Stable bundles and the first eigenvalue of the Laplacian

Discretization of vector bundles and rough Laplacian

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space coming from a discretization process. As an application, we obtain a comparison of the first positive eigenvalue of the rough Laplacian with the holonomy.

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.

Eigenvalues of the Laplacian on Riemannian manifolds

For a bounded domain $Ω$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of $L^2(Ω)$ in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.

Quantization of the Laplacian operator on vector bundles I

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.

Bounds to the first eigenvalues of weighted p-Steklov and (p,q)-Laplacian Steklov problems

We consider the Steklov problem associated with the weighted p-Laplace operator and $(p,q)$-Laplacian on submanifolds with the boundary of Euclidean spaces and prove Reilly-type upper bounds for their first eigenvalues.

First eigenvalue of the $p$-Laplacian on Kähler manifolds

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kähler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.

共形平坦黎曼流形上Dirichlet Laplacian特征值的普遍不等式

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。

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.

The spectral density function for the Laplacian on high tensor powers of a line bundle

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.

关于子流形Hodge拉普拉斯算子第一个特征值的研究

On the first eigenvalue of the Hodge Laplacian of submanifolds

本文证明在空间形式中闭子流形的第一个p-特征值的严格下界等号成立仅可能在正曲率的球面成立。

Upper bounds on the first eigenvalue for the $p$-Laplacian

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$Δ_{p}u=-λ|u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian.

子流形上 Laplacian 和 biharmonic 算子特征值的不等式

Inequalities for eigenvalues of Laplacian and biharmonic operators on submanifolds

本文研究了完备流形紧致区域上 Laplacian 和 biharmonic 算子的特征值，利用 Sobolev 型不等式分别建立了这两种算子特征值的若干新不等式。

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.

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.

The first positive eigenvalue of the sub-laplacian on cr spheres

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 $\subset$ C n+1 , achieves its maximum when $\θ$ is the standard contact form.

Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the $k^{\text{th}}$ eigenvalue and for isoparametric minimal hypersurfaces in the unit sphere, an explicit upper bound of the $(n+3)^{\text{th}}$ eigenvalue of the Laplacian is obtained. Furthermore, we generalize the Reilly's result on the first eigenvalue of the Laplacian.

关于连接拉普拉斯算子和主丛上 $G$-简单性的谱的普遍简单性

On the generic Simplicity of the spectrum for Connection Laplacian and $G$-simplicity on Principal Bundles

本文证明了连接拉普拉斯算子在光滑向量丛上的谱的普遍简单性，并展示了主丛上拉普拉斯-贝尔特拉米算子的 $G$-简单性。

Estimates for the first eigenvalues of Bi-drifted Laplacian on smooth metric measure space

In this paper, we obtain lower bounds for the first eigenvalue to some kinds of the eigenvalue problems for Bi-drifted Laplacian operator on compact manifolds (also called a smooth metric measure space) with boundary and $m$-Bakry-Emery Ricci curvature or Bakry-Emery Ricci curvature bounded below. We also address the eigenvalue problem with Wentzell-type boundary condition for drifted Laplacian on smooth metric measure space.

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.

$p$-Laplacian first eigenvalues controls on Finsler manifolds

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian metric on $M$ and $β$ an appropriate $1$-form on $M$, it is shown that the first eigenvalue $λ_{1,p}(M,F)$ of the Finslerian $p$-Laplacian defined by the Finsler metric $F$ is controled by the first eigenvalue $λ_{1,p}(M,g)$ of the Riemannian $p$-Laplacian defined on $(M,g)$. Fi…

Sharp Estimates for the First Eigenvalues of the Bi-drifting Laplacian

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.

Estimates of eigenvalues of Laplacian by a reduced number of subsets

Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce "input" in Chung-Grigor'yan-Yau's inequality in the setting of Alexandrov spaces satisfying CD$(0,\infty)$. We will also discuss a related conjecture for some universal inequality among eigenvalues of Laplacian.

