Eigenvalue estimates for Kato-type Ricci curvature conditions

Abstract

We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature. This generalizes all earlier results on $L^p$-curvature assumptions. Moreover, we introduce the Kato condition on compact manifolds with boundary with respect to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat kernel and lower bounds for all Neumann eigenvalues, what provides a first insight in handling variable Ricci curvature assumptions in this case.

21 pages. The proof of Theorem 1.3 is replaced by an easier one

References in corpus (4)

• Geometric inequalities for manifolds with Ricci curvature in the Kato class
• Geometric and spectral estimates based on spectral Ricci curvature assumptions
• Almost positive Ricci curvature in Kato sense -- an extension of Myers' theorem
• Manifolds with Ricci curvature in the Kato class: heat kernel bounds and applications