Related papers: The Drift Laplacian and Hermitian Geometry

Let $M^{4n}$ be a complete quaternionic K\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\lambda_1(M)$ of the Laplacian which is $\lambda_1(M)\le (2n+1)^2$. If the…

A long-standing conjecture in non-K\"ahler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be K\"ahler when $c\neq 0$ and must be…

In analogy with classical results in Riemannian geometry, we establish estimates for the first eigenvalue of the Laplace-de Rham operator on complete balanced Hermitian manifolds in terms of either the holomorphic Ricci curvature or the…

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…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjoestrand asymptotic law, and can be viewed as a CR version of Korevaar's…

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of…

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on…

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\Omega\subset M$ be a suitable domain, and let $\lambda(\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove several…

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a…

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the $(1,1)$ curvature form of the Levi-Civita connection represents the first Aeppli-Chern…

A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(1)Sp(1) component of the qc-Ricci curvature on a compact seven dimensional quaternionic contact manifold is…

We provide a lower bound for the first eigenvalue of the Laplace-Beltrami operator on a closed orientable hypersurface minimally embedded in an orientable compact Riemannian manifold with Ricci curvature bounded below by a positive…

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the…

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We…

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\Sigma$ does not separate $M$ then $\Sigma$ is totally…

Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1…

Let $\Omega$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted…

We show that for n dimensional manifolds whose the Ricci curvature is greater or equal to n-1 and for k in {1,...,n+1}, the k-th eigenvalue for the Laplacian is close to n if and only if the manifold contains a subset which is…