First Eigenvalue Of Laplacian Research Articles

Lower bounds for the first eigenvalue of Laplacian on graphs

We establish a lower bound for the first eigenvalue of the Dirichlet Laplacian on locally finite graphs, which extending a previous result. We also provide an improved lower bound for the first eigenvalue for finite graphs with non-negative Ricci curvature.

Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-\Delta+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator $-\Delta+cR$ which generalizes some results of Cao \cite{Cao2007}. For $c=0$, We derive monotonicity of the first eigenvalue of Laplacian which generalizes some results of Ma \cite{Ma2006}. Moreover, we prove that when $(M_{3}, g_{0})$ is a closed three manifold with positive Ricci curvature, the eigenvalue of the Laplacian diverges as $t \rightarrow T$ on a limited maximal time in terval $[0, T)$, which generalizes some results of Cerbo and Fabrizio \cite{Fabrizio2007}.

Eigenvalue problems on Riemannian manifolds with a modified Ricci tensor

In this paper, we study the eigenvalue problems on a Riemannian manifold with a modified Ricci tensor. We obtain some sharp lower bound estimates for the first eigenvalue of Laplacian. We also prove some rigidity theorems for the Riemannian manifold with some suitable conditions.

Sharp upper bound for the first eigenvalue

Let \(M\) be a closed hypersurface in a noncompact rank-1 symmetric space \((\overline{\mathbb{M }}, ds^2)\) with \(-4 \le K_{\overline{\mathbb{M }}} \le -1,\) or in a complete, simply connected Riemannian manifold \(\mathbb M \) such that \(0 \le K_\mathbb{M } \le \delta ^2\) or \(K_\mathbb{M } \le k\) where \(k = -\delta ^2\) or 0. In this paper we give sharp upperbounds for the first eigenvalue of laplacian of \(M\).

Isoperimetric inequalities, the torsion problem, and the first eigenvalue of Laplacian of multiply-connected, compact surfaces with boundary

Isoperimetric inequalities, the torsion problem, and the first eigenvalue of Laplacian of multiply-connected, compact surfaces with boundary

Linear approximation of the first eigenvalue on compact manifolds

For compact, connected Riemannian manifolds with Ricci curvature bounded below by a constant, what is the linear approximation of the first eigenvalue of Laplacian? The answer is presented with computer assisted proof and the result is optimal in certain sense.

A Metric Deformation and the First Eigenvalue of Laplacian on 1-Forms

A Metric Deformation and the First Eigenvalue of Laplacian on 1-Forms

A metric deformation and the first eigenvalue of Laplacian on 1-forms

We search for a higher-dimensional analogue of Calabi’s example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on S n {S^n} , n ≥ 5 {n} \geq 5 , where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on 1 1 -forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an ( S n {S^n} , some metric) to an ( S n − 1 × S 1 {S^{n - 1}} \times {S^1} , some metric).