Related papers: Eigenvalues under the backward Ricci flow on local…

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…

Let $\lambda(t)$ be the first eigenvalue of $-\Delta+aR\, (a>0)$ under the backward Ricci flow on locally homogeneous 3-manifolds, where $R$ is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of $\lambda(t)$. In…

Let $\Delta_\varphi = \Delta -\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $d\mu = e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the…

We derive, under a technical assumption, the first variation formula for the eigenvalues of the Laplacian on a closed manifold evolving by the Ricci flow and give some applications.

In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing…

In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar…

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $d\mu=e^{-\phi(x)}d\nu$ be the weighted measure and $\Delta_{p,\phi}$ be the weighted $p$-Laplacian. In this article we will investigate monotonicity for the first…

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…

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…

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding…

In this paper, we prove that the first eigenvalues of $-\Delta + cR$ ($c\geq \frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.

We study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate.

In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near…

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…

In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic…

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that…

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first…

On closed Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded from below and bounded gradient of the potential function, we obtain lower bounds for all positive eigenvalues of the Beltrami-Laplacian instead of the drifted…

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…

We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also…