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.

Quantitative Sobolev Extensions and the Neumann Heat Kernel for Integral Ricci Curvature Conditions

“…Moreover, such estimates give the opportunity to provide quantitative lower bounds on the first (non-zero) Neumann eigenvalue of (r , H , K )-regular subsets. Recently, the third author and G. Wei obtained in [30] gradient and Neumann eigenvalue estimates assuming only the interior rolling r -ball condition, a lower bound on the second fundamental form, and a Kato-type condition on the negative part of the Ricci curvature defined by the Neumann heat semigroup. It is known that the latter condition is more general than assuming κ M ( p, R) is small.…”

“…It is known that the latter condition is more general than assuming κ M ( p, R) is small. We refer to [4,5,26,27,29,30] for more information about Kato-type curvature assumptions. While those results are very general, it is hard to check that the assumptions are indeed satisfied for compact manifolds with boundary and small κ M ( p, R).…”

Quantitative Sobolev extensions and the Neumann heat kernel for integral Ricci curvature conditions

“…Moreover, such estimates give the opportunity to provide quantitative lower bounds on the first (non-zero) Neumann eigenvalue of (r, H, K)-regular subsets. Recently, the third author and G. Wei obtained in [RW20] gradient and Neumann eigenvalue estimates assuming only the interior rolling r-ball condition, a lower bound on the second fundamental form, and a Katotype condition on the negative part of the Ricci curvature defined by the Neumann heat semigroup. It is known that the latter condition is more general than assuming κ M (p, R) is small.…”

“…Proof of Corollary 1.5. According to [RW20], there exists an explicit constant ε = ε(n, r, H, K) > 0 such that if…”

Manifolds with bounded integral curvature and no positive eigenvalue lower bounds

“…Estimates (1.6) and (1.7) recover (1.2) and (1.3) in the limit where Ric ≥ 0 and Ric ≥ (n − 1)K > 0, respectively, so they are in that sense sharp. Further extensions of (1.7) to a Kato-type condition on the Ricci curvature (an even weaker curvature assumption) recently appeared in the work of Rose-Wei [14].…”