Spectrum of the Laplacian on radial graphs
Rodrigo Bezerra de Matos, Jose Fabio B. Montenegro
证明了若$M$是实径向函数图$\mathbb{R}^{n+1}$中的完备超曲面，则M上Laplace算子的谱是区间$[0,\infty)$ .
We prove that if $M$ is a complete hypersurface in $\mathbb{R}^{n+1}$ which is graph of a real radial function, then the spectrum of the Laplace operator on M is the interval $[0,\infty)$.
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Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values at every point. We shall see that T is a function of two different eigenvalues corresponding to the given pair of radial eigenfunctions. On the other hand, we shall see that at every fixed point and for the value already assumed by a radial eigenfunction at the fixed point, ther…
On the essential spectrum of the Laplacian and the drifted Laplacian
本文研究了完备黎曼流形上的Laplacian $Δ$和漂移Laplacian $Δ_f$的$L^2$本质谱，并给出了加权测度$e^{-f}d\;vol_g$ .证明了漂移Laplacian $Δ_f$的本质谱是$[0,+\infty) $，只要Bakry-Émery曲率张量$Ric_f$是非负的，且$f$具有次线性增长.当$Ric_f \geq 1/2 g$和$|\nabla f|^2 \leq f$时，我们证明了拉普拉斯算子的本质谱也是$[0,+\infty)$。在这些结果的证明过程中，$f$ -体积生长估计起着重要的作用，可能是独立的兴趣。
We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian $Δ^0$ on functions and the Laplacian $Δ^1$ on 1-forms. We examine the nature of the singularity as the geodesic distance $r$ tends to zero of radial eigen-functions and 1-forms. Via duality, our results give rise to corresponding results for radial vector fields. Many of our results extend to the context of spaces which are harmonic with respect to a single point.
Remark about the spectrum of the p-form Laplacian under a collapse with curvature bounded below
给出了曲率有界的p-型Laplacian在某类坍缩中小的正特征值个数的一个下界.
On the spectrum of the twisted Dolbeault Laplacian on line bundles over Kähler manifolds
本文利用Dirac算子技巧建立了紧Kähler流形上全纯线丛上扭Dolbeault Laplacian第一特征值的一个精确下界.
Spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces
证明了在任意渐近双曲曲面上，Lichnerowicz Laplacian $Δ_L$的本质谱包含射线$[{1/4},+\infty[$ .此外，如果数量曲率是常数，则-2和0是无穷维特征值。此外，如果不等式$<Δu, u>_{L^2}\geq \frac14||u||^2_{L^2}$对所有光滑紧支函数$u$成立，则谱中没有其他值。
We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a consequence, we give estimates for the weighted mean curvature of complete noncompact hypersurfaces into weighted manifolds.
On the spectrum of the twisted Dolbeault Laplacian over Kähler manifolds
本文利用Dirac算子技巧，在紧致Kähler流形上的负度向量丛上建立了被Hermitian-Einstein联络扭曲的Dolbeault Laplacian的第一特征值的精确下界.
Absence of singular continuous spectrum for some geometric Laplacians
我们提供了两个例子的谱分析技术的薛定谔经营者适用于几何拉普拉斯算子。特别是，我们展示了如何适应的方法分析膨胀拉普拉斯算子的余维2角的完整流形上找到这些运营商的奇异连续谱的情况下，其纯点谱的基本几何行为的描述，和量子共振理论。
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。
A sharp estimate for the bottom of the spectrum of the Laplacian on Kähler manifolds
在完备的非紧Kähler流形上，如果里奇曲率由下而上由$-2(m+1)$有界，则Laplacian谱的底由上而下由$m^2$有界.然后，我们表明，如果这个上界是实现，然后流形有最多两个端点。这些结果改进了P. Li和J. Wang在\cite {L-W3}和\cite{L-W}中在对分曲率假设下的结果.
