Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.     

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch–Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.

     

The operator-valued Marcinkiewicz multiplier theorem and maximal regularity

Given a closed linear operator on a UMD-space, we characterize maximal regularity of the non-homogeneous problem with periodic boundary conditions in terms of R-boundedness of the resolvent. Here A is not necessarily generator of a -semigroup. As main tool we prove an operator-valued discrete multiplier theorem. We also characterize maximal regularity of the second order problem for periodic, Dirichlet and Neumann boundary conditions. Received: 21 December 2000; in final form: 12 June 2001 / Published online: 1 February 2002     

The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions

Let Ω be a bounded C² domain in ?ⁿ and ? ?Ω → ?^m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?^m with f|_?Ω = ? and with the graph of f a minimal submanifold in ?^n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?^m satisfies 8nδ sup_Ω |D²ψ| + √2 sup_?Ω |Dψ| < 1, then the Dirichlet problem for ψ|_?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc.     

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

On the Existence of Completely Saturated Packings and Completely Reduced Coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W.Kuperberg: every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every >0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than but which also admits a tiling.     

A polynomial dual simplex algorithm for the generalized circulation problem

This paper presents a polynomial-time dual simplex algorithm for the generalized circulation problem. An efficient implementation of this algorithm is given that has a worst-case running time of O(m ²(m+nlogn)logB), where n is the number of nodes, m is the number of arcs and B is the largest integer used to represent the rational gain factors and integral capacities in the network. This running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem. Received: June 1998 / Accepted: June 27, 2001?Published online September 17, 2001     

Submanifolds in de sitter space-time satisfying ΔH=λH

In [3] the author initiated the study of submanifolds whose mean curvature vectorH is an eigenvector of the Laplacian Δ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. The classification of surfaces with ΔH=λH in a Euclidean 3-space was done by the author in 1988. Moreover, in [4] the author classified such submanifolds in hyperbolic spaces. In this article we study this problem for space-like submanifolds of the Minkowski space-timeE ₁ ^m when the submanifolds lie in a de Sitter space-time. As a result, we characterize and classify such submanifolds in de Sitter space-times.     

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

Affine complete locally convex hypersurfaces

An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n+1)-space is Euclidean complete for n≥2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R ³ must be an elliptic paraboloid. Oblatum 16-VI-2001 & 27-II-2002?Published online: 29 April 2002     

The n-centre problem of celestial mechanics for large energies

We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations. Received October 16, 2000 / final version received June 18, 2001?Published online August 15, 2001     

Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·)) ^1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p ^1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t↦p(x+td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a self-concordant barrier for its hypograph, with barrier parameter O(m ²). Our approach is direct, and shows, for example, that the function −mlog(det(·)−1) is an m ²-self-concordant barrier on a natural domain. Such barriers suggest novel interior point approaches to convex programs involving hyperbolic means. Received: December 2, 1999 / Accepted: February 2001?Published online September 3, 2001     

A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem

This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n _s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m _b independent conserved quantities, where m _b=n _s when no reaction is possible and m _b n _e +1 when reaction is possible and n _e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the reaction tangent-plane criterion, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m _b(n _s-1) dimensions) to a finite sequence of local optimization steps inK(n _s-1) -space, K m _b, and global optimization steps in (n _s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.     

Le genre de Mislin des espaces rationnellement équivalents à un produit de deux sphères de dimensions différentes

 Zabrodsky exact sequences are algebraic tools which express the genus set of a space X in term of its self-maps, when X has the rational homotopy type of a co-ℋ-space or an ℋ-space. Explicit examples show these methods can't be generalized to the class of all simply connected finite CW-complexes. We however construct a Zabrodsky exact sequence for those three cells CW-complexes rationally equivalent to the product of two spheres S ^k ×S ⁿ , n>k≥2. We deduce, from results of Morisugi-Oshima, the genus of some spherical bundles. Received: 17 March 2001 / Revised version: 8 August 2001     

Expansions of Dirichlet to Neumann operators under perturbations

We obtain an asymptotic expansion of the Dirichlet to Neumann operator (DNO) for the Dirichlet problem on perturbations of the unit disk. We write our result in terms of pseudodifferential operators which themselves have expansions in the perturbation parameter. For a given power of the perturbation parameter, m > 0, and a given order, n < 0, we give an algorithm which allows for the expansion of the symbol of the DNO up to mth power in the perturbation parameter, with error terms belonging to symbols of order n.     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002