The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature

Geometric inequalities for manifolds with Ricci curvature in the Kato class

“…Recents papers have emphasized how a control on the Kato constant of the Ricci curvature can be useful in order to control some geometrical quantities for closed or complete Riemannian manifolds ( [5,17,20,25,47,48,49,58,59]). For a closed Riemannian manifold (M, g), we will explain how the works of Qi S. Zhang and M. Zhu [59], together with some classical ideas, can be used in order to obtain geometric and topological estimates based on the Kato constant of the Ricci curvature.…”

Limits of manifolds with a Kato bound on the Ricci curvature

“…Observe that the uniform Kato bound (3) with a function satisfying (4) is guaranteed as soon as one has an appropriate estimate on the L p norm of the Ricci curvature. This is due to C. Rose and P. Stollmann [RS17]: thanks to their work, it is possible to show that if p > n/2 and ε(p, n) is small enough, then the following estimate…”

Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

“…Furthermore, it was shown that under suitable conditions, the Hodge-deRham Laplacian is dominated by a Schrödinger operator generated by the Laplace-Beltrami plus a suitable potential depending on Ricci curvature, as we will use later. Especially in Riemannian geometry, this fact has been used extensively to study geometric and topological properties of manifolds as well as properties of the semigroup and corresponding heat kernel of generalized Schrödinger operators on vector bundles, see [5,9,[12][13][14][22][23][24][25] and the references therein. For a recent survey, see Sect.…”