Analyzing non-degenerate 2-forms with riemannian metrics

We give a variational proof of the harmonicity of a symplectic form with respect to adapted riemannian metrics, show that a non-degenerate 2-form must be Kähler whenever parallel for some riemannian metric, regardless of adaptation, and discuss kählerness under curvature assumptions.     

Vertical sectional curvature and K-contactness

We prove that a contact form with riemannian characteristic flow is K-contact. We also present a purely riemannian hypothesis which implies the existence of a K-contact form with a prescribed unit Killing vector field as characteristic vector field. Our hypothesis is weaker than that previously presented by Hatakeyama, Ogawa and Tanno.     

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.     

The hausdorff dimension of the harmonic class on negatively curved surfaces

We study the regularity of the Hausdorff dimension of the harmonic class of a surface M of negative curvature as a function of the riemannian metric. We prove that it is a C^{r− 3} function of the metric in the Banach manifold of C^r riemannian metrics on M. We also prove regularity results for some asymptotic quantities associated to the Brownian motion on M.     

Applications harmoniques, applications pluriharmoniques et existence de 2-formes parallèles non nulles

In this paper we study harmonic maps from a compact riemannian manifold equiped with a non trivial parallel 2-form, to a K?hler manifold of strongly negative curvature tensor or a riemannian manifold of strictly negative complex sectional curvature. In a first part we set up some rigidity results of Siu type. Then we obtain an upper bound for the rank of such maps in terms of the rank of the 2-form and deduce some vanishing theorems. Received: March 4, 1996     

La topologie à l'infini des variétés à géométrie bornée et croissance linéaire

We study the topology at infinity of a non compact riemannian manifold with bounded geometry and linear growth-type.     

A Kato type inequality for Riemannian submersions with totally geodesic fibers

A Kato type inequality is proved for a riemannian submersion with totally geodesic fibers. It compares such a situation with the trivial fibration and generalizes analogous results for vector bundles.     

Exponential maps of Sobolev metrics on loop groups

We find conditions for the exponential map on a weak riemannian Hilbert manifold to be a nonlinear Fredholm map of index zero and apply the result to left-invariant Sobolev metrics on loop groups.

     

Gradient estimates for potentials of invertible gradient–mappings on the sphere

McCann showed that, if the potential of a gradient-mapping, on a compact riemannian manifold, is c-convex, the length of its gradient cannot exceed the diameter of the manifold. We improve this bound in two different manners on the constant curvature spheres, under assumptions on the relative density of the image-measure of the riemannian volume. One proof, with the standard metric, relies on the Brenier–McCann optimal measure-transport property; the other, purely pde, ignores it. This work is thought of as a preliminary step toward a second derivatives estimate which would yield smooth optimal measure-transport (open). Mathematics Subject Classification (2000) Primary 35J60, 35B45; Secondary 28C99, 49Q99     

Sur la cohomologie LP des variétés

The main purpose of this Note is to indicate vanishing and non vanishing theorems for L^p cohomology of riemannian manifolds, especially when the dimension is half the dimension of Riemannian symmetric space.     

Cut locus of the B-spheres

This paper studies the geometry of a family of homogeneous riemannian metrics on odd-dimensional spheres. For some metrics of the family the cut locus coincides with the first conjugate locus. For the others these two sets do not coincide but intersect.     

Probleme aux limites pour les hypersurfaces maximales lorentziennes

We prove in this paper the existence of maximal spatial submanifolds with prescribed boundary conditions in a lorentzian manifoldV=R×U whereU is a complete riemannian manifold.     

On the curvature of homogeneous Kähler metrics of bounded domains

Summary There is an infinite sequence of conditions {H_k} that the curvature tensor of a harmonic space must satisfy. The first one H₁ is the Einstein condition, which is satisfied by a large number of non-harmonic riemannian manifolds. On the other hand, the conditions H_k, for k >1,seem to be much more restrictive. In this paper, we examine conditions H ₁ and H ₂ on the riemannian manifolds obtained endowing a bounded domain inC ⁿ with all possible homogeneous Kähler metrics.     

Inégalités de hardy sur les variétés riemanniennes non-compactes

We give a criterium and its applications in order that a riemannian manifold satisfy a inequality similar to Hardy's inequality on Rⁿ.     

Two-dimensional geodesic flows having first integrals of higher degree

We present a family of riemannian metrics on two-sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2. They also have the property that all geodesics are closed. Received January 18, 1998 / Published online March 12, 2001     

Integration by parts formulas and dilatation vector fields on elliptic probability spaces

Summary. By coupling two arbitrary riemannian connections Γ and Γ˜ on a riemannian manifold M, we perform the stochastic calculus of ɛ-variation on the path space P(M) of the manifold M. The method uses direct calculations on Ito’s stochastic differential equations. In this context, we obtain intertwinning formulas with the Ito map for first order operators on the path space P(M) of M. By a judicious choice of the second connection Γ˜ in terms of the connection Γ, we can prolongate the intertwinning formulas to second order differential operators. Thus, we obtain expressions of heat operators on the path space P(M) of a riemannian manifold M endowed with an arbitrary connection. The integration by parts of the laplacians on P(M) leads us to the notion of dilatation vector field on the path space. Received: 18 April 1995 / In revised form: 18 March 1996     

A Vanishing Theorem for the Heat Content of a Riemannian Manifold

The paper considers the asymptotic behaviour of the heat contentof a riemannian manifold M, when Dirichlet conditions are imposedon a small compact subset of M. 2000 Mathematics Subject Classification58J35, 35K05.     

The relative volume growth of minimal submanifolds

The volume growth of certain well-defined subsets of minimal submanifolds in riemannian spaces are compared with the volume growth of balls and spheres in space forms of constant curvature. Work partially supported by a DGICYT Grant No. PB97-1425 and Fundacio Caixa Castello Grant No. 0I027.     

Almost invariant submanifolds for compact group actions

We define a C ¹ distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C ¹-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros of sections of extended normal bundles. Received September 14, 1999 / final version received November 29, 1999     

The Exponential Map on the Free Loop Space is Fredholm

We study the exponential map on Hilbert riemannian manifolds and show that on the free loop space it is a nonlinear Fredholm map of index zero. Among corollaries we obtain that only one type of conjugate points exists in . This answers a question of W. Klingenberg. Submitted: December 1996, revised version: March 1997.