Renata Grimaldi

Calibrations and isoperimetric profiles

We equip many noncompact nonsimply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly nongeneric property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new results on the profile of surfaces of revolution and some hardware know-how.

Remplissage et surfaces de révolution

The filling function of a Riemannian manifold is either linear or at least quadratic. This fact was originally discovered by M. Gromov in 1985. We address the question of the existence of further obstructions. We give a partial answer: every superadditive and superquadratic function is asymptotic to the filling function of a surface of revolution. A function which furthermore satisfies a natural second-order differential inequation is equal to a filling function.

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

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

The Ends of Manifolds with Bounded Geometry, Linear Growth and Finite Filling Area

We prove that simply connected open manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Croissance lineaire et géométrie bornée

We show that on a noncompact manifold which has ‘finite topology at infinity’, there exists a Riemannian metric with ‘bounded geometry’ and linear growth-type.

Sulla geometria asintotica delle foglie di una foliazione

Pour les feuilles (non-compactes) d’un feuilletageF d’une variété différentiable compacteV ily a une «géométrie riemannienne asymptotique» bien définie à quasiisométrie près.La géométrie euclidiennen-dimensionnelle peut être réaliséc comme une telle géométrie asymptotique.

On systolic growth-type

Abstract On a noncompact riemannian product ( W n − 1 × S 1 , ℷ 1 ⊕ ℷ 2 ) we construct a new “warped product” type metric ℷ with the same growth-type as ℷ 1 ⊕ ℷ 2 but with arbitrarily high systolic growth.