Complete space-like hypersurfaces with constant scalar curvature

Let M ⁿ be a complete space-like hypersurface with constant normalized scalar curvature R in the de Sitter space S ^{n + 1} ₁ and denote . We prove that if the norm square of the second fundamental form of M ⁿ satisfies , then either and M ⁿ is a totally umbilical hypersurface; or , and, up to rigid motion, M ⁿ is a hyperbolic cylinder . Received: 8 February 2001 / Revised version: 27 April 2001     

There exists no (15,5,4) RBIBD†*

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 357–362, 2001     

Bounding the crossing number of a graph in terms of the crossing number of a minor with small maximum degree

We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface Σ is at least one fourth the crossing number of M in Σ. We use this result to show that every graph embedded on the torus with representativity r ≥ 6 has Klein bottle crossing number at least ⌊2r/3⌋²/64. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 168–173, 2001     

K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S ². Oblatum 23-II-2001 & 9-V-2001?Published online: 20 July 2001     

Extension of C R Structures on three dimensional compact pseudoconvex C R manifolds

Let M be a smooth compact orientable pseudoconvex C R manifold of real dimension three and assume that there is a smooth function λ which is strictly subharmonic in any direction where the Levi-form vanishes on M. Then we extend the given C R structure on M to an integrable almost complex structure on the concave side of M. As an application, if M is a non-compact pseudoconvex C R manifold of real dimension three, we prove that the given C R structure on M can be locally extended to an integrable almost complex structure on the concave side of M. Partially supported by Korea research foundation Grant (KRF-2001-015-DP0016) and by R14-2002-044-01000-0 from KRF     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type

 Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact K?hler manifold, then its Euler number χ(M) satisfies (−1) ^m χ(M)>0. Received: 25 September 2001 / Published Online: 16 October 2002     

Invariant subsets of rank 1 manifolds

It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M. Received: 6 July 2000 / Revised version: 11 October 2001     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

On the dynamics of certain actions of free groups on closed real analytic manifolds

Let M be a closed connected real analytic manifold; let be a free group on two generators. The set of analytic actions of on M endowed with Taken‚s topology contains a nonempty open subset whose corresponding actions share three properties: (a) they have every orbit dense, (b) they leave invariant no geometric structure on M, (c) any homeomorphism conjugating two of them is analytic. Received: October 3, 2001     

Axiomatization of abelian-by-G groups for a finite group G

We show that, for each finite group G, there exists an axiomatization of the class of abelian-by-G groups with a single sentence. In the proof, we use the definability of the subgroups M ⁿ in an abelian-by-finite group M, and the Auslander-Reiten sequences for modules over an Artin algebra. Received: 15 March 1996 / Published online: 18 July 2001     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

Complexity of degenerations of modules

A module M over an associative algebra A over an algebraically closed field k is said to degenerate to a module N if N belongs to the closure of the isomorphism class of M in the algebraic variety of d-dimensional A-modules, . We associate a non-negative integer to a degeneration , its complexity, and study its properties. Received: January 30, 2001     

Asymptotique de la norme $L^2$ d'un cycle géodésique dans les revêtements de congruence d'une variété hyperbolique arithmétique

Let M be an arithmetic hyperbolic manifold and be a codimension 1 geodesic cycle. In this paper, we study the asymptotic growth of the -norm of the lifts of F in the congruence tower above M. We obtain an explicit value for the growth rate of this norm. In particular, we provide a new proof of a celebrated result of Millson [Mi] on the homology of the arithmetic hyperbolic manifolds. The method is quite general and gives a new way of getting non zero homology classes in certain locally symmetric spaces. Received: 20 April 2001; in final form: 26 September 2001 / Published online: 28 February 2002     

Dimension minimale des orbites d'une action de ℝn sur une variété de contact

By differentiability we means C ^∞ differentiability. Recall that the span of a manifold M is the maximum number of linearly independent vector fields in every point. The aim of this paper is to relate the span of M with the minimal dimension of the orbits of a differentiable action ϕ:ℝ ⁿ ×M→M that keeps a contact structure.

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Received: 19 July 2000 / Revised version: 20 April 2001     

Discs in Stein manifolds containing given discrete sets

Denote by the open unit disc in . We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map such that ; if the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of with given one dimensional submanifolds in M. Received: 19 June 2000; in final form: 29 November 2000 / Published online: 19 October 2001     

On wedge extendability of CR-meromorphic functions

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a -smooth embedded Cauchy-Riemann generic manifold M (CR functions on ) and more generally, we consider holomorphic functions defined in wedgelike domains attached to . Our main result establishes the wedge- and the L¹-removability of E under the hypothesis that the -dimensional Hausdorff volume of E is zero and that M and are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everywhere locally minimal M to which every CR-meromorphic function on M extends meromorphically. Received: 7 September 2000; in final form: 20 December 2001 / Published online: 6 August 2002     

Roots of mappings on nonorientable surfaces¶and equations in free groups

Let f:M ₁→M ₂ be a continuous map and c:M ₁→M ₂ a constant map between closed (not necessarily orientable) surfaces. By definition the pair (f,c) has the Wecken property if f can be deformed into a map f' such that the number of coincidence points of (f',c) is the same as the number of essential coincidence classes of (f,c) and, hence, every essential coincidence class consists of exactly one point. When both surfaces are orientable the problem to determine all maps which have the Wecken property was solved in [14]. Let A(f) denote the absolute degree as defined in [6] or [15] and . Here we show that a map f has the Wecken property iff either the Euler characteristic or . In free groups there are solved certain quadratic equations closely related to the root problem. Received: Received: 18 January 2001 / Revised version: 27 November 2001