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We obtain an Euclidean volume growth results for complete Riemannianmanifolds satisfying a Euclidean Sobolev inequality and a spectral typecondition on the Ricci curvature. We also obtain eigenvalue estimates, heatkernel estimates, Betti number estimates for closed manifolds whose Riccicurvature is controlled in the Kato class.

Geometric inequalities for manifolds with Ricci curvature in the Kato class