Some q-convexity properties of coverings of complex manifolds

Let C be a nowhere dense compact analytic subset of a 2-dimensional complex manifold X. A result of Napier is the construction of a neighborhood V of C such that for any covering space in which is holomorphically convex, there is a C ^∞ exhaustion function φ on which is strictly plurisubharmonic on π^-1(V) away from the compact irreducible components of . Colţoiu and Vajaitu obtained a q-convex version of this result for C a q-dimensional fiber of a proper surjective submersion X→Y. The goal of this paper is a similar version for Cprojective.     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

A strict Positivstellensatz for enveloping algebras

Let G be a connected and simply connected real Lie group with Lie algebra . Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of is proved. An erratum to this article is available at .     

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Finite domination, Novikov homology and nonsingular closed 1-forms

Let X be a finite connected CW-complex and ρ: a regular covering space with free abelian covering transformation group. For ξ ∈ H¹ (Xℝ) we define the notion of ξ-contractibility of X. This notion is closely related to the vanishing of the Novikov homology of the pair (X,ξ). We show that finite domination of is equivalent to X being ξ-contractible for every nonzero ξ with ρ*ξ =0  ∈ H¹(; ℝ). If M is a closed connected smooth manifold the condition that M is ξ-contractible is necessary for the existence of a nonsingular closed 1-form representing ξ. Also ξ-contractibility guarantees the definition of the Latour obstruction τ_L(M,ξ) whose vanishing is then sufficient for the existence of a nonsingular closed 1-form provided  dim M≥6. Now if ρ: is a finitely dominated regular ℤ^k-covering space we get that τ_L(M,ξ) is defined for every nonzero ξ with ρ*ξ=0 and the vanishing of one such obstruction implies the vanishing of all such τ_L(M,ξ).     

Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system where 0<a<1/2 and δ γ=β ² ∈ (0,a), and the following combinatorial problem: (*) Given K points Q ₁ , . . . , Q _K ∈R ^N with minimum distance 1, find out the maximum number of times that the minimum distance 1 can occur. More precisely, we show that for any given positive integer K, there is a δ _K >0 such that for 0<δ<δ _K , there exists a standing-wave solution (u _δ ,ν _δ ) to the FitzHugh-Nagumo system with the property that u _δ has K spikes Q ^δ ₁,. . .,Q ^δ _K and approaches an optimal configuration in (*), where .     

Hausdorff measures, capacities and compact composition operators

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E_ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on . First author is partially supported by Plan Nacional I+D grant no. BFM2003-00034, and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 . Second author is partially supported by Plan Nacional I+D grant no. BFM2002-00571 and Junta de Andalucía RNM-314.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Quelques résultats sur les déformations équivariantes des courbes stables

Let G be a finite group, let g≥2 and g ′ ≥ 0 be integers. We introduce the algebraic stack classifying the stable curves of genus g endowed with an action of G faithful in each geometric fiber and such that the quotient of each fiber is a semi-stable curve of genus g′. We study the completion of the local rings of this algebraic stack. They are closely related to universal equivariant deformation rings R_C,G of stable curves endowed with a faithful action of G. A useful tool for this purpose is a local-global principle generalizing the one obtained by Bertin and Mézard in [BM00]. We then use the results we already proved in [Mau03b] and [Mau03a] to describe some properties of the space (purity, dimension).     

On the Circumference of 3-Connected Quasi-Claw-Free Graphs

A graph G is quasi-claw-free if it satisfies the property: d(x,y)=2 ⇒ there exists such that . In the paper, we prove that the circumference of a 3-connected quasi-claw-free graph G on n vertices is at least min{4δ−2,n} and G is hamiltonian if n≤5δ−5.     

Mean convex hulls and least area disks spanning extreme curves

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in such that for any asymptotic curve , there is a canonical mean convex hull containing all minimal planes spanning Γ. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C³-smooth curve in the boundary of C³-smooth mean convex domain in ℝ³, there exist a unique least area disk spanning the curve.     

Consistent Cycles in Graphs and Digraphs

Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle of Γ is called G-consistent whenever there is an element of G whose restriction to is the 1-step rotation of . Consistent cycles in finite arc-transitive graphs were introduced by J. H. Conway in his public lectures at the Second British Combinatorial Conference in 1971. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general setting of arbitrary groups of automorphisms of graphs and digraphs. Supported in part by ``Ministrstvo za šolstvo, znanost in šport Republike Slovenije', bilateral project BI-USA/04-05/38.     

Positive Projective Space of a <Emphasis Type="Italic">C</Emphasis>*-Algebra with a Tracial Conditional Expectation

Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient where G⁺ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|²a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow ℙ⁺ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in ℙ⁺, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space ℙ⁺ with the given geodesic distance is non positively curved.     

Zeros of Asai-Eisenstein series

We generalize a result of Max Deuring on the zeros of Zeta-function of quadratic forms to Asai's non-holomophic Eisenstein series E(z,s) of the Hilbert modular group. We prove that inside the rectangular and −1≤Re(s) ≤ 2 the function E(z,s) has only simple zeros on the line Re(s)=1/2 and two simple real zeros, if |N(y)| is large. The research was supported by a fellowship within the Post-doc-Program of the DAAD (German Academic Exchange Service)     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

Toward equivariant Iwasawa theory, III

Let l be an odd prime number and K _∞/k a Galois extension of totally real number fields, with and K _∞/k _∞ finite, where k _∞ is the cyclotomic -extension of k. The ``main conjecture' of equivariant Iwasawa theory, as formulated in [RW2], is, up to its uniqueness statement, reduced to the existence of a nonabelian pseudomeasure whenever G _∞=G(K _∞/k) is an l-group and Iwasawa's μ-invariant vanishes. This follows from combining the validity of the conjecture in the maximal order case with special congruences. The main tool of proof is a generalization of the Taylor-Oliver integral group logarithm so that it applies to the setting of Iwasawa theory. We acknowledge financial support provided by NSERC and the University of Augsburg.     

Mukai flops and deformations of symplectic resolutions

We prove that two projective symplectic resolutions of are connected by Mukai flops in codimension 2 for a finite sub-group G <Sp(2n). It is also shown that two projective symplectic resolutions of are deformation equivalent.     

Distribution of harmonic sums and Bernoulli polynomials modulo a prime

For a fixed integer s≥1, we estimate exponential sums with harmonic sums individually and on average, where H_s(n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H_s(n). For example, our estimates imply that for any ɛ>0, the set {H_s(n):n<p^1/2+ɛ} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as with max{n_ν|ν=1,. . . , 7}≤p^11/12+ɛ, and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values B_p−r(n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche. During the preparation of this paper, F. L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I. S. was supported in part by ARC grant DP0211459.     

A schlichtness theorem for envelopes of holomorphy

Let Ω be a domain in . We prove the following theorem. If the envelope of holomorphy of Ω is schlicht over Ω, then the envelope is in fact schlicht. We provide examples showing that the conclusion of the theorem does not hold in , n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in whose envelopes are multiply sheeted.