Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.     

Hypercyclic Pairs of Coanalytic Toeplitz Operators

A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A ⁿ B ^k x : n, k ≥ 0} is dense in . If f, g ∈H ^∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M ^* _f , M ^* _g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H ^∞(G) such that the pair (M ^* _f , M ^* _g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.     

The fraction of the bijections generating the near-ring of 0-preserving functions

Let 〈G, +〉 be a finite (not necessarily abelian) group. Then M₀(G) := {f : G → G| f (0) = 0} is a near-ring, i.e., a group which is also closed under composition of functions. In Theorem 4.1 we give lower and upper bounds for the fraction of the bijections which generate the near-ring M₀(G). From these bounds we conclude the following: If G has few involutions and the order of G is large, then a high fraction of the bijections generate the near-ring M₀(G). Also the converse holds: If a high fraction of the bijections generate M₀(G), then G has few involutions (compared to the order of G). Received: 10 January 2005     

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.     

Topological rings,homogeneous functions,and primeness

For any group G such that G is a right R-module for some ring R, the elements of R act on G as endomorphisms and we obtain the near-ring of R-homogeneous maps on G: M_R(G) = {f: G → G|f(ga) = f(g)a for all a ∈ R, g ∈ G}. In the special case that R is a topological ring and G is a topological R-module, we study N_R(G): = {f ∈ M_R(G)|f is continuous}. In particular, we investigate primeness of the near-ring N_R(G) of continuous homogeneous maps on G.     

Obstruction theory and coincidences of maps between nilmanifolds

Let f, g : M N be two maps between two compact nilmanifolds with dim M dim N = n. In this paper, we show that either the Nielsen coincidence number N(f, g) = 0 or N(f, g) = R(f, g) where R(f, g) denotes the Reidemeister number of f and g. Furthermore, we show that if N(f, g) > 0 then the primary obstruction o_n(f, g) to deforming f and g to be coincidence free on the n-th skeleton of M is non-trivial.Received: 30 April 2004; revised: 20 July 2004     

Invariant functions in Denjoy–Carleman classes

Let V be a real finite dimensional representation of a compact Lie group G. It is well known that the algebra of G-invariant polynomials on V is finitely generated, say by σ ₁, . . . , σ _p . Schwarz (Topology 14:63–68, 1975) proved that each G-invariant C ^∞-function f on V has the form f = F(σ ₁, . . . , σ _p ) for a C ^∞-function F on . We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that f and F lie in the same Denjoy–Carleman class C ^M (with M = (M _k )). For finite groups G and (more generally) for polar representations V, we show that for each G-invariant f of class C ^M there is an F of class C ^N such that f = F(σ ₁, . . . , σ _p ), if N is strongly regular and satisfies

Gauge-Invariant Characterization of Yang–Mills–Higgs Equations

Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n⁺, n⁻) and let E → M be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n⁺, n⁻) × G-invariant quadratic Lagrangians on J¹(C × _M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n⁺, n⁻) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory. Submitted: May 19, 2005; Accepted: April 25, 2006     

Bands of invariantly extensible measures

Given two σ-algebrasU ⊂A, invariant under a fixed semigroupG of transformations, the following subsetC of the lattice coneM (U) _G ofG-invariant finite measures onU is shown to be (the positive part of) a band inM (U) _G : AG-invariant measure μ belongs toC iff the setexM Bμ) _G of extremalG-invariant extensions of μ toB is non-empty and eachG-invariant extensionv of μ admits a barycentric decompositionv=→v′ρ(dv′) with some representing probability ρ onexM U μ) _G .—Any band of extensible measures allows to study the corresponding extension problem locally.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

The Problem of Prescribed Critical Functions

Let (M,g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere Sⁿ. We prove that a smooth function f on M is a critical function for a metric g conformal to g if and only if there exists x ∈ M such that f(x) > 0.Mathematics Subject Classifications (2000): 53C21, 46E35, 26D10.     

Separating maps on weighted function algebras on topological groups

Let G ₁ and G ₂ be locally compact groups and let ω ₁ and ω ₂ be weight functions on G ₁ and G ₂, respectively. For i = 1, 2, let also C ₀(G _i , 1/ω _i ) be the algebra of all continuous complex-valued functions f on G _i such that f/ω _i vanish at infinity, and let H: C ₀(G ₁, 1/ω ₁) → C ₀(G ₂, 1/ω ₂) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C ₀(G ₁, 1/ω ₁) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C ₀(G ₁, 1/ω ₁), where φ: G ₂ → ℂ is a non-vanishing continuous function and h: G ₂ → G ₁ is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

Coincidences of maps into homogeneous spaces

Let f,g:X→M be maps between two closed connected orientable n-manifolds where M=G/K is the homogeneous space of left cosets of a compact connected Lie group G by a finite subgroup K. In this note, we obtain a simple formula for the Lefschetz coincidence number L(f,g) in terms of topological degree, generalizing some previously known formulas for fixed points. Our approach, by means of Nielsen root theory, also allows us to give a simpler and more geometric proof of the fact that all coincidence classes of f and g have coincidence index of the same sign. Received: 3 March 1998 / Revised version: 29 June 1998     

Invariance of the Nayatani metrics for Kleinian manifolds

The Nayatani metric g _N is a Riemannian metric on a Kleinian manifold M which is compatible with the standard flat conformal structure. It is known that, for M corresponding to a geometrically finite Kleinian group, g _N has large symmetry: the isometry group of (M, g _N ) coincides with the conformal transformation group of M. In this paper, we prove that this holds for a larger class of M. In particular, this class contains such M that correspond to Kleinian groups of divergence type.      

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

Flows, fixed points and rigidity for Kleinian groups

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G ₁, G ₂ cocompact Kleinian groups, any quasiconformal map pairing a G ₁-invariant pattern to a G ₂-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math.GT/08094449, 2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G ₁ and a quasiconformal conjugate h ^?1 G ₂ h of a cocompact group G ₂. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence.