Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry

Weak collapsing and geometrisation of aspherical 3-manifolds

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds.Comment: Improved version in Englis

Variétés hyperboliques de petit volume [d'après D. Gabai, R. Meyerhoff, P. Milley, ...]

International audienc

Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions

We give examples of open 3-manifolds and 3-orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2-suborbifolds) with nonnegative Euler characteristic.Comment: 17 pages, 5 figure