Showing papers in "Geometric and Functional Analysis in 2008"

Special Cube Complexes

Abstract: We introduce and examine a special class of cube complexes. We show that special cube-complexes virtually admit local isometries to the standard 2-complexes of naturally associated right-angled Artin groups. Consequently, special cube-complexes have linear fundamental groups. In the word-hyperbolic case, we prove the separability of quasiconvex subgroups of fundamental groups of special cube-complexes. Finally, we give a linear variant of Rips’s short exact sequence.

The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals

Abstract: We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtain the rank-one case of Lieb’s fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher-rank case, giving two new proofs of the general-rank case of Lieb’s theorem.

From the Mahler Conjecture to Gauss Linking Integrals

Abstract: We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K) = (Vol K)(Vol K°) of the volume of a symmetric convex body $$K \in {\mathbb{R}}^{n}$$ and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $$K^{\diamond} \subseteq K \times K^{\circ}$$ is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) n γ n , where γ n is a monotonic factor that begins at 4/π and converges to $${\sqrt2}$$ . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c n . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S n-1 and in hyperbolic space H n-1.

Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

Abstract: We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are as homeomorphic to the two-dimensional sphere Our methods rely on the study of certain random geodesic laminations of the disk

Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

Abstract: We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher.

Log Canonical Thresholds of Del Pezzo Surfaces

Abstract: We study global log canonical thresholds of del Pezzo surfaces. All varieties are assumed to be defined over $${\mathbb{C}}$$ .

Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Abstract: Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F λ and G λ of Heckman and Opdam, and sharp estimates of the particular functions F 0 and G 0. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel.

Number variance of random zeros on complex manifolds

Abstract: We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set \(U \subset {\mathbb{C}}^m\) with smooth boundary is asymptotic to \(N^{{m-1}/2} u_{mm} {\rm Vol}(\partial U)\), where \( u_{mm}\) is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on \({\mathbb{C}}^{m}\). Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kahler manifold.

Fine Properties of the Integrated Density of States and aQuantitative Separation Property of the Dirichlet Eigenvalues

Abstract: We consider one-dimensional difference Schrodinger equations $$[H(x,\omega)\varphi](n)\equiv - \varphi(n-1)-\varphi(n+1)+V(x+n\omega)\varphi(n) = E \varphi (n), n \, \in \, \mathbb {Z}, \, x, \omega \, \in \, [0,1]$$ with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V 0(x) of degree k 0, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states $$\mathcal{N}$$ is Holder $$\frac{1}{2k_0}-k$$ continuous for any k > 0. Moreover, we show that $$\mathcal{N}$$ is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem $$H(x, \omega)\varphi = E \varphi$$ on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval $$(E - \eta, E + \eta) \, \rm{with}\, \eta \, \asymp \, N^{-1+\delta}, 0 < \delta \ll 1$$ , does not exceed $$N\eta^{ \frac{1}{2k_0}-k}$$ , k > 0. Moreover, for $$ \omega\, otin\, \Omega(\varepsilon), \rm {mes}\, \Omega(\varepsilon) \, < \,\varepsilon \, {\rm and}\,\, E \, otin \, \mathcal{E}_{\omega} (\varepsilon), \, {\rm mes}\, \mathcal{E}_{\omega} (\varepsilon) \, < \, \varepsilon$$ , this averaged number does not exceed exp $$((\log \varepsilon^{-1})^A)\eta N$$ , for any $$ {\eta} > N^{-1+b}, b > 0$$ . For the integrated density of states $$\mathcal{N}(\cdot)$$ of the problem $$H(x, \omega) \varphi =E \varphi$$ this implies that $$ \mathcal{N}(E + \eta)- \mathcal{N}(E - \eta) \leq {\rm exp}((\log \varepsilon^{-1})^A)\eta$$ for any $$E \, otin \, {\mathcal{E}_\omega}(\varepsilon)$$ . To investigate the distribution of the Dirichlet eigenvalues of $$H(x, \omega)\varphi = E \varphi$$ on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants $$f_N(\cdot,\omega, E)$$ with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus $$\mathcal{A}_\rho = \{z \, \in \, \mathbb{C} : 1-\rho < |z| < 1+\rho \}$$ and show also that no more than 2k 0 of them fall into any disk of radius exp $$(-(\log N)^A), A {\gg}1$$ . In addition, we obtain the lower bound $$e^{-{N}^{\delta}}$$ (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.

Einstein Metrics with Prescribed Conformal Infinity on 4-Manifolds

Abstract: This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.

Derived Equivalences by Quantization

Abstract: We assume given a smooth symplectic (in the algebraic sense) resolution $X$ of an affine algebraic variety $Y$, and we prove that, possibly after replacing $Y$ with an etale neighborhood of a point, the derived category of coherent sheaves on $X$ is equivalent to the dervied category of finitely generated left modules over a non-commutative algebra $R$, a non-commutative resolution of $Y$ in a sense close to that of M. Van den Bergh. We also prove some applications, such as: two resolutions are derived-equivalent; every resolution $X$ admits a "resolution of the diagonal"; the cohomology groups of the fibers of the map $X \to Y$ are spanned by fundamental classes of algebraic cycles.

