An estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphs

A note on lower bounds estimates for the Neumann eigenvalues of manifolds with positive Ricci curvature

We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain new lower bounds for the Neumann eigenvalues which are consistent with Weyl's asymptotics.

Ricci膨胀子上漂移拉普拉斯算子的谱

Spectrum of the drift Laplacian on Ricci expanders

该文研究Ricci膨胀子上漂移拉普拉斯算子的谱，证明势函数为真时谱离散且该假设不可去，拓展势函数渐近行为结果，算出刚性膨胀/收缩子上全部特征值，还探究了特定刚性Ricci膨胀子的第二特征值。

Eigenvalue estimates for Beltrami-Laplacian under Bakry-Émery Ricci curvature condition

On closed Riemannian manifolds with Bakry-Émery 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 Laplacian. The lower bound of the $k$th eigenvalue depends on $k$, Bakry-Émery Ricci curvature lower bound, the gradient bound of the potential function, and the dimension and diameter upper bound of the manifold, but the volume of the manifold is not involved. Especially, these results apply to closed manifolds with Ricci curvature bounded from below.

Rigidity phenomena on lower $N$-weighted Ricci curvature bounds with $\varepsilon$-range for non-symmetric Laplacian

Lu-Minguzzi-Ohta have introduced the notion of a lower $N$-weighted Ricci curvature bound with $\varepsilon$-range, and derived several comparison geometric estimates from a Laplacian comparison theorem for weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for non-symmetric Laplacian induced from vector field potential.

Gradient estimates, Bakry-Emery Ricci curvature and ellipticity for unbounded graph Laplacians

In this article, we prove gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians which satisfy an ellipticity assumption. As applications, we study completeness and finiteness of stochastically complete graphs under Bakry-Emery curvature bounds.

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows.

Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of \cite{Oden-Sung-Wang99} to $L^p$-Ricci curvature assumptions, $p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest.

Eigenvalues of the Laplacian under the Ricci Flow

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.

Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.

Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs

In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs.

Evolution of the first eigenvalue of weighted $p$-Laplacian along the Ricci-Bourguignon flow

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $dμ=e^{-ϕ(x)}dν$ be the weighted measure and $Δ_{p,ϕ}$ be the weighted $p$-Laplacian. In this article we will investigate monotonicity for the first eigenvalue problem of the weighted $p$-Laplace operator acting on the space of functions along the Ricci-Bourguignon flow on closed Riemannian manifolds. We find the first variation formula for the eigenvalues of the weighted $p$-Laplacian on a closed Riemannian manifold evolving by the Ricci-Bourguignon flow and we obtain various monotonic quantities. At the end we find som…

Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds

In this paper, we consider the non-linear general $p$-Laplacian equation $Δ_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

共形里奇曲率与谱估计

Conformal Ricci Curvature and Spectral Estimates

该研究针对Shen和Ye的共形里奇张量，推导了有界区域第一狄利克雷特征值的优化上界，还得到完备非紧流形底谱等相关精确估计，实现了拉普拉斯算子与共形拉普拉斯算子谱的精确比较。

Zero and negative eigenvalues of the conformal Laplacian

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound o…

具有从下方有界的修正Ricci曲率的完备流形上Witten拉普拉斯算子和加权p-拉普拉斯算子的特征值比较定理

Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below

在具有从下方有界的修正Ricci曲率的完备流形上，建立Witten拉普拉斯算子和加权p-拉普拉斯算子在测地球上的第一狄利克雷特征值的程氏型特征值比较定理。

伪拉普拉斯算子的小特征值

Small eigenvalues of pseudo-Laplacians

本文扩展了Otal-Rosas对超曲面小特征值的界限到伪拉普拉斯算子，并将Colin de Verdière的谱理论扩展到具有多个孔的超曲面。

Conformal mappings and first eigenvalue of Laplacian on surfaces

In this note we give a simple relation between conformal mapping and the first eigenvalue of Laplacian for surfaces in Euclidean spaces.

Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Ba…

On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking Ricci solitons

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Y. Sano in [12], and by B. Andrews and L. Ni in [1]. Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.

积分里奇曲率下p-拉普拉斯算子第一特征值的尖锐下界

Sharp lower bound of the first eigenvalue of the $p$-Laplacian under integral Ricci curvature

该研究针对p≥2的闭流形，在小积分里奇曲率条件下，结合Sobolev截断的全局椭圆估计与一维比较剖面的严格扰动，证明了p-拉普拉斯算子第一非零特征值的尖锐Zhong-Yang型下界。

Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are…

Lower order eigenvalues of the poly-Laplacian with any order on spherical domains

We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.

Sub-Laplacian eigenvalue bounds on CR manifolds

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 bounds for Laplace eigenvalues of conformal metrics.

The Ricci curvature on simplicial complexes

We define the Ricci curvature on simplicial complexes by modifying the definition of the Ricci curvature on graphs, and we prove the upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial complexes using the Ricci curvature.

Lower diameter bounds for compact shrinking Ricci solitons

It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.

