Local limit theorems for free groups

In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map . This is obtained via an analysis involving subshifts of finite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus is also discussed. Received: 20 December 2000 / Revised version: 23 March 2001 / Published online: 24 September 2001     

A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.     

Wedge removability of metrically thin sets and application to the CR-meromorphic extension

We give a wedge removability theorem for metrically thin sets of two codimensional Hausdorff null measure. Following [22], this removability theorem combined with the wedge removability theorem of [21] for closed subsets of two codimensional manifolds, gives a CR-meromorphic extension theorem in the greater codimensional case. Received: 28 August 1999; in final form: 10 April 2000 / Published online: 17 May 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     

Revisiting Hardy's theorem for the Heisenberg group

We establish several versions of Hardy's theorem for the Fourier transform on the Heisenberg group. Let be the Fourier transform of a function f on and assume where is the heat kernel associated to the sublaplacian. We show that if then whenever . When we replace the condition on f by where is the Fourier transform of f in the t-variable. Under suitable assumptions on the ‘spherical harmonic coefficients’ of we prove: (i) when a=b; (ii) when a > b there are infinitely many linearly independent functions f satisfying both conditions on and . Received: 9 January 2001 / in final form: 17 April 2001 Published online: 1 February 2002     

The $H^{\infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001     

A Bernstein theorem for special Lagrangian graphs

We obtain a Bernstein theorem for special Lagrangian graphs in for arbitrary n only assuming bounded slope but no quantitative restriction. Received: 18 January 2001 / Accepted: 7 June 2001 / Published online: 12 October 2001 The second-named author is grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and support and also 973 project in China.     

Rational points on hyperelliptic curves and an explicit Weierstrass preparation theorem

 By using the so-called elliptic curve Chabauty method, N. Bruin [1], V. Flynn and J. Wetherell [6] have extended Chabauty's method to some cases where the rank of the Jacobian may not be less than the genus. The main tool in these methods is a theorem of Strassman on p-adic zeros of power series in one variable, and is applicable only if certain Jacobians are of rank less than or equal to 1. In the present paper, we give an explicit generalization of Strassman's theorem to several variables, enabling us to treat cases where the rank is greater than 1. We apply this to find all the rational points on a hyperelliptic curve of rank and genus equal to 4. Received: 2 January 2001     

Geometry and algebra of real forms of complex curves

Let Y be a complex algebraic curve and let be the set of all real algebraic curves with complexification , such that the real points divide . We find all such families [Y]. According to Harnak theorem a number of connected components of satisfies by the inequality , where g is the genus of Y. We prove that and these estimates are exact. Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002     

Log-concave and spherical models in isoperimetry

We derive several functional forms of isoperimetric inequalities, in the case of concave isoperimetric profile. In particular, we answer the question of a canonical and sharp functional form of the Lévy—Schmidt theorem on spheres. We use these results to derive a comparison theorem for product measures: the isoperimetric function of is bounded from below in terms of the isoperimetric functions of . We apply this to measures with finite dimensional isoperimetric behaviors. All the previous estimates can be improved when uniform enlargement is considered. Submitted: December 2000, Revised: May 2001.     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001     

Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol’nikov, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver’s theorem. Oblatum 17-IV-2001 & 12-IX-2001?Published online: 19 November 2001 An erratum to this article is available at .     

Pointwise theorems for amenable groups

In this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along F?lner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups. We also prove a generalization of the Shannon-McMillan-Breiman theorem to all discrete amenable groups. --> Oblatum 10-I-2000 & 9-V-2001?Published online: 13 August 2001     

An effective criterion for ballistic behavior of random walks in random environment

 We investigate multi-dimensional random walks in random environment. Specifically, we provide an effective criterion which can be checked by inspection of the environment in a finite box, and implies a ballistic strong law of large numbers, a central limit theorem and several large deviation controls. With the help of this criterion, we provide an example of a ballistic multi-dimensional walk which does not fulfill the criterion introduced by Kalikow in [4]. The present work complements the results of [9], where a certain condition (T) was introduced, and confirms the interest of this condition. Received: 5 October 2000/ Revised version: 24 May 2001 Published online: 22 February 2002     

On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori

Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. In particular, for any marking the existence of limits of the quadratic growth rate is shown using elementary methods (avoiding Ratner's theorem). We study the growth rate limit as function of the marking. Precise formulas for this function in the case of two marked points are given and the sets where the growth rate function is maximal and continuous are described. For rational two markings the index of the Veech group in is calculated in two different ways. Submitted: August 2001, Revised: March 2002.     

Martingales and large deviations for binary search trees

We establish an almost sure large deviations theorem for the depth of the external nodes of binary search trees (BSTs). To achieve this, a parametric family of martingales is introduced. This family also allows us to get asymptotic results on the number of external nodes at deepest level. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 112–127, 2001     

A limit theorem for sequences generated by Weyl transformation: Disappearance of dependence

 We prove a limit theorem for certain pseudo-random sequences generated by Weyl transformations, thereby extending a result of Sugita in [3]. Received: 8 June 2001 / Revised version: 4 March 2002 / Published online: 22 August 2002     

Optimal Lipschitz extensions and the infinity laplacian

We reconsider in this paper boundary value problems for “infinity Laplacian” PDE and the relationships with optimal Lipschitz extensions of the boundary data. fairly elegant new proofs, which clarify and simplify previous work, and may be characterized by a comparison principle with appropriate cones. We in comparison with cones directly implies the variational principle associated Liouville theorem for subsolutions bounded above by planes. Received: 29 May 2000 / Accepted: 12 June 2000 / Published online: 23 April 2001     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

The scalar curvature problem on S^n: an approach via Morse theory

We prove the existence of positive solutions for the equation on , where is the Laplace-Beltrami operator on is the critical Sobolev exponent, and is a small parameter. The problem can be reduced to a finite dimensional study which is performed via Morse theory. Received: 29 September 1999 / Accepted: 11 May 2001 / Published online: 19 October 2001