Related papers: Optimal lower eigenvalue estimates for Hodge-Lapla…

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical…

Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO(M) --> M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept…

In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $\lambda_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general…

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we…

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=…

Let $M$ be an $m (\ge2)$-dimensional closed orientable submanifold in an $n$-dimensional complete simply-connected Riemannian manifold $N$, where the sectional curvature of $N$ is bounded above by $\delta$. When $\delta<0$, inspired by…

In this work, optimal rigidity results for eigenvalues on K\"ahler manifolds with positive Ricci lower bound are established. More precisely, for those K\"ahler manifolds whose first eigenvalue agrees with the Ricci lower bound, we show…

We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a…

Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Writing his approach in terms of…

We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the…

Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields…

We establish inequalities for the eigenvalues of Schr\"odinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related…

In this paper, we establish a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of a complete, non-compact Riemannian manifold with non-negative Ricci curvature.

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of…

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we…

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of dimension $2m+1\geq 5$. We prove that the equality holds iff the manifold is equivalent to…

We prove various comparison theorems of the $i$-th eigenvalue $\lambda_i$ of the Laplacian on fibred Riemannian manifolds by using fiberwise spherical and Euclidean (or hyperbolic) symmetrization. In particular we generalize the…

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies…

Let $(M, g, f, \tau)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2\tau}$ and let $R$ denote its scalar curvature. For a confined function $V$ on $M$, we obtain a lower bound for the lowest eigenvalue of the…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has…