Zhong–Yang type eigenvalue estimate with integral curvature condition

Olivé, Xavier Ramos; Seto, Shoo; Wei, Guofang; Zhang, Qi S.

doi:10.1007/s00209-019-02448-w

Cited by 11 publications

(10 citation statements)

References 24 publications

(21 reference statements)

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“…We show the necessary comparison estimate for the eigenfunction w and the function J depending on the Kato condition. Then we apply to V = 2(τ − 1)ρ − and proving Proposition 2.5 and 2.9, replacing Propositions 2.2 and 2.3 in [ROSWZ19],…”

Section: Proofs Of Theorem 11mentioning

confidence: 90%

“…With those choices, we can finish the proof of the proposition as in [ROSWZ19]. Consider the function w 2 := w w−1 .…”

Section: Proofs Of Theorem 11mentioning

confidence: 95%

“…In this paper, we prove all the eigenvalue estimates mentioned above assuming only a Kato-type condition on the negative part of Ricci curvature. First, we generalize the Zhong-Yang type eigenvalue estimate obtained in [ROSWZ19] to the Kato condition.…”

Section: Another Generalization Of Pointwise Ricci Curvature Lower Bo...mentioning

confidence: 99%

“…The proof of Theorem 1.1 is adapted from [ROSWZ19]. In [ROSWZ19] the key estimate is to control the auxilary function J which is the solution of the following problem,…”

Section: Proofs Of Theorem 11mentioning

confidence: 99%

“…Moreover, Aubry showed an optimal Lichnerowicz-type estimate [Aub07]. Recently, in [ROSWZ19] an optimal Zhong-Yang type estimate was also obtained.…”

Section: Introductionmentioning

confidence: 98%

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Eigenvalue estimates for Kato-type Ricci curvature conditions

Rose,

Wei

2020

Preprint

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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.

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Section: Proofs Of Theorem 11mentioning

confidence: 90%

“…With those choices, we can finish the proof of the proposition as in [ROSWZ19]. Consider the function w 2 := w w−1 .…”

Section: Proofs Of Theorem 11mentioning

confidence: 95%

Section: Another Generalization Of Pointwise Ricci Curvature Lower Bo...mentioning

confidence: 99%

“…The proof of Theorem 1.1 is adapted from [ROSWZ19]. In [ROSWZ19] the key estimate is to control the auxilary function J which is the solution of the following problem,…”

Section: Proofs Of Theorem 11mentioning

confidence: 99%

“…Moreover, Aubry showed an optimal Lichnerowicz-type estimate [Aub07]. Recently, in [ROSWZ19] an optimal Zhong-Yang type estimate was also obtained.…”

Section: Introductionmentioning

confidence: 98%

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Eigenvalue estimates for Kato-type Ricci curvature conditions

Rose,

Wei

2020

Preprint

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Comparison Geometry for Integral Radial Bakry-Émery Ricci Tensor Bounds

2021

Potential Anal

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

2022

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We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative estimate to obtain uniform Neumann heat kernel upper bounds and gradient estimates for positive solutions of the Neumann heat equation assuming integral Ricci curvature conditions and geometric conditions on the possibly non-convex boundary. Those estimates also imply quantitative lower bounds on the first Neumann eigenvalue of the considered domains.

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Zhong–Yang type eigenvalue estimate with integral curvature condition

Abstract: We prove a sharp Zhong-Yang type eigenvalue lower bound for closed Riemannian manifolds with control on integral Ricci curvature.

Cited by 11 publications

References 24 publications

Eigenvalue estimates for Kato-type Ricci curvature conditions

Eigenvalue estimates for Kato-type Ricci curvature conditions

Comparison Geometry for Integral Radial Bakry-Émery Ricci Tensor Bounds

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

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