Hall’s Theorem for Limit Groups

Abstract: A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela.

Universal Sampling and Interpolation of Band-Limited Signals

Abstract: We ask if there exist discrete “universal” sets \(\bigwedge\) of given finite density such that every signal f with bounded spectrum of small measure can be recovered from the samples \(f(\lambda),\,\lambda \in \bigwedge\). We prove that uniqueness in this problem can be achieved in general situation. On the other hand, for stable reconstruction it is crucial whether the spectrum is compact or dense.

Multiple Saddle Connections on Flat Surfaces and the Principal Boundry of the Moduli Spaces of Quadratic Differentials

Abstract: We describe typical degenerations of quadratic differentials thus describing “generic cusps” of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from “generic” degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differential which is close to a “cusp” from the corresponding point at the principal boundary.

Scattering Theory for Radial Nonlinear Schrödinger Equations on Hyperbolic Space

Abstract: We study the long time behavior of radial solutions to nonlinear Schrodinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions.

Independent Sets in Graph Powers are Almost Contained in Juntas

Abstract: Let G = (V;E) be a simple undirected graph. Define G n , the n-th power of G, as the graph on the vertex set V n in which two vertices (u1;:::;un) and (v1;:::;vn) are adjacent if and only if ui is adjacent to vi in G for every i. We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G n . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e., it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [18]. This principle has already been shown in [11] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.

End-Point Equations and Regularity of Sub-Riemannian Geodesics

Abstract: For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x1, x2. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.

Homological Invariance For Asymptotic Invariants and Systolic Inequalities

Abstract: We show that the systolic constant, the minimal volume entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value for the real projective space) and derive an inequality between the minimal volume entropy and the systolic constant.

L 2 -Cohomology for Von Neumann Algebras

Abstract: We study L2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L2-cohomology and show how the study of the first L2-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. In particular, we show that the first L2-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra.

A characterization of the duality mapping for convex bodies

Abstract: We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. The property that the duality interchanges pairwise intersections and convex hulls of unions is sufficient for a characterization, up to a trivial exception and the composition with a linear transformation.

The De Rham Decomposition Theorem For Metric Spaces

Abstract: We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.

Superrigidity, Generalized Harmonic Maps and Uniformly Convex Spaces

Abstract: We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs rely on certain notions of harmonic maps and the study of their existence, uniqueness, and continuity.

A Generalization Of Reifenberg’s Theorem In {\mathbb{R}}^3

Abstract: In 1960 Reifenberg proved the topological disc property. He showed that a subset of R-n which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Holder image of the unit ball in R-m. In this paper we prove that a subset of R-3 which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-Holder deformation of a minimal cone. We also prove an analogous result for more general cones in R-n.

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

Abstract: The main result of this paper is that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not Kahler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2.

Boolean Functions with small Spectral Norm

Abstract: Let \(f : {\mathbb{F}}^{n}_{2} \rightarrow \{0, 1\}\) be a boolean function, and suppose that the spectralnorm\(\|f\|_{A} := \sum_{r} \mid \widehat{f}(r)\mid\) of f is at most M. Then \(\mathop {f = \sum\limits^{L}_{j=1}\pm 1_{{H}_{j}}},\) where \(L \leq 2^{{{2}^{CM}}^{4}}\) and each Hj is a subgroup of \({\mathbb{F}}^{n}_{2}\) . This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups.

Reconstruction of function fields

Abstract: For k an algebraic closure of the finite field $$\mathbb{F}_p$$ , l prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-l-quotient $${\mathcal{G}}_{K}$$ of its absolute Galois group – in fact already from the second central descending series quotient of $${\mathcal{G}}_{K}$$ .

Countable Primitive Groups

Abstract: We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product.

Deformations of Local Systems and Eisenstein Series

Abstract: Let $X$ be a (smooth and complete) curve and $G$ a reductive group. In [BG] we introduced the object that we called "geometric Eisenstein series". This is a perverse sheaf $\bar{Eis}_E$ (or rather a complex of such) on the moduli stack $Bun_G(X)$ of principal $G$-bundles on $X$, which is attached to a local system $E$ on $X$ with respect to the torus $\check{T}$, Langlands dual to the Cartan subgroup $T\subset G$. In loc. cit. we showed that$\bar{Eis}_E$ corresponds to the $\check{G}$-local system induced from $E$, in the sense of the geometric Langlands correspondence. In the present paper we address the following question, suggested by V. Drinfeld: what is the perverse sheaf on $Bun_G(X)$ that corresponds to the universal deformation of $E$ as a local system with respect to the Borel subgroup $\check{B}\subset \check{G}$? We prove, following a conjecture of Drinfeld, that the resulting perverse sheaf if the classical, i.e., non-compactified Eisenstein series.

Trees, Valuations and the Green–Lazarsfeld Set

Abstract: We study the Green–Lazarsfeld set from the point of view of geometric group theory and compare it with the Bieri–Neumann–Strebel invariant. Applications to the study of fundamental groups of Kahler manifolds are given.