Geometric and spectral estimates based on spectral Ricci curvature assumptions

Classical results in differential geometry such as the Lichnerowicz and Bonnet-Myers theorems or isoperimetric estimates relate the Ricci curvature of a manifold to its analytic and topological properties. Originally, those estimates rely on sharp Ricci curvature lower bounds, and during the last years they have been generalized to integral curvature bounds. This talk will consider even more general Ricci curvature assumptions implying generalizations of the classical estimates. Namely, we show that, in a certain sense, relative boundedness of the Ricci curvature suffices to prove a Lichnerowicz and Bonnet-Myers type theorem. If time allows, we will also discuss isoperimetric estimates based on Kato-type assumptions.