Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Closed geodesics on compact nilmanifolds with Chevalley rational structure

We study the distribution of closed geodesics on nilmanifolds Γ \ N arising from a 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact semisimple Lie algebra on a real finite dimensional vector space U. We determine sufficient conditions on the semisimple Lie algebra for Γ \ N to have the density of closed geodesics property where Γ is a lattice arising from a Chevalley rational structure on .     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

On some types of radical classes

Let be an infinite cardinal. We denote by the collection of all -representable Boolean algebras. Further, let be the collection of all generalized Boolean algebras B such that for each b ∈ B, the interval [0, b] of B belongs to . In this paper we prove that is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized MV-algebras. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information.     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

The Kostant–García quantization of the bundle of connections

We shall call quantum states of a principal bundle π : P → M with structure group a semi-simple Lie group G, the elements of certain space of sections of the adjoint bundle , associated to the G-bundle of connections . An inner product of sections of is defined for which is a Hilbert space such that the Gauge group gau(P) of the given bundle represents in a family of self-adjoint operators. This work crystallizes some heuristic considerations, on the unitary representations of Gauge algebras, of Garcia in the already a classical article (J. Differ. Geom. 12, 209–227, 1977).     

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

We consider one-dimensional difference Schr?dinger equations 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 is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (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. Received: February 2006, Accepted: December 2007     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Numerically satisfactory solutions of Kummer recurrence relations

Pairs of numerically satisfactory solutions as for the three-term recurrence relations satisfied by the families of functions , , are given. It is proved that minimal solutions always exist, except when and z is in the positive or negative real axis, and that is minimal as whenever . The minimal solution is identified for any recurrence direction, that is, for any integer values of and . When the confluent limit , with fixed, is the main tool for identifying minimal solutions together with a connection formula; for , is the main tool to be considered.     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Intersection homology Künneth theorems

Cohen, Goresky, and Ji showed that there is a Künneth theorem relating the intersection homology groups to and , provided that the perversity satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem relating to and for all choices of and . Furthermore, we prove that the Künneth theorem still holds when the biperversity p, q is “loosened” a little, and using this we recover the Künneth theorem of Cohen–Goresky–Ji.     

On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Carleson Measures for the Bloch Space

In this paper we study the positive Borel measures μ on the unit disc in for which the Bloch space is continuously included in , 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure μ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for μ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator , defined by , maps continuously into the Bergman space A ¹, . Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekollé-Bonami -condition, then the measure defined by is a p-Bloch-Carleson-measure. We also consider the Banach space of those functions f which are analytic in and satisfy , as . The Bloch space is contained in . We describe the p-Carleson measures for and study weighted composition operators and a class of integration operators acting in this space. We determine which of these operators map continuously to the weighted Bergman space and show that they are automatically compact. This research is partially supported by several grants from “the Ministerio de Educación y Ciencia, Spain” (MTM2005-07347, MTM2007-60854, MTM2006-26627-E, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from “La Junta de Andalucía” (FQM210 and P06-FQM01504); from “the Academy of Finland” (210245) and from the European Networking Programme “HCAA” of the European Science Foundation.     

Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, a nontrivial additive character, and a character on . Then ψ defines an Artin-Schreier sheaf on the affine line , and χ defines a Kummer sheaf on the n-dimensional torus . Let be a Laurent polynomial. It defines a k-morphism . In this paper, we calculate the weights of under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

Parabolic Raynaud bundles

Let X be an irreducible smooth projective curve defined over the field of complex numbers, a finite set of closed points and N ≥ 2 a fixed integer. For any pair , there exists a parabolic vector bundle on X, with parabolic structure over S and all parabolic weights in , that has the following property: Take any parabolic vector bundle of rank r on X whose parabolic points are contained in S, all the parabolic weights are in and the parabolic degree is d. Then is parabolically semistable if and only if there is no nonzero parabolic homomorphism from to .     

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007