Lower Bounds of the First Closed and Neumann Eigenvalues of Compact Manifolds with Positive Ricci Curvature

A Lower Bound of the First Eigenvalue of a Closed Manifold with Negative Lower Bound of the Ricci Curvature

Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.

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.

Prescribing symmetric functions of the eigenvalues of the Ricci tensor

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.

Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend X. Cao's methods of eigenvalues\cite{C} and impro…

$p$-拉普拉斯算子沿里奇流的第一个特征值

First eigenvalue of the $p$-Laplace operator along the Ricci flow

研究了闭流形上里奇流中$p$-拉普拉斯算子第一个特征值的连续性、单调性和可微性，证明其在一定曲率条件下几乎处处可微，并在无曲率假设下推导出$p$-特征值比较定理。

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。

Neumann Li-Yau gradient estimate under integral Ricci curvature bounds

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left( \oint_{B(x,D)} |Ric^-|^p dy \right)^{\frac{1}{p}} < K \end{equation} for $K(n,p)$ small enough, $p>n/2$, where $diam({\bf M})\leq D$. The boundary of ${\bf M}$ is not necessarily convex, but it needs to satisfy the interior rolling $R-$ball condition.

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.

The Weyl-type asymptotic formula for biharmonic Stekloff eigenvalues with Neumann boundary condition in Riemannian manifolds

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our r…

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 …

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature II

Let $L=Δ-\nablaφ\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $dμ=e^{-φ}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla φ|\leqθ$ and $\infty$-dimensional Bakry-Émery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975) 289-297).

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.

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 curvature and eigenvalue estimates for the magnetic Laplacian on manifolds

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

Variation of the first eigenvalue of $(p,q)$-Laplacian along the Ricci-harmonic flow

In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotic quantities by imposing some conditions on initial manifold.

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。

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.

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.

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.

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。

Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $Ω$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $μ_2(Ω)$ with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case when $μ_2(Ω)$ is not a simple eigenv…

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 $D>0$.

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.

Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau's conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.

Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds

Let $λ(t)$ be the first eigenvalue of $-Δ+aR\, (a>0)$ 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 $λ(t)$. In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, $λ^{+}(t)$ approaches zero, where $λ^{+}(t)=\max\{λ(t),0\}$.

First eigenvalue for p-Laplacian with mixed boundary conditions on manifolds

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the…

A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples of any complex dimension n > 1, if one eigenvalue is negative and the other positive or zero, and of any complex dimension n > 2, if the both eigenvalues are negative; there are non-homogeneous examp…

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 …

Two balls maximize the third Neumann eigenvalue in hyperbolic space

We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument

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 …

Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE$_1(κ,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under time changes with arbitrary $ψ\in{\mathrm Lip}_b(X)$, and $\bullet$ which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $Y\subset X$. In the latter case, the distribution-valued Ricci bound will be given by the signed measure $κ= k\, m_Y + \ell\,σ_{\pa…

Pólya conjecture for the Neumann eigenvalues

For a given bounded domain $Ω\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} where $μ_k$ is the $k$-th Neumann eigenvalue of the Laplacian for $Ω$.

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…

Conformal deformations of the smallest eigenvalue of the Ricci tensor

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the the…

第一Neumann特征值与宽度

The first Neumann eigenvalue and the width

本文针对有界凸平面区域，证明了第一Neumann特征值可由直径和宽度给出的精确下界，揭示了特征值与区域几何尺度之间的定量关系。

关于球面上第一个正Neumann特征值的等周不等式

On the isoperimetric inequality for the first positive Neumann eigenvalue on the sphere

研究证明，在固定面积的球面单连通域中，测地盘是第一个非平凡Neumann特征值的唯一最大值解。

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

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.

平面凸域上拉普拉斯算子Neumann特征值的两个新通用不等式

Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain

本文提出两个新通用不等式，用于平面凸域中拉普拉斯算子的Neumann特征值，扩展了相关理论的应用范围。

Neumann Isoperimetric Constant Estimate for Convex Domains

We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.

收缩Ricci孤立子的特征值刚性

Rigidity of eigenvalues of shrinking Ricci solitons

本文研究收缩Ricci孤立子的特征值刚性，证明了第n个特征值接近下界时孤立子为平凡高斯孤立子，在非坍塌条件下得到第(n-1)、(n-2)个特征值的类似结果，并给出一般k的第k个特征值的几乎刚性结论，部分结果拓展了紧流形上的相关定理至非紧孤立子情形。

Eigenvalues estimate for the Neumann problem on bounded domains

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As application, we get upper bounds for the Neumann spectrum which is clearly in agreement with the Weyl law and which is analogous to Buser's upper bounds of the spectrum of a closed Riemannian manifold with lower b…

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.

秩为1对称空间中有界区域上Neumann特征值的等周不等式

Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

本文证明了紧致与非紧致秩为1对称空间中有界区域上Neumann Laplacian低阶特征值的精确等周不等式，推广了Wang和Xia关于双曲空间的工作及Aithal和Santhanam的Szegö-Weinberger不等式。

空间形式中测地三角形第一非零Neumann特征值的依赖直径的尖锐下界

A Sharp Diameter-Dependent Lower Bound for the First Nonzero Neumann Eigenvalue of Geodesic Triangles in Space Forms

该研究针对二维空间形式的测地三角形，证明了其第一非零Neumann特征值的依赖直径的尖锐下界，还推导了非锐角球面三角形的热点定理及等腰球面三角形的相关性质。

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.

实空间形式中具有常Ricci特征值的超曲面

Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms

本文证明实空间形式中连通超曲面具有常Ricci特征值当且仅当曲率齐性，从而完成分类，推广了Einstein超曲面的分类，并得到曲率非齐性流形不能等距浸入余维一实空间形式的推论。

共形与内蕴上界对于Neumann和Steklov特征值调和平均值的上界

Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues

本文研究了Neumann和Steklov特征值调和平均值的共形与内蕴上界，并给出了闭子流形在空间形式中的最优上界。

Higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues

In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-Wang\cite{HHW} and Hassannezhad-Miclo \cite{HM}. Then, some Raulot-Savo-type estimates on the eigenvalues of the DtN maps introduced are derived.