Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

Zeta functions of three-dimensional <Emphasis Type="Italic">p</Emphasis>-adic Lie algebras

We give an explicit formula for the subalgebra zeta function of a general three-dimensional Lie algebra over the p-adic integers . To this end, we associate to such a Lie algebra a ternary quadratic form over . The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form. We acknowledge support from the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation.     

Anisotropic version of a theorem of H. Hopf

Given a positive function F on S ² which satisfies a convexity condition, we define a function for surfaces in which is a generalization of the usual mean curvature function. We prove that an immersed topological sphere in with = constant is the Wulff shape, up to translations and homotheties.      

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Semilinear subelliptic problems with critical growth on Carnot groups

We prove existence and multiplicity of solutions for the semilinear subelliptic problem with critical growth in Ω, u = 0 on ∂Ω, where is a sublaplacian on a Carnot group , 2* = 2Q/(Q − 2) is the critical Sobolev exponent for and Ω is a bounded domain of .     

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric g _h on given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (). By following the ideas and techniques by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group , arXiv:math.DG/0512547) we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot–Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C ² compact, connected, embedded surfaces in () with empty singular set and constant mean curvature H such that is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (). A. Hurtado has been partially supported by MCyT-Feder research project MTM2004-06015-C02-01. C. Rosales has been supported by MCyT-Feder research project MTM2004-01387.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-estimates for Sobolev functions in terms of ∇ and Δ

Several L ^∞-estimates are obtained for in terms of and , where are determined by m. If p = 2, then the estimates are given with explicit constants. However, if p ≠ 2, it is difficult to derive explicit constants except in two simple cases. Applicability to PDE’s is illustrated.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Sous-Groupes Algébriques du Groupe de Cremona

We give a complete classification of maximal algebraic subgroups of the Cremona group Bir() and provide algebraic varieties that parametrize the conjugacy classes.

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Résumé  Résumé Nous donnons une classification complète des sous-groupes algébriques maximaux du groupe de Cremona Bir() et explicitons les variétés qui paramètrent les classes de conjugaison.

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Soutenu par le fonds national suisse de la recherche scientifique (FNRS).     

On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

Density property for hypersurfaces $$UV = P({\bar X})$$

We study hypersurfaces of given by equations of form where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.      

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism , where B _n ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded (N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue (N, k) of (N, k); it is a graded Lie algebra with an action of , and we give a lower bound for the character of its space of generators.      

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

Constant mean curvature hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing spherical building blocks

The techniques developed by Butscher (Gluing constructions amongst constant mean curvature hypersurfaces of ) for constructing constant mean curvature (CMC) hypersurfaces in by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These balancing conditions are analogous to those that must be satisfied in the “classical” context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all.