An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

The cylinder over the Koras–Russell cubic threefold has a trivial Makar-Limanov invariant

In 1996 Makar-Limanov established that the Koras–Russell cubic threefold

Arboreal Galois representations

Let be the absolute Galois group of , and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of into the automorphism group of T, particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of . We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of . Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of .      

A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π₁(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B _t }, satisfying certain explicit hypotheses, |π⁻¹(B _t )| is asymptotic to vol(B _t ). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.      

On (k,\varepsilon)-saddle submanifolds of Riemannian manifolds

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of $(k,\varepsilon)The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of -saddle, -parabolic, and -convex submanifolds ( ). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( ). It follows that the definition of -saddle submanifolds is equivalent to the existence of -asymptotic subspaces in the tangent space. We prove non-embedding theorems of -saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, , and for submanifolds with extrinsic curvature (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendon?a and Zhou.      

Finite groups with minimal 1-PIM

Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).     

Convexes Hyperboliques et Quasiisométries

For any m ≥ 3, we construct properly convex open sets Ω in the real projective space whose Hilbert metric is Gromov hyperbolic but is not quasiisometric to the hyperbolic space . We show that such examples cannot exist for m = 2. Some of our examples are divisible, i.e. there exists a discrete group Г of projective transformations preserving Ω with a compact quotient Г\Ω. The open set Ω is strictly convex but the group Г is not isomorphic to any cocompact lattice in the isometry group of .     

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .     

Diameter preserving bijections between Grassmann spaces over Bezout domains

Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation. Project 10671026 supported by National Natural Science Foundation of China.     

Homology of planar polygon spaces

In this paper, we study topology of the variety of closed planar n-gons with given side lengths . The moduli space where , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces as functions of the length vector . We also find sharp upper bounds on the sum of Betti numbers of depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.      

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.     

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

Polyhedral hyperbolic metrics on surfaces

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space and a group G of isometries of such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.      

Quantization for a fourth order equation with critical exponential growth

For concentrating solutions weakly in H ²(Ω) to the equation on a domain with Navier boundary conditions the concentration energy is shown to be strictly quantized in multiples of the number .     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

Quasiperiodic Group Factorizations

If is any ring or semi-ring (e.g., ) and G is a finite abelian group, two elements a, b of the group (semi-)ring are said to form a factorization of G if ab = rΣ _g∈G g for some . A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b ₀ + ... + b _m−1 of m elements of such that ab _h = g ^h ab ₀ for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when and r = 1 but Sands [15] found a counterexample for the group . Here we show however that all factorizations of abelian groups are quasiperiodic when and that all factorizations of cyclic groups or of groups of the type are quasiperiodic when . We also give some new examples of non-quasiperiodic factorizations with for the smaller groups and . Received: May 12, 2006. Revised: October 3, 2007.