Jun Ling
We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous estimates.
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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.
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.
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 estimates.
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 lower bounds and splitting for modified Ricci flow
该研究针对修正Ricci流，证明了滑移拉普拉斯算子特征值的精确下界，并给出了等号成立时的分裂定理，丰富了Ricci流相关理论。
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.
Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci curvature equation $\mathrm{Ric}(G)=T$ for a Riemannian metric $G$ on $M$ subject to the boundary condition $G_{\partial M}=R$ (the notation $G_{\partial M}$ here stands for the metric induced by $G$ on $\partial M$). Imposing a standard assumption on $M_1$, we describe a set of requirements on $T$ and $R$ that guarantee global and local solvability.
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.
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…
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.
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。
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。
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.
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…
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
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.
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).
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.
Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds
本文研究了非紧黎曼流形上 $p$-拉普拉斯算子第一个狄利克雷特征值的估计问题，得出了具有非负里奇曲率的完整非紧流形上的精确下界结果。
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。
New inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space
本文基于对狄利克雷拉普拉斯算子特征值通用不等式的研究，证明了超几何空间上相关特征值的新不等式，并验证了陈猜想在两种特殊有界域中的成立，允许ε损失。
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.
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 …
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 $r\le 0$.
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。
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.
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.
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.
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.
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.
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.
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。
On Finsler metric measure manifolds with integral weighted Ricci curvature bounds
本文研究具有积分加权Ricci曲率界的Finsler度量测度流形，建立多个比较定理，得到体积增长估计、Gromov预紧性及Dirichlet等周常数估计，并应用于特征值与调和函数梯度估计。
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 $D>0$.
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.
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.
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 $λ^{+}(t)=\max\{λ(t),0\}$.
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。
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.
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…
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 …
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.
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 …
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…
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…
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.
In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.
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。
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.
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.
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。
The first Dirichlet eigenvalue and the width
该研究针对非负Ricci曲率、平均凸边界的测地球，提出定量显式不等式，用第一Dirichlet特征值与其精确下界的谱间隙界定测地球宽度，拓展了相关谱几何结论。
We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector).
Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds
本文研究了共形平坦黎曼流形上Dirichlet Laplacian的特征值问题，并在双曲空间$\mathbb{H}^n(-1)$上建立了特征值的普遍不等式。
Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound
本文在下界里奇曲率限制条件下，研究了闭合黎曼流形上霍奇拉普拉斯算子的特征值估计，并建立了连接拉普拉斯算子的特征值界及微分形式的全局皮卡尔不等式。
We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an $L^\infty$-drift term. We make no assumptions that the operator is self-adjoint or that the drift has any additional regularity. In the case where the operator is self-adjoint, this establishes a lower bound on the spectrum without assuming a lower bound for the Bakry-Emery Ricci tensor. Put colloquially, this result states that no matter which way the wind blows, heat will diffuse at a definite rate depending only on the geometry of the underlying space and the maximal wind speed.
Metrics with positive Ricci eigenvalue sums on $\mathbb{T}^k\times \mathbb{S}^m$
本文构造了环面与球面乘积上任意接近标准度量的正 Ricci 特征值和度量，否证了 Wolfson 猜想，并给出了二正 Ricci 曲率的局部障碍。
A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds
该研究在正里奇曲率流形中，通过曲率界、法向注入半径和超曲面几何特性，推导出嵌入超曲面第一个非零特征值的下界估计，扩展了Choi和Wang的经典结果到非极小情况。
We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of Bér…
In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.
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