Eigenvalues of Weighted-Laplacian under the extended Ricci flow

Let $Δ_φ= Δ-\nabla φ\nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $dμ= e^{-φ} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of $Δ_φ$ and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case $n\geq 3$. Our results are natural extension of some known results …

Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.

Sharp estimate on the first eigenvalue of the p-Laplacian on compact manifold with nonnegative Ricci curvature

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 proof is based on a gradient comparison theorem. We will also charachterize the equality case in the estimate.

Eigenvalues under the Ricci flow of model geometries

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 monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.

具有非负 Ricci 曲率和凸边界的流形上的第一 Steklov 特征值

The first Steklov eigenvalue on manifolds with nonnegative Ricci curvature and convex boundary

本文为具有非负 Ricci 曲率与凸边界的紧致 Riemannian 流形的第一非零 Steklov 特征值建立了新的下界，并在更弱的几何假设下获得了相关结果。

Eigenvalues and entropies under the harmonic-Ricci flow

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 harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eigenvalues of Laplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of the shrinker and expanding entropies of the harmonic-Ricci flow.

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.

Monotonicity of eigenvalues of geometric operaters along the Ricci-Bourguignon flow

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-Δ+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator $-Δ+cR$ which generalizes some results of Cao \cite{Cao2007}. For $c=0$, We derive monotonicity of the first eigenvalue of Laplacian which generalizes some results of Ma \cite{Ma2006}. Moreover, we prove that when $(M_{3}, g_{0})$ is a closed three manifold with positive Ricci curvature, the eigenvalue of the Laplacian diverges as $t \rightarrow T$ on a limited …

Eigenvalue estimates for Kato-type Ricci curvature conditions

We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature. This generalizes all earlier results on $L^p$-curvature assumptions. Moreover, we introduce the Kato condition on compact manifolds with boundary with respect to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat kernel and lower bounds for all Neumann eigenvalues, what provides a first insight in handling variable Ricci curvature assumptions in …

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 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.

First Eigenvalues of Geometric Operators under the Ricci Flow

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

具有常Ricci特征值的曲率非齐次子流形的例子

Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues

该研究构造了两类具有常Ricci特征值的曲率非齐次黎曼流形，分别源自Einstein扭曲积和黎曼Schwarzschild-Tangherlini流形，明确了其特征值数量及等距浸入/嵌入的最小余维数。

Ricci flow and the determinant of the Laplacian on non-compact surfaces

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow conver…

Monotonicity formulas of eigenvalues and energy functionals along the rescaled List's extended Ricci flow

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-functional which can be seen as a generalized $\mathcal{F}$-functional corresponding with steady Ricci breathers, and $\mathcal{W}_k$-functional which generalizes $\mathcal{W}$-functional corresponding with expanding Ricci breathers.

Approximating coarse Ricci curvature on submanifolds of Euclidean space

For an embedded submanifold $Σ\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $Σ$ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [AW19]. Specifically, we prove Proposition 3.2 in [AW19].

Negative eigenvalues of the Ricci operator of solvable metric Lie algebras

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property.

Eigenvalue estimates of the Dirac operator depending on the Ricci tensor

We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.

具有正Ricci下界的Kähler流形上特征值的刚性

The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound

本文研究了正Ricci下界Kähler流形的特征值刚性，证明复射影空间是第一特征值重数最大的唯一流形且存在重数间隙，并在Kähler--Einstein情形下得到近似刚性结果。

关于欧几里得凸域上拉普拉斯算子狄利克雷本征值的一些通用不等式

Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domain

本文建立了欧几里得凸域上拉普拉斯算子狄利克雷本征值的两个通用不等式，探讨了相关数学性质。

Some Asymptotic Behavior of the first Eigenvalue along the Ricci Flow

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.

Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.

关于欧几里得凸域上拉普拉斯算子Neumann特征值分布的注记

A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain

本文研究了欧几里得凸域上拉普拉斯算子Neumann特征值的分布，建立了两个通用不等式。

A Lower Bound of The First Eigenvalue of a Closed Manifold with Positive Ricci Curvature

We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.

Analysis of weighted Laplacian and applications to Ricci solitons

We study both function theoretic and spectral properties of the weighted Laplacian $Δ_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper bound of the bottom spectrum of $Δ_f$ in terms of the lower bound of $Ric_{f}$ and the linear growth rate of $f.$ We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of $f$ is bounded…

Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

$W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the $W$-entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemanni…

3-Manifolds with Constant Ricci Eigenvalues $(λ, λ, 0)$

We consider complete Riemannian $3$-manifolds whose Ricci tensors have constant eigenvalues $(λ, λ, 0)$. When $π_1$ is finitely generated, we classify the topology of such manifolds by showing that they have a free fundamental group if non-trivial and that every free group is obtained. We give a description up to isometry, when the metric is locally irreducible or when it is analytic.

欧氏空间凸域上拉普拉斯算子Neumann特征值的一个普适不等式

A universal inequality for Neumann eigenvalues of the Laplacian on a convex domain in Euclidean space

该研究针对欧氏空间有界凸域上的拉普拉斯算子Neumann特征值，得到了新的上界，并借此推导出相应的普适不等式。