Related papers: Eigenvalue estimates for 3-Sasaki structures

We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the…

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the…

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we…

Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues…

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…

Let $\Sigma$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $\Lambda=\max\limits_{\Sigma}|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the…

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the…

For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the…

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature…

In this paper, we study the first eigenvalue of the Laplace--Beltrami operator on the Lawson minimal surfaces $\xi_{m,k}$ embedded in the unit three-sphere $\mathbb{S}^3$. Motivated by Yau's conjecture on the first eigenvalue of closed…

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric…

Cosmological models where spatial sections are the Poincar\'e dodecahedral space D have been recently invoked to give an account of the lower modes of the angular anisotropies of the cosmic microwave background. Further explorations of this…

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq…

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…

A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51, 2013] for the Laplace operator.…

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging…

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show…

We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. Our results generalize earlier work concerning the degeneration of hyperbolic surfaces.

In this paper, we investigate eigenvalues of Laplacian on a bounded domain in an $n$-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to A. D. Melas [15].…

This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented.…