Showing papers in "Mathematical Research Letters in 2002"

Almost Conservation Laws and Global Rough Solutions to a Nonlinear Schrödinger Equation

Abstract: We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrodinger equation in Hs(R) when n = 2, 3 and s > 4 7 , 5 6 , respectively.

$Q$-Curvature and Poincaré Metrics

Abstract: This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This gives an easy proof of the result of Graham-Zworski that the log coefficient in the volume expansion of a Poincare metric is a multiple of the integral of the Q-curvature, and leads to a definition of a non-local version of the Q-curvature in odd dimensions.

An instability property of the nonlinear Schrödinger equation on $S^{d}$

Abstract: We consider the NLS on spheres. We describe the nonlinear evolutions of the highest weight spherical harmonics. As a consequence, in contrast with the flat torus, we obtain that the flow map fails to be uniformly continuous for Sobolev regularity above the threshold suggested by a simple scaling argument.

L^p bounds for the function of Marcinkiewicz

Abstract: The above operator was introduced by E.M. Stein in [7] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions. By using the L boundedness of the 1-dimensional Marcinkiewicz function, Stein showed that μΩ is bounded on L(R) for 1 < p < ∞ whenever Ω is odd. For a general kernel function Ω, the L boundedness of μΩ has been established under various conditions on Ω. For example, Stein proved that μΩ is bounded on L(R) for 1 < p ≤ 2 if Ω ∈ Lip(Sn−1). Benedek, Calderon and Panzone proved in [2] that the L boundedness of μΩ holds for 1 < p < ∞ under the condition that Ω ∈ C1(Sn−1). In 1972 T. Walsh showed that the L boundedness of μΩ can still hold even if Ω is quite rough. Theorem 1 (Walsh [11]). Suppose that p ∈ (1,∞), r = min{p, p′}, and Ω ∈ L(log L)(log log L)2(1−2/r )(Sn−1). Then μΩ is bounded on L(R). When p = 2, the condition in Theorem 1 is simply Ω ∈ L(log L)1/2(Sn−1), which was shown by Walsh to be optimal in the sense that the exponent 1/2 in L(log L) cannot be replaced by any smaller numbers. On the other hand, Walsh did not consider his condition to be in any sense optimal when p = 2. Indeed, by comparing with the result of Calderon and Zygmund on singular integrals, one is naturally led to the question whether the condition Ω ∈ L(log L)1/2(Sn−1) is also sufficient for the L boundedness of μΩ even when p = 2. This problem, which was formally proposed by Y. Ding in [4], is resolved by our next theorem.

The pressure function for products of non-negative matrices

Abstract: Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, ..., M_m$. The pressure function reduces to $P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

The Grothendieck ring of varieties is not a domain

Abstract: Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.

K3 surfaces via almost-primes

Abstract: Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard lattices are not isomorphic but their transcendental lattices are Hodge isometric, or equivalently, their derived categories are mutually equivalent. After reviewing some finiteness result, we also give an explicit formula for the cardinality of the isomorphism classes of projective K3 surfaces having derived categories equivalent to the one of X with Picard number 1 in terms of the degree of X.

Minimal Hypersurfaces with Finite Index

Abstract: In an article of Cao-Shen-Zhu [C-S-Z], they proved that a complete, immersed, stable minimal hypersurface M of R with n ≥ 3 must have only one end. When n = 2, it was proved independently by do Carmo-Peng [dC-P] and FischerColbrie-Schoen [FC-S] that a complete, immersed, oriented stable minimal surface in R must be a plane. Later Gulliver [G] and Fischer-Colbrie [FC] proved that if a complete, immersed, minimal surface in R has finite index, then it must be conformally equivalent to a compact Riemann surface with finitely many punctures. Fischer-Colbrie actually proved this for minimal surfaces in a complete manifold with non-negative scalar curvature. In any event, a corollary is that if a complete, immersed, oriented minimal surface in R has finite index then it must have finitely many ends. The purpose of this paper is to generalize this result for finitely many ends to higher dimensional minimal hypersurfaces in Euclidean space (see Theorem 5). In fact, we will also show that the first L-Betti number of such a manifold must be finite. The strategy of Cao-Shen-Zhu was to utilize a result a Schoen-Yau [S-Y] asserting that a complete, stable minimal hypersurface of R cannot admit a non-constant harmonic function with finite Dirichlet integral. Assuming that M has more than one end, Cao-Shen-Zhu constructed a non-constant harmonic function with finite Dirichlet integral. This approach very much fits into the scheme studied by the first author and Tam in [L-T]. In fact, the authors showed that the number of non-parabolic ends of any complete Riemannian manifold is bounded above by the dimension of the space of bounded harmonic functions with finite Dirichlet integral. The proof of Cao-Shen-Zhu can be modified to show that each end of a complete, immersed, minimal submanifold must be non-parabolic. Due to this connection

Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

Abstract: Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of $\psi$ will grow as $\lambda^{1/2}$ as $\lambda \to \infty$. We prove an upper bound of the form $\|\psi \|_2^2 \leq C\lambda$ for any Riemannian manifold, and a lower bound $c \lambda \leq \|\psi \|_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $\lambda$. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate.

Balance point and Stability of Vector Bundles Over a Projective Manifold

Abstract: In this paper, we show that stability of a vector bundle over a projective manifold is equivalent to the embedding of the base manifold induced by the global sections into the Grassmanian can be moved to a balanced place in the sense of Donaldson, thus verifying a conjecture of Donaldson.

Willmore Submanifolds in a sphere

Abstract: Let x : M → S n+p be an n-dimensional submanifold in an (n + p)- dimensional unit sphere S n+p , x : M → S n+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: � M (S − nH 2 ) n 2 dv, where S = � α,i,j (h α ) 2 is the square of the length of the second fundamental form, H is the mean curvature of M. In (13), author proved an integral inequality of Simons' type for n-dimensional compact Willmore hypersurfaces in S n+1 and gave a characterization of Willmore tori. In this paper, we generalize this result to n-dimensional compact Willmore submanifolds in S n+p . In fact, we obtain an integral inequality of Simons' type for compact Willmore submanifolds in S n+p and give a characterization of Willmore tori and Veronese surface by use of our integral inequality.

Birkhoff normal forms in semi-classical inverse problems

Abstract: Iantchenko, A.; Sjostrand, J.; Zworski, M., (2002) 'Birkhoff normal forms in semi-classical inverse problems', Mathematical Research Letters 9(3) pp.337-362 RAE2008

Eigenfunction and Bochner Riesz estimates on manifolds with boundary

Abstract: The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the $L^2\to L^\infty$ mapping properties of the operators $\chi_\lambda$ which project onto unit bands of the spectrum of the square root of the Laplacian. These generalize the recent estimates of D. Grieser for individual eigenfunctions. We use an idea of Grieser of estimating away from the boundary using earlier estimates of Seeley, Pham The Lai, and H\"ormander, and then proving estimates near the boundary using a maximum principle. Our proof of the Bochner Riesz estimates is related to an argument for the Euclidean case of C. Fefferman, and one for the boundaryless Riemannian case that is due to the author. Here, unlike in the boundaryless Riemannian case we cannot use parametrices for the wave equation, and instead lie on a more economical argument that only uses the eigenfunction estimates and the finite propagation speed of the wave equation.

Convergence versus integrability in Poincare-Dulac normal form

Abstract: We show that, to find a Poincare-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that any local analytic vector field which is integrable in the non-Hamiltonian sense admits a local convergent Poincare-Dulac normalization. These results generalize the main results of our previous paper from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.

