The asymptotic Plateau problem in Gromov hyperbolic manifolds

We solve the asymptotic Plateau problem in every Gromov hyperbolic Hadamard manifold (X,g) with bounded geometry. That is, we prove existence of complete (possibly singular) k-dimensional area minimizing surfaces in X with prescribed boundary data at infinity, for a large class of admissible limit sets and for all . The result also holds with respect to any riemannian metric on X which is lipschitz equivalent to g. Received: 23 January 2001 / Accepted: 25 October 2001 Published online: 28 February 2002     

On the exceptional set in Goldbach’s problem in short intervals

Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + X ^θ] which are not a sum of two primes is 《 X ^θ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions. Current address: Department of Mathematics, 20014 University of Turku, Finland. Author’s address: Department of Mathematics, University of London, Royal Holloway, Egham, Surrey TW20 0EX, UK     

Optimal and one-complemented subspaces

Let X be a real Banach space and let V ⊂ X be a closed linear subspace. In [4, Prop. 5] it has been proven that if X is strictly convex, reflexive and smooth and V is an optimal subset of X then V is one-complemented in X. In this note we would like to extend this result to non-smooth Banach spaces. In particular, we show that any existence subspace of c,c _o and l ₁ is one-complemented. Also some results concerning non-smooth Musielak-Orlicz sequence spaces equipped with the Luxemburg norm will be presented. The first author was supported by Polish State Committee for Scientific Research Grant KBN 1 PO3A O10 26. Authors’ addresses: Grzegorz Lewicki, Department of Mathematics, Jagiellonian University, 30-059 Kraków, Reymonta 4, Poland; Giulio Trombetta, Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende, Cosenza, Italy     

Symmetric Groups and Lattices

This paper deals with various problems in lattice theory involving local extrema. In particular, we construct infinite series of highly symmetric spherical 3-designs which include some of the examples constructed in [9] in dimensions 5 and 7. We also construct new types of dual-extreme lattices.Received June 29, 2002; in final form January 14, 2003 Published online May 16, 2003     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds. Received 25 April 2001     

Hypersurfaces of prescribed Gauß curvature in exterior domains

We prove an existence theorem for convex hypersurfaces of prescribed Gau? curvature in the complement of a compact set in Euclidean space which are close to a cone. Received: 23 February 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

On Sums of Units

It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in particular to rings of integers in finite extensions of the rationals. On the other hand there are many infinite algebraic extensions of the rationals in which every integer is a sum of two units.     

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group closely related to the order of the higher K-theory group of the ring of integers in L. Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002     

Growth Series and Random Walks on Some Hyperbolic Graphs

 Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs. Received 19 September 2001; in revised form 23 December 2001