Related papers: Eigenvalue Estimate for the basic Laplacian on man…

We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The…

In this paper, as a continuation of [30], we consider the Gromov-Hausdorff convergence and collapsing in the family of compact Riemannian manifolds with boundary satisfying lower bounds on the sectional curvatures of interior manifolds,…

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct…

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected…

We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero…

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions…

We study an eigenvalue problem for the Laplacian on a compact K\"{a}hler manifold. Considering the $k$-th eigenvalue $\lambda_{k}$ as a functional on the space of K\"{a}hler metrics with fixed volume on a compact complex manifold, we…

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) $L_\sigma$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $\sigma dv_g$. First, we obtain upper…

In this short survey, we derive some weyl-type universal inequalities of eigenvalues of the Laplacian on a closed Riemannian manifold of nonnegative Ricci curvature. We also give upper bounds for the $L_{\infty}$ norm of eigenfunctions of…

Upper bounds of the first non-trivial eigenvalue $\lambda_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $\lambda_1$ depends on…

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…

On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on $p$-form for $0<p<n$. In particular, we prescribe the multiplicity of the first eigenvalues.…

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$…

We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded…

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.

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta\in\R\setminus\{0\}$ the Robin parameter), we consider a shape minimization problem for a function of the first eigenvalues if $\beta>0$ and a shape…

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest…

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic,…