On polynomial eigenfunctions for a class of differential operators

Abstract: The main topic of this doctoral thesis is asymptotic properties of zeros in polynomial families arising as eigenfunctions to exactly-solvable differential operators. The study was initially inspired by a number of striking results from computer experiments performed by G. Masson and B. Shapiro for a more restrictive class of operators. Our research is also motivated by a classical question going back to S. Bochner on a general classification of differential operators possessing an infinite sequence of orthogonal eigenpolynomials. In general however, the sequence of eigenpolynomials of an exactly-solvable operator is not an orthogonal system and it can therefore not be studied by means of the extensive theory known for such systems. Our study can thus be considered as the first steps to a natural generalization of the asymptotic behaviour of the roots of classical orthogonal polynomials. Exactly-solvable operators split into two major classes: non-degenerate and degenerate. We prove that in the former case, as the degree tends to infinity, the zeros of the eigenpolynomial are distributed according to a certain probability measure which is compactly supported on a tree and which depends only on the leading term of the operator. Computer experiments indicate the existence of a limiting root measure in the degenerate case too, but that it is compactly supported (conjecturally on a tree) only after an appropriate scaling which is conjectured (and partially proved) in this thesis. One of the main technical tools in this thesis is the Cauchy transform of a probability measure, which in the considered situation satisfies an algebraic equation. Due to the connection between the asymptotic root measure and its Cauchy transform it is therefore possible to obtain detailed information on the limiting zero distribution.

On Vafa’s theorem for tensor categories

Abstract: In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper math/0109196 of S.Gelaki and the author.

Sharp counterexamples for Strichartz estimates for low regularity metrics

Abstract: On the other hand, in [3] there were constructed for each α < 1 examples of A with coefficients of regularity C for which the same estimates fail to hold. The first author then showed in [1] that, in space dimensions 2 and 3, the estimates do hold if the coefficients of A are C. The second author subsequently showed in [4] that the estimates hold for C metrics in all space dimensions, and that for operators with C coefficients, such estimates hold provided that γ is replaced by γ + σ/p, where σ = 1−α 3+α . Indeed, [5] showed that such estimates hold under the condition that 1 + α derivatives of the coefficients belong to LtL ∞ x , which is important for applications to quasilinear wave equations. The counterexamples of [3] do not coincide with the estimates established by [4], however. In this paper we remedy this gap, by producing examples of time independent C metrics which show that the results established in [4] are indeed best possible. (Strictly speaking, we produce a family of examples, one at each frequency, which show that if the Strichartz estimates hold with a constant C depending only on the C norm of the coefficients, and the assumption that the coefficients are pointwise close to the euclidean metric, then the loss of σ/p derivatives is sharp.) We remark that this construction also produces examples of C metrics which show that the closely related spectral projection estimates for C metrics established by the first author [2] are best possible. For the spectral projection estimates, however, the counterexamples of [3] were already sharp.

The rational cohomology ring of the moduli space of abelian 3-folds

Abstract: The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the moduli space of smooth projective curves of genus 3, and (2) the stratified Morse theory of Goresky and MacPherson, which we use to compute the homology of the jacobian locus (in the rank 3 Siegel upper half plane), or equivalently of the extended Torelli group in genus 3. In the revised version, we also compute the rational cohomology of the Satake compactification of A_3.

Generic absoluteness and the continuum

Abstract: Let Hω2 denote the collection of all sets whose transitive closure has size at most א1. Thus, (Hω2 ,∈) is a natural model of ZFC minus the power-set axiom which correctly estimates many of the problems left open by the smaller and better understood structure (Hω1 ,∈) of hereditarily countable sets. One of such problems is, for example, the Continuum Hypothesis. It is largely for this reason that the structure (Hω2 ,∈) has recently received a considerable amount of study (see e.g. [15] and [16]). Recall the well-known Levy-Schoenfield absoluteness theorem ([10, §2]) which states that for every Σ0−sentence φ(x, a) with one free variable x and parameter a from Hω2 , if there is an x such that φ(x, a) holds then there is such an x in Hω2 , or in other words, (Hω2 ,∈) ≺1 (V,∈). (1)

Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions

Abstract: Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree $i$ has dimension exactly dim(R/I) - i. This Cohen-Macaulay characterization reduces to the Eagon-Reiner theorem by Alexander duality when R is a polynomial ring. The proof exploits a graded ring-theoretic generalization of the Zeeman spectral sequence, thereby also providing a combinatorial topological version for polyhedral cell complexes, involving no commutative algebra.

