Ricerc@Sapienza

We will investigate the local topological rigidity fenomenon for 3 dimensional Riemannian manifolds under restraints of the volume entropy. A Riemannian manifold M is said to be locally topologically rigid if any other manifold M' sufficiently close to M with respect to the Gromov-Hausdorff distance is diffeomorphic to M. This property is too strong to hold in the class of Riemannian manifolds without further assumptions, hence we restrict ourselves to the class of Riemannian 3-manifolds whose volume entropy is bounded from above.
Local topological rigidity has already been investigated by Cheeger and Colding, who proved it for Riemannian manifolds of any dimension with a uniform lower bound on the Ricci curvature.
The volume entropy can be thought as a weak, large scale, version of the curvature, hence the upper bound on the entropy is a natural weakening of the (negative) lower bound on the Ricci curvature, this motivates the assumption of the upper bound on the entropy. In this setting only large scale information is provided by the assumption so one needs to recover topological information on a small scale by other means. This is the reason why we restrict to 3 dimensional manifolds.
In this setting partial results are already known: in a recent paper due to Cerocchi and Sambusetti the local topological rigidity is shown to hold for torsionless non-geometric Riemannian 3-manifolds under the assumption of the volume entropy upper bound.
For the other eight geometric cases instead the question is still open, and to be investigated in this project.
In particular we will prove that local topological rigidity holds for torsionless 3-manifolds of hyperbolic type, due to their very rich geometry. On the other hand, for less negatively curved geometries, we will try to recover weaker results, or the same result under stronger assumptions. It is an interesting open question that will be explored to determine which hypothesis are the most adapt to work in this direction.