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.      

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.     

Stable constant mean curvature hypersurfaces in the real projective space

In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in ; and (iii) the quotient to of the hypersurface obtained as the product of two spheres of dimensions k and n − k, with k = 1,..., n − 1, and radii r and , respectively, with .     

The Poisson boundary of lamplighter random walks on trees

Let be the homogeneous tree with degree q + 1 ≥ 3 and a finitely generated group whose Cayley graph is . The associated lamplighter group is the wreath product , where is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter “operates at bounded range”. Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577.     

Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in whose asymptotic boundaries are closed codimension-1 submanifolds in . We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S ³ and Λ(G) is the union of all complex projective lines in which are tangent to S ³ at a point in L(G).     

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 .     

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.     

Regular quantizations and covering maps

Let $\tilde{M} \rightarrow MLet be a holomorphic (unbranched) covering map between two compact complex manifolds, with . We prove that if and M both admit regular K?hler forms and ω respectively then, up to homotheties, and (M, ω) are biholomorphically isometric. This work was supported by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006     

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.      

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

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.      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

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.