Heat flow for p-harmonic maps with small initial data

We study developing singularities for surfaces of rotation with free boundaries and evolving under volume-preserving mean curvature flow. We show that singularities form a finite, discrete set along the axis of rotation. We prove a monotonicity formula and conclude that type I singularities are asymtotically cylindrical.     

Uniqueness of harmonic maps with small energy

Harmonic maps from B ₁ (0, ℝ³) to a smooth compact target manifold N with uniformly small scaled energy (see assumption (2) below) are shown to be unique for their boundary values. Received: 12 May 1997     

A comparison of the eigenvalues of the Dirac and Laplace operators on a two-dimensional torus

We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples. Received: 10 December 1998     

A Poincaré type inequality for Hessian integrals

In this paper we show that Hessian integrals , , can be estimated by those of higher order. The result extends a variant of the Poincaré inequality corresponding to the cases . The proof depends on solving a related non-linear parabolic initial boundary value problem. Received January 15, 1997 / Accepted March 1997     

On SS-rigid groups and A. Weil's criterion for local rigidity. I

The fundamental group Γ of a compact complete affine manifold is represented as an affine crystallographic subgroup of . L.S. Auslander conjectured that Γ is virtually solvable. Our purpose is to find the algebraic condition on Γ which leads affirmative answer to the conjecture. Received: 26 May 1997 / Revised version: 17 December 1997     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998     

Volume preserving flows with cyclic winding numbers groups and without periodic orbits on 3-manifolds

We construct examples of volume preserving non-singular C ¹ vector fields on closed orientable 3-manifolds, which have cyclic winding numbers groups with respect to the preserved volume element, but have no periodic orbits. Received: 17 January 1998 / Revised version: 31 March 1998     

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     

Sur l'erreur d'approximation par éléments finis en utilisant la méthode des plaquettes splines

Résumé. On établit des majorations de l'erreur d'approximation par éléments finis à partir de données de Lagrange pour des fonctions appartenant à un espace de Sobolev d'ordre convenable, lorsque les degrés de liberté sont approchés à l'aide de la méthode des plaquettes splines introduite par A. Le Méhauté (cf. [13], [14], [15]). Les résultats obtenus s'appliquent notamment à la construction de surfaces de classe . Received May 29, 1995 / Revised version received August 20, 1995     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

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     

Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 1997