The heat flow and harmonic maps between complete manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application, we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing energy density.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

Asymptotic behaviour of families of real curves

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case. Received: 5 May 1998 / Revised version: 20 March 1999     

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space. Received: 25 January 2001 / Accepted: 23 February 2001 / Published online: 25 June 2001     

On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities. (Received 2 March 1999)     

Solvability of the Dirichlet Problem for the Poisson Equation on Some Noncompact Riemannian Manifolds

The behavior of solutions of the Poisson equation on noncompact Riemannian manifolds of a special form is studied. Sharp conditions for the unique solvability of the Dirichlet problem on the reconstruction of solutions of the Poisson equation from continuous boundary data at infinity are found.     

Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten

In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of R _n(n ? 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R ₃ with vanishing Dirichlet or Neumann boundary condition.     

On the Stokes and Navier-Stokes System for Domains with Noncompact Boundary in Lq-spaces

We consider the L^q-theory of weak solutions of the Stokes and Navier-Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRⁿ, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut-off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.     

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S ¹-valued function defined on the boundary of a bounded regular domain of R ⁿ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S ¹-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension. Received December 3, 1998 / final version received May 10, 1999     

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmüller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmüller space AT (X). The work was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science Foundation of China (Grant No. 10401036).     

A Dirichlet problem for harmonic forms inC 1-domains

A Dirichlet problem for harmonic forms in openC ¹-domains is here investigated with boundary data of classL ^p .Work partially supported by MURST 60% and 40%.     

A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann

This paper is the last of a series devoted to the solution of Alexandrov’s problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that isometries of a geodesically complete connected at infinity proper Busemann space X are characterized as follows: If a bijection f: X → X and its inverse f ⁻¹ preserve distance 1, then f is an isometry.     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Désingularisation de métriques d’Einstein. I

We find a new obstruction for a real Einstein 4-orbifold with an A ₁-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

We show that the Dirichlet problem for the minimal hypersurface equation defined on arbitrary C ² bounded domain Ω of an arbitrary complete Riemannian manifold M is solvable if the oscillation of the boundary data is bounded by a function \({\mathcal{C}}\) that is explicitely given and that depends only on the first and second derivatives of the boundary data as well as the second fundamental form of the boundary \({\partial\Omega}\) and the Ricci curvature of the ambient space M. This result extends Theorem 2 of Jenkins-Serrin (J Reine Angew Math 229:170–187,1968) about the solvability of the Dirichlet problem for the minimal hypersurface equation defined on bounded domains of the Euclidean space. We deduce that the Dirichlet problem for the minimal hypersurface equation is solvable for any continuous boundary data on a mean convex domain. We also show existence and uniqueness of the Dirichlet problem with boundary data at infinity—exterior Dirichlet problem—on Hadamard manifolds.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x| ≥ 1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x| ≤ 1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N≥3. Solutions are sought in a space E ¹(U) of finite energy functions that decay at infinity. This space contains H ¹(U) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis.     

A note on weighted Banach spaces of holomorphic functions

For every open subset G of and for every continuous, strictly positive weight v on G, the Banach space of all the holomorphic functions f on G such that vanishes at infinity on G, endowed with the natural weighted sup-norm, is isomorphic to a closed subspace of the Banach space c₀; hence it is reflexive if and only if it is finite dimensional.Received: 30 September 2002     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999