Florent Balacheff

Isosystolic inequalities for optical hypersurfaces

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encloses a volume

Sur la Systole de la Sphère au Voisinage de la Métrique Standard

V

Systolic volume of homology classes

, then it carries a periodic characteristic whose action is at most

A LOCAL OPTIMAL DIASTOLIC INEQUALITY ON THE TWO-SPHERE

\sqrt{V/3}

Volume entropy, systole and stable norm on graphs

. This result is deduced from an interesting dual version of Minkowskis lattice-point theorem: if the origin is the unique integer point in the interior of a planar convex body, the area of its dual body is at least 3/2.

Measurements of Riemannian two-disks and two-spheres

We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth Riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, admits the standard metric g0 as a critical point, although it does not achieve the conjectured global minimum: we show that for each tangent direction to the space of metrics at g0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase.

On the systole of the sphere in the proximity of the standard metric)

Given an integer homology class of a finitely presentable group, the systolic volume quantifies how tight could be a geometric realization of this class. In this paper, we study various aspects of this numerical invariant showing that it is a complex and powerful tool to investigate topological properties of homology classes of finitely presentable groups.

Sur la forme de la boule unité de la norme stable unidimensionnelle

We prove a local optimal inequality on the two-sphere between the area and the diastole — defined by a minimax process on the one-cycle space — in a neighborhood of the singular metric made of two equilateral triangles glued along their boundaries.

Systolic geometry and simplicial complexity for groups

We study some new isoperimetric inequalities on graphs. We etablish a relation between the volume entropy (or asymptotic volume), the systole and the first Betti number of weighted graphs. We also find bounds for the volume, associated to some special measure, of the unit ball for the stable norm of graphs.

Optimalité systolique infinitésimale de l’oscillateur harmonique

We prove that any Riemannian two-sphere with area at most 1 can be continuously mapped onto a tree in a such a way that the topology of fibers is controlled and their length is less than 7.6. This result improves previous estimates and relies on a similar statement for Riemannian two-disks.