Lacunary partition functions

Abstract: A function on the natural numbers which is almost always 0 is called lacunary. Euler’s pentagonal number theorem and other identities for q-series involving theta or false theta functions give examples of partition functions which are trivially lacunary. A number of less obvious examples have been found by Serre, Gordon and Robins, and Andrews, Dyson, and Hickerson using the arithmetic of quadratic fields. We combine the theory of Bailey chains and estimates involving quadratic forms to demonstrate that infinite families of lacunary partition functions are widespread.

Continuity of the measure of the spectrum for discrete quasiperiodic operators

Abstract: We study discrete Schrodinger operators (H�,�ψ)(n) = ψ(n − 1) + ψ(n + 1) + f(αn+θ)ψ(n) on l 2 (Z), where f(x) is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of H�,� to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of H�,� are positive. For the almost Mathieu operator (f(x) = 2λcos2πx) it follows that the measure of the spectrum is equal to 4|1 − |λ|| for all real θ, λ 6 ±1, and all irrational α.

The Castelnuovo-Mumford regularity of an integral variety of a vector field on projective space

Abstract: The Castelnuovo-Mumford regularity r of a complex, projective variety V is an upper bound for the degrees of the hypersurfaces necessary to cut out V. In this note we give a bound for r when V is left invariant by a vector field on the ambient projective space. More precisely, assume V is arithmetically Cohen-Macaulay, for instance, a complete intersection. Assume as well that V projects to a normal-crossings hypersurface, which is the case when V is a curve with at most ordinary nodes. Then we show that r

Divergence points of deformed empirical measures

Abstract: We introduce and develope a unifying multifractal framework based on deformations of empirical measures. This framework (1) unifies and extends many results in multifractal analysis of local characteristics of dynamical systems and “fractal” measures and (2) provides a systematic basis for the detailed study of divergence points.

Spin manifolds, Einstein metrics, and differential topology

Abstract: We show that there exist smooth, simply connected, fourdimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy re nement of the Seiberg-Witten invariant [4, 3], in conjunction with curvature estimates previously proved by the second author [17]. These methods also easily allow one to construct examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have in nitely many other smooth structures for which no Einstein metric can exist.

On the surjectivity of the power maps of algebraic groups in characteristic zero

Abstract: In this paper we study the surjectivity of the power maps gg n for algebraic groups over an algebraically closed field of characteristic zero. We describe certain necessary and sufficient conditions for surjectivity to hold, and using these results we determine the set of n for which it holds in the case of simple algebraic groups. The results are also applied to study the exponentiality of algebraic groups.

A bernstein type result for special lagrangian submanifolds

Abstract: Let Σ be a complete minimal Lagrangian submanifold of C n . We iden- tify several regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σ lies in one of these regions, then Σ is an affine space.

Affine Spheres and Kähler-Einstein Metrics

Abstract: In this note, we introduce a straightforward correspondence between some natural affine Kahler metrics on convex cones and natural metrics on certain hypersurfaces asymptotic to the boundary of these cones. Recall an affine Kahler metric is a Riemannian metric locally given by the Hessian of a potential function φ, i.e. gij = ∂2φ ∂xi∂xj . Note that this metric is well defined only up to affine coordinate changes. Affine Kahler metrics are sometimes called Hessian metrics. See e.g. [4, 12]. The centroaffine second fundamental form provides a Riemannian metric on a hypersurface H ⊂ R, if H is the radial graph of a function − 1 u , with u negative and convex. The formula for this metric is − 1 u ∂2u ∂ti∂tj , and if u transforms as section of a certain line bundle, this metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone

Scattering Theory and Adiabatic Decomposition of the $\z$-determinant of the Dirac Laplacian

Abstract: In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac Laplacian. Theorem 1.1 of this paper extends the re- sult of our earlier work (see (8) and (9)), which covered the case of the invertible tangential operator. The presence of the non-trivial kernel of the tangential op- erator requires careful analysis of the small eigenvalues of the Dirac Laplacian, which employs elements of scattering theory. 1. Statement of the Result Let D : C ∞ (M; S) → C ∞ (M; S) denote a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a closed manifold M of di- mension 2k +1.Assume that we have a decomposition of M as M1 ∪M2 , where M1 and M2 are compact manifolds with boundary so that M = M1 ∪ M2 ,M 1 ∩ M2 = Y = ∂M1 = ∂M2 .