Showing papers in "Mathematische Annalen in 2011"

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

Abstract: We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R n , n ≥ 1, so that the following inequalities hold for all $${u \in C_{0}^{\infty}(B)}$$ : $$\label{one} \int\limits_{B}V(x)| abla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$$ $$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)| abla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)| abla u|^2dx.$$ This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vazquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.

Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words

Abstract: We construct irreducible representations of affine Khovanov–Lauda–Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the ‘cuspidal modules’ are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.

Automorphic lifts of prescribed types

Abstract: We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.

Monotonic properties of the least squares mean

Abstract: We settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares mean and derive our results in this more general setting.

Einstein solvmanifolds: existence and non-existence questions

Abstract: The aim of this paper is to study the problem of which solvable Lie groups admit an Einstein left invariant metric. The space \({\mathcal{N}}\) of all nilpotent Lie brackets on \({\mathbb{R}^n}\) parametrizes a set of (n + 1)-dimensional rank-one solvmanifolds \({\{S_{\mu}:\mu\in\mathcal{N}\}}\), containing the set of all those which are Einstein in that dimension. The moment map for the natural GL n -action on \({\mathcal{N}}\), evaluated at \({\mu\in\mathcal{N}}\), encodes geometric information on S μ and suggests the use of strong results from geometric invariant theory. For instance, the functional on \({\mathcal{N}}\) whose critical points are precisely the Einstein S μ ’s, is the square norm of this moment map. We use a GL n -invariant stratification for the space \({\mathcal{N}}\) and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds S μ . As an application, we obtain criteria to decide whether a given nilpotent Lie algebra can be the nilradical of a rank-one Einstein solvmanifold or not. We find several examples of \({\mathbb{N}}\)-graded (even 2-step) nilpotent Lie algebras which are not. A classification in the 7-dimensional, 6-step case and an existence result for certain 2-step algebras associated to graphs are also given.

On the energy functional on Finsler manifolds and applications to stationary spacetimes

Abstract: In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais–Smale condition under the completeness of the Finsler metric. Moreover, we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover, we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows one to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.

Operator log-convex functions and operator means

Abstract: We study operator log-convex functions on (0, ∞), and prove that a continuous nonnegative function on (0, ∞) is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditions related to operator means are given for such functions. Operator log-concave functions are also discussed.

Sharp convex Lorentz–Sobolev inequalities

Abstract: New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L p Minkowski problem. New L p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.

Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in $${\mathbb{R}^N}$$

Abstract: We discuss the existence of a maximizer for a maximizing problem associated with the Trudinger–Moser type inequality in \({\mathbb{R}^N(N\geq 2)}\). Different from the bounded domain case, we obtain both of the existence and the nonexistence results. The proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences.

Existence and uniqueness of global weak solution to a two-phase flow model with vacuum

Abstract: In this paper, we study a two-phase liquid–gas model with constant viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously. Just as in Evje and Karlsen (Commun Pure Appl Anal 8:1867–1894, 2009) and Evje et al. (Nonlinear Anal 70:3864–3886, 2009), the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We use a new technique to get the upper and lower bounds of gas and liquid masses n and m. Then we get the global existence of weak solution by the line method. Also, we obtain the uniqueness of the weak solution.

Big rational surfaces

Abstract: We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of $${\mathbb{P}^2}$$ at distinct points lying on a (possibly reducible) cubic.

Foliations of asymptotically flat manifolds by surfaces of Willmore type

Abstract: The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.

Hyperbolicity, CAT(-1)-spaces and the Ptolemy inequality

Abstract: Using a four points inequality for the boundary of CAT(-1)-spaces we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.

Varieties fibered by good minimal models

Abstract: Let f : X → Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map with relative dimension no more than three, then X has a good minimal model.

Primary decomposition and the fractal nature of knot concordance

Abstract: For each sequence $${\mathcal{P}=(p_1(t),p_2(t),\dots)}$$ of polynomials we define a characteristic series of groups, called the derived series localized at $${\mathcal{P}}$$ . These group series yield filtrations of the knot concordance group that refine the (n)-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. The new filtrations allow us to distinguish between knots whose classical Alexander polynomials are coprime and even to distinguish between knots with coprime higher-order Alexander polynomials. This provides evidence of higher-order analogues of the classical p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set.

