Related papers: Zhong-Yang type eigenvalue estimate with integral …

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…

Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound…

We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.

We develop geometric analysis on weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted…

We consider the class of closed Riemannian $n$-manifolds with Ricci curvature and injectivity radius bounded below by uniform constants, and an upper bound on the diameter. We establish a uniform upper bound for the eigenvalues of the Hodge…

The aim of this paper is give a simple proof of some results in \cite{Jun Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the sharp lower bound of the first eigenvalue in the Laplacian operator on compact…

We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.

In this paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature…

We give new estimates on the lower bounds for the first closed or Neumann eigenvalue for a compact manifold with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results improve the previous…

We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on $L^p$ norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian $\lambda_1$.…

On closed Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded from below and bounded gradient of the potential function, we obtain lower bounds for all positive eigenvalues of the Beltrami-Laplacian instead of the drifted…

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and…

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For…

We establish Willmore-type inequalities for bounded domains in complete non-compact Riemannian manifolds, under either asymptotic or integral Ricci curvature bounds. Those results recover a recent inequality of Jin-Yin arXiv:2402.02465.

Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral…

Let $\Sigma$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the…

We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature…

In this note, we extend the rigidity of Cheng-Yau gradient estimate in \cite{HXY} to surfaces with lower Ricci curvature bound. Motivated by these sharp Cheng-Yau gradient estimates, pointwise Cheng-Yau gradient estimates for higher…

We prove that a Bishop-Gromov inequality gives a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.