Dr. Christian Rose (MPI Leipzig)
The Kato condition on the negative part of the Ricci curvature turned out to be an appropriate generalization of Lp- curvature conditions that can be used to investigate geometric and topological properties of compact Riemannian manifolds using perturbation theory of Dirichlet forms. A question that seems to be a hard task is whether the Kato condition on the Ricci curvature below a positive threshold of a complete Riemannian manifold already implies its compactness. This would be generalizing the classical Myers theorem, stating that a Riemannian manifold with uniformly positive Ricci curvature is compact and its fundamental group is finite. I will show that a complete Riemannian manifold is compact provided the negative part of Ricci curvature below a positive number is in the generalized Kato class in the sense of Stollmann and Voigt and that the manifold has asymptotically non-negative Ricci curvature. The finiteness of the fundamental group follows eventually from other recent results with G. Carron.