Bonnet-Myers theorem for Kato-type Ricci curvature conditions

11.12.2019, 13:45 Uhr  –  Campus Golm, Haus 9, Raum 2.09.2.22
Forschungsseminar Diskrete Spektraltheorie

Dr. Christian Rose (MPI Leipzig)

The Kato condition on the negative part of the Ricci curvature turned out to be an appropriate generalization of L^p- 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.