加权Laplace在积分Ricci曲率条件下的特征值估计

本文研究了积分Ricci曲率条件下加权Laplace算子的第一特征值估计的问题.利用Bochner公式与加权Reilly公式等处理特征值问题的方法,获得了加权Laplace在积分Ricci曲率条件下第一特征值估计下界的估计.     

带权流形上的纽曼特征值估计

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.     

双漂移拉普拉斯特征值的最优估计（英文）

本文研究了四类双漂移拉普拉斯算子的特征值问题.利用带权Reilly公式,当m-权重Ricci曲率满足一定条件时,得到了紧致带边光滑度量测度空间上四类双漂移拉普拉斯算子的第一非零特征值的最优估计.推广了双调和算子特征值的相应结果.     

正Ricci曲率的紧流形上第一特征值下界的新估计

将研究Ricci曲率以非负常数为下界的紧致黎曼流形上第一(闭的,Dirichlet,或Neumann)特征值下界,并给出第一特征值新的下界估计,以及Ling的估计~([16])一个容易的证明.虽然仍使用Ling的某些方法,但是该文的证明避免了试验函数奇性的产生,并且在很大程度上简化了Ling的计算,这或许提供了估计特征值的一种新方式.     

紧Riemann流形的高阶特征值估计

对Ricci曲率具负下界的紧Riemann流形,本文获得了热方程正解优化的梯度估计及Harnack不等式,证明了高阶特征值下界定量估计的猜想．     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz箅子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界...     

紧致Riemann流形上Laplace算子第一特征值的估计

本文证明的主要定理是:设M是具非负Ricci曲率的紧致黎曼流形,则其Laplace算子之第一特征值-λ₁满足:λ₁≥π²/d²,此处d为M之直径.此估计改进了Yau与Li的新近结果,达到了关于第一特征值的最佳估计.     

Dirac-Witten算子特征值的下界估计

在某些条件下,我们获得了Lorentzian流形中紧致spin类空超曲面（无边或带边）的Dirac-Witten算子特征值的一个优化下界估计.该估计依赖于超曲面的数量曲率、平均曲率以及旋量诱导的能量动量张量.在极限情形下,我们发现类空超曲面或者是极大的且具有正数量曲率的Einstein流形,或者是具有非零常平均曲率的Ricci平坦流形.     

Estimate and monotonicity of the first eigenvalue under the Ricci flow

In this paper, we first derive a monotonicity formula for the first eigenvalue of on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces.     

Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds

In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace–Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.     

Evolution of the first eigenvalue of the clamped plate on manifold along the Ricci flow

ABSTRACT

In this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds.     

First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow

In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator (p ?? 2) along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma??s results [Ann. Glob. Anal. Geom., 29, 287?C292 (2006)].     

First eigenvalue of the <Emphasis Type="Italic">p</Emphasis>-Laplace operator along the Ricci flow

In this article, 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 and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first p-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a p-eigenvalue comparison-type theorem when its Euler characteristic is negative.     

Kähler non-collapsing,eigenvalues and the Calabi flow

We first prove a new compactness theorem of Kähler metrics, which confirms a prediction in [17]. Then we establish several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed for the Kähler–Ricci flow in [22] to obtain several new small energy theorems of the Calabi flow.     

Harnack inequalities for graphs with non-negative Ricci curvature

We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.     

Conformally Flat Algebraic Ricci Solitons on Lie Groups

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segrè type {(1... 1 2)} with a unique eigenvalue (equal to 0).     

Time-dependent Sobolev inequality along the Ricci flow

In this article, we get a time-dependent Sobolev inequality along the Ricci flow in a more general situation than those in Zhang (A uniform Sobolev inequality under Ricci flow. Int Math Res Not IMRN 2007, no 17, Art ID rnm056, 17 pp), Ye (The logarithmic Sobolev inequality along the Ricci flow. arXiv:0707.2424v2) and Hsu (Uniform Sobolev inequalities for manifolds evolving by Ricci flow. arXiv:0708.0893v1) which also generalizes the results of them. As an application of the time-dependent Sobolev inequality, we get a growth of the ratio of non-collapsing along immortal solutions of Ricci flow.     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.