Stability and Perturbations of the Domain for the First Eigenvalue of the 1-Laplacian

We discuss stability of the first eigenvalue of the 1-Laplacian under perturbations of the domain.

黎曼曲面上半正线丛的Bochner拉普拉斯算子与Bergman核展开

Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

本研究推广了Montgomery关于线丛高张量幂上Bochner拉普拉斯算子的结果，特化到黎曼曲面时得到半正线丛的Bergman核展开与几何量子化结果，证明借助了Bochner拉普拉斯算子与次黎曼拉普拉斯算子的关联。

The first nonzero eigenvalue of the $p$-Laplacian on Differential forms

we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.

Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.

黎曼流形上 Dirichlet Laplacian 特征值的估计

Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds

本文重新探讨了完备黎曼流形有界域上 Dirichlet Laplacian 的特征值问题，基于 Li-Yau 和 Yang 不等式等经典结果推导出特征值上下界，并对射影空间及其极小子流形给出了特征值下界的显式估计。

Remark on the paper "Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds" Q.-M. Cheng and L. Zeng [arXiv:1304.3212]

Why the results of that article [arXiv:1304.3212] are immediate consequences of known ones.

关于单位球面中紧致子流形的p-Laplacian第一特征值的一个注记

A remark on the first eigenvalue of the p-Laplacian on compact submanifolds in the unit sphere

本文针对3/2<p<2建立了奇异p-laplacian的积分不等式，并由此推导出单位球面中极小子流形和预定数量曲率子流形p-laplacian第一特征值的下界。

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。

Some sharp Hodge Laplacian and Steklov eigenvalue estimates for differential forms

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.

A new characterization of the CR shpere and the sharp eigenvalue estimate for the Kohn Laplacian

Motivated by a sharp eigenvalue estimate for the Kohn Laplacian, we prove a theorem that characterizes the CR sphere in terms of the existence of a non-trivial complex-valued function satisfying a certain overdetermined system.

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.

On Lower Bounds of the First Eigenvalue of Finsler-Laplacian

By using Bochner technique and gradient estimate, we give the lower bound estimates of the first eigenvalue of Finsler-Laplacian on Finsler manifolds. These results generalize the corresponding famous theorems in the Riemannian geometry.

Self-adjointness of the Gaffney Laplacian on vector bundles

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

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.

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。

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.

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。

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.

First eigenvalue of the $p$-Laplacian under integral curvature condition

We give various estimates of the first eigenvalue of the $p$-Laplace operator on closed Riemannian manifold with integral curvature conditions.

A lower bound for eigenvalues of the poly-Laplacian with arbitrary order

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an $n$-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter. In particular, the result of Melas is included here.

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.

Semiclassical eigenvalue asymptotics for the Bochner Laplacian of a positive line bundle on a symplectic manifold

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic expansions for its low-lying eigenvalues.

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.

Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

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。

关于非紧流形上 $p$-拉普拉斯算子第一个狄利克雷特征值的估计

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。

超几何空间上狄利克雷拉普拉斯算子特征值的新不等式

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。

Explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds

We establish an explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and orthogonal Ricci curvature.

Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.

Principal eigenvalue problem for infinity Laplacian in metric spaces

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the $\infty$-eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space.

三维曲面的图拉普拉斯算子特征值灵活性

Flexibility of eigenvalues for graph Laplacians arising from genus 3 surfaces

本文研究三维曲面图拉普拉斯算子的小特征值近似，通过分析四顶点图的特征值集，揭示了特征值在不同曲面分解下的灵活性。

Lower bounds for the first eigenvalue of $p$-Laplacian on Kähler manifolds

We study the eigenvalue problem for the $p$-Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for $p\in (1, 2]$. Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact Kähler manifolds with smooth boundary for $p\in (1, \infty)$. Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved in […