Braid groups and Kleinian singularities

Abstract: We establish faithfulness of braid group actions generated by twists along an ADE configuration of 2-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland’s description of a space of stability conditions associated to a Kleinian singularity.

On the dimension of divergence sets of dispersive equations

Abstract: We refine results of Carleson, Sjogren and Sjolin regarding the pointwise convergence to the initial data of solutions to the Schrodinger equation. We bound the Hausdorff dimension of the sets on which convergence fails. For example, with initial data in $${H^1(\mathbb{R}^{3})}$$ , the sets of divergence have dimension at most one.

Kähler–Einstein submanifolds of the infinite dimensional projective space

Abstract: This paper consists of two main results. In the first one we describe all Kahler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kahler–Einstein metric with negative scalar curvature which admits a Kahler immersion into the infinite dimensional complex projective space.

Generalized Bernstein–Reznikov integrals

Abstract: We find a closed formula for the triple integral on spheres in \({\mathbb{R}^{2n} \times \mathbb{R}^{2n} \times \mathbb{R}^{2n}}\) whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein–Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures.

Homogenization of elliptic boundary value problems in Lipschitz domains

Abstract: In this paper we study the L p boundary value problems for $${\mathcal{L}(u)=0}$$ in $${\mathbb{R}^{d+1}_+}$$ , where $${\mathcal{L}=-{\rm div} (A abla )}$$ is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x d+1 and satisfies some minimal smoothness condition in the x d+1 variable, we show that the L p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L p Dirichlet problem for 2 − δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x d+1 variable, these results extend directly from $${\mathbb{R}^{d+1}_+}$$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.

Koszul homology and syzygies of Veronese subalgebras

Abstract: A graded K-algebra R has property N p if it is generated in degree 1, has relations in degree 2 and the syzygies of order ≤ p on the relations are linear. The Green–Lazarsfeld index of R is the largest p such that it satisfies the property N p . Our main results assert that (under a mild assumption on the base field) the cth Veronese subring of a polynomial ring has Green–Lazarsfeld index ≥ c + 1. The same conclusion also holds for an arbitrary standard graded algebra, provided $${c\gg 0}$$ .

The GL(2,C) McKay correspondence

Abstract: In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of $${{\rm GL}(2,\mathbb{C})}$$ , it means that the endomorphism ring of the special CM $${\mathbb{C}}$$ [[x, y]] G -modules can be used to build the dual graph of the minimal resolution of $${\mathbb{C}^{2}/G}$$ , extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of $${{\rm SL}(2,\mathbb{C})}$$ to all finite subgroups of $${{\rm GL}(2,\mathbb{C})}$$ .

The Cauchy–Riemann equations on product domains

Abstract: We establish the L 2 theory for the Cauchy–Riemann equations on product domains provided that the Cauchy–Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Abstract: We consider the Hill operator $$Ly = - y^{\prime \prime} + v(x)y, \quad0 \leq x \leq \pi,$$ subject to periodic or antiperiodic boundary conditions, with potentials v which are trigonometric polynomials with nonzero coefficients, of the form Then the system of eigenfunctions and (at most finitely many) associated functions is complete but it is not a basis in $${L^2 ([0,\pi], \mathbb{C})}$$ if |a| ≠ |b| in the case (i), if |A| ≠ |B| and neither −b 2/4B nor −a 2/4A is an integer square in the case (iii), and it is never a basis in the case (ii) subject to periodic boundary conditions.

The universality of ℓ1 as a dual space

Abstract: Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a $${{\mathcal L}_\infty}$$ space Z whose dual is isomorphic to l 1. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.

A sofic group away from amenable groups

Abstract: We give an example of a sofic group, which is not a limit of amenable groups.

Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds

Abstract: In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetric spaces.

Random groups do not split

Abstract: We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on simplicial trees. This implies that their Gromov boundaries, defined at density less than \({\frac{1}{2}}\) , are Menger curves.

The equivalence between pointwise hardy inequalities and uniform fatness

Abstract: We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.

The geometry of the Gauss–Picard modular group

Abstract: We give a construction of a fundamental domain for $${{\rm PU}(2,1,\mathbb{Z} [i])}$$ , that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers $${\mathbb{Z} [i]}$$ . We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.