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## Compact manifolds with negative part of the Ricci curvature in the Kato class

### Mittwoch, 2020-10-28 11:30Uhr

The Ricci curvature encodes much geometric and analytic information of the underlying Riemannian manifold. For classes of compact Riemannian manifolds with a prescribed uniform lower bound on the Ricci curvature and an upper bounded diameter, quantitative estimates on the eigenvalues of the Laplace-Beltrami operator, the associated heat kernel, or on the isoperimetric constants can be derived. The resulting estimates depend heavily on the prescribed lower bound of the Ricci curvature. In view of geometric ows where metrics are deformed and the Ricci curvature possibly develops singularities, the obtained estimates become valueless even if there is only a small region where the singularity appears. For this reason, people became interested in relaxing the uniform lower Ricci curvature bound assumptions to integral conditions on the negative part of the Ricci curvature. Besides the commonly imposed $L^p$-assumptions, a part of my research is focussed on the implications of the even more general Kato condition, which appears naturally, e.g., in the theory of the Ricci ow. In this talk, I will present geometric and analytic properties of classes of compact Riemannian manifolds whose negative part of the Ricci curvature satisfies such a Kato condition and relate the results to recent work on manifolds with $L^p$-Ricci curvature assumptions.