44 research outputs found
Measurements of Riemannian two-disks and two-spheres
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
Stable norms of non-orientable surfaces
We study the stable norm on the first homology of a closed, non-orientable
surface equipped with a Riemannian metric. We prove that in every conformal
class there exists a metric whose stable norm is polyhedral. Furthermore the
stable norm is never strictly convex if the first Betti number of the surface
is greater than two
(On the systole of the sphere in the proximity of the standard metric)
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 g 0 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 g 0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increas
Systolic geometry and simplicial complexity for groups
Twenty years ago Gromov asked about how large is the set of isomorphism
classes of groups whose systolic area is bounded from above. This article
introduces a new combinatorial invariant for finitely presentable groups called
{\it simplicial complexity} that allows to obtain a quite satisfactory answer
to his question. Using this new complexity, we also derive new results on
systolic area for groups that specify its topological behaviour.Comment: 35 pages, 9 figure
Minimal length product over homology bases of manifolds
Minkowski's second theorem can be stated as an inequality for -dimensional
flat Finsler tori relating the volume and the minimal product of the lengths of
closed geodesics which form a homology basis. In this paper we show how this
fundamental result can be promoted to a principle holding for a larger class of
Finsler manifolds. This includes manifolds for which first Betti number and
dimension do no necessarily coincide, a prime example being the case of
surfaces. This class of manifolds is described by a non-vanishing condition for
the hyperdeterminant reduced modulo of the multilinear map induced by the
fundamental class of the manifold on its first -cohomology group
using the cup product.Comment: 24 page