tz approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space l2(S(∞)). In contrast with the latter representation, the generalized regular representations Tz are highly reducible and have a rich structure. We prove that any Tz admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z1, z2, the spectral types of the representations \(T_{z_1}\) and \(T_{z_2}\) are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.

Néron models, Lie algebras, and reduction of curves of genus one

Abstract: Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.

On the homotopy types of compact Kähler and complex projective manifolds

Abstract: The celebrated Kodaira theorem [6] says that a compact complex manifold is projective if and only if it admits a Kahler form whose cohomology class is integral. This suggests that Kahler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on a Kahler class. This point of view, together with the many restrictive conditions on the topology of Kahler manifolds provided by Hodge theory (the strongest one being the formality theorem [4]), would indicate that compact Kahler manifolds and complex projective ones cannot be distinguished by topological invariants. This is supported by the results known for Kahler surfaces, for which a much stronger statement is known, as a consequence of Kodaira’s classification : recall first that two compact complex manifolds X and X ′ are said to be deformation equivalent if there exist a family π : X → B, where B is a connected analytic space and π is smooth and proper, and two points b, b′ ∈ B such that Xb ∼= X, Xb′ ∼= X ′.

Toric reduction and a conjecture of Batyrev and Materov

Abstract: This paper grew out of our efforts to understand the Toric Residue Mirror Conjecture formulated by Batyrev and Materov in [2]. This conjecture has its origin in Physics and is based on a work by Morrison and Plesser [14]. According to the philosophy of mirror symmetry, to every manifold in a certain class one can associate a dual manifold, the so-called mirror, so that the intersection numbers of the moduli spaces of holomorphic curves in one of these manifolds are related to integrals of certain special differential forms on the other. While at the moment this mirror manifold is only partially understood, there is an explicit construction due to Victor Batyrev [1], in which the two manifolds are toric varieties whose defining data are related by a natural duality notion for polytopes. Let us recall the setting of the conjecture of Batyrev and Materov. Let t be a d-dimensional real vector space endowed with an integral structure: a lattice Γt ⊂ t of full rank. We denote by Γ∗t the embedded dual lattice {v ∈ t∗; 〈v, γ 〉 ∈ Z for all γ ∈ Γt}. Consider two convex polytopes, Π ⊂ t and Π ⊂ t∗, containing the origin in their respective interiors, and related by the duality

Gerbes of chiral differential operators. II. Vertex algebroids.

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 0 Recollections and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 1 Vertex algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 2 From vertex algebras to vertex algebroids . . . . . . . . . . . . . . . . . . . . . 622 3 Category of vertex algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 4 Cofibered structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 5 Computation of a cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 6 Characteristic classes of vector bundles . . . . . . . . . . . . . . . . . . . . . . . 643 7 Torseurs and gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 8 Gerbes of vertex algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 9 Chiral differential operators on curves and sheaves of Virasoro algebras . . . . . 657 10 Enveloping algebra of a vertex algebroid . . . . . . . . . . . . . . . . . . . . . . 664 11 Calabi–Yau structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 12 Virasoro and Calabi–Yau structures . . . . . . . . . . . . . . . . . . . . . . . . . 671 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

Lattice point problems and values of quadratic forms

Abstract: For d-dimensional ellipsoids E with d≥5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order $\mathcal{O}(r^{d-2})$ for general ellipsoids and up to an error of order o(r d-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d≥9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m∈ℤ d of positive definite irrational quadratic forms Q of dimension d≥5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.

Chow rings of toric varieties defined by atomic lattices

Abstract: We study a graded algebra \(D=D(\mathcal{L},\mathcal{G})\) over ℤ defined by a finite lattice ℒ and a subset \(\mathcal{G}\) in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and \(\mathcal{G}\). We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Grobner basis of the relation ideal of D and a monomial basis of D.

Poincaré series of a rational surface singularity

Abstract: Recently there was found a new method to compute the (generalized) Poincare series of some multi-index filtrations on rings of functions. First the authors had elaborated it for computing the Poincare series of the filtration on the ring OC2,0 of germs of functions of two variables defined by irreducible components of a plane curve singularity. The corresponding formula (announced in [2]) was proved in [3] by another method. The new method uses the notion of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions. This notion is similar to (and inspired by) the notion of the motivic integration. Here we apply this method for computing the Poincare series of a (natural) multi-index filtration on the ring of germs of functions on a rational surface singularity. We explicitly calculate the coefficients of this series. Let (S, 0) be a germ of an (isolated) rational surface singularity and let π : (X,D) → (S, 0) be its minimal resolution. Here X is a smooth surface, π is a proper analytic map which is an isomorphism outside of D = π−1(0), and the exceptional divisor D is the union of irreducible components Ei (i = 1, . . . , r) transversal to each other, each component Ei is isomorphic to the projective line CP1.

Propagation of singularities for the wave equation on conic manifolds

Abstract: For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.

Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

Abstract: If \(\mathfrak{g}\) is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of \(\mathfrak{g}\), then the absolute value of the kth coefficient of the \(\dim\mathfrak{g}\) power of the Euler product may be given by the dimension of a subspace of \(\wedge^k\mathfrak{g}\) defined by all abelian subalgebras of \(\mathfrak{g}\) of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson’s 2rank theorem on the number of abelian ideals in a Borel subalgebra of \(\mathfrak{g}\), an element of type ρ and my heat kernel formulation of Macdonald’s η-function theorem, a set Dalcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when \(\mathfrak{g}= \text{Lie}\,\mathit{Sl}(m,\mathbb{C})\)), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac–Moody Lie algebra.

Absolutely indecomposable representations and Kac-Moody Lie algebras

Abstract: A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.

Hard Lefschetz theorem for nonrational polytopes

Abstract: The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.

p-adic multiple zeta values I.

Abstract: Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values We introduce (one variable) p-adic multiple polylogarithms by Coleman’s p-adic iterated integration theory We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms We consider the (formal) p-adic KZ equation and introduce the p-adic Drinfel’d associator by using certain two fundamental solutions of the p-adic KZ equation We show that our p-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear as coefficients of the p-adic Drinfel’d associator We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation

The Shannon-McMillan theorem for ergodic quantum lattice systems

Abstract: We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on ℤν-lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.