Partial collapsing and the spectrum of the Hodge Laplacian
本文的目的是计算Hodge-de Rham算子在坍缩流形的一部分扰动下的极限谱，该流形是由两个具有相同边界的流形粘合在一起而得到的.它似乎发生在一般问题的爆破圆锥奇点介绍\cite{Maz}和\cite{Row1,Row2}。
Spectrum of the Laplacian on manifolds with Spin(9) holonomy
考虑具有Spin（9）完整性的非紧完备流形，在谱底的不同条件下证明了一个单端结果和一个分裂型定理。证明了任何具有有限狄利克雷积分的调和函数必是Cayley-调和函数，从而得到了一个唯一结果.第二部分利用Busemann函数建立了一个分裂型定理。
The Wave Trace Invariants of the Spectrum of the $G$-Invariant Laplacian
给定一个紧李群$G$作用于其上的紧无边界黎曼流形$Y$，在$Y$上总有一个度量使得该作用是等距的。假设$Y$配备有这样的度量，回想一下，$G$不变拉普拉斯算子是普通拉普拉斯算子对沿着$G$的轨道恒定的函数空间的限制。在本文中，作者分析了波迹的$G$不变拉普拉斯算子，并表明，这种波迹的奇异性发生在某些测地弧的长度-那些正交的轨道和其端点有关的行动$G$。这定义了群作用的长度谱的概念。进一步，利用流形上的等距群作用和叶状流形之间的深层联系，计算了长度谱中任意非零个数下这些奇点的渐近性。
An optimal inequality between scalar curvature and spectrum of the Laplacian
对于黎曼闭自旋流形，在一定的拓扑假设（$\hat{A}$ -亏格非零或Gromov-Lawson意义下的可扩性）下，给出了关于Laplacian第一特征值的数量曲率下确界的最优上界.主要困难在于奇维情形的研究。另一方面，我们研究了具有非零$\hat{A}$ -亏格的闭自旋黎曼流形的等式情形。本文改进了K. 1988年的小野。
Heat kernel estimates and the essential spectrum on weighted manifolds
考虑完备非紧光滑黎曼流形$M$，它具有权测度和相应的漂移拉普拉斯算子.证明了只要$M$上的$q$ -Bakry-Émery Ricci张量有下界，则可以由一族相关翘曲积空间上的拉普拉斯算子的热核的上界估计得到漂移拉普拉斯算子的热核的上界估计.我们应用这些结果研究了$M$上漂移拉普拉斯算子的本质谱。
On the Spectrum of weighted Laplacian operator and its application to uniqueness of Kähler Einstein metrics
利用Chen的弱C^{1，1}测地线，给出了Fano流形上Kähler Einstein度量的Bando-Mabuchi唯一性定理的一个新证明.与光滑情形不同，测地线上的谱的缺乏使得我们不能直接使用加权Laplacian算子的谱公式。然而，我们可以用光滑测地线来逼近弱测地线，然后证明一个特征函数序列将收敛到加权拉普拉斯算子的第一特征空间。
We give upper bounds for the bottom of the essential spectrum of properly immersed minimal submanifolds of $\mathbb{R}^{n}$ in terms of their volume growth. Our result improves the extrinsic version of Brook's essential spectrum estimate given by Ilias-Nelli-Soret in \cite[Cor.3]{ins}.
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。
We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.
Adiabatic limits and the spectrum of the Laplacian on foliated manifolds
我们提出了一些最近的结果的频谱的微分形式的拉普拉斯黎曼叶状流形上的度量时，周围的流形上的吹在垂直于叶的方向（在绝热极限）的行为。
In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4λ_1 > h^2$, where $λ_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow.
On the essential spectrum of the Laplacian and vague convergence of the curvature at infinity
本文证明了在一定的体积增长条件下，如果$n$维黎曼流形有一个端点，且端点上里奇曲率低于一个非正常数$(n-1)K$的部分的平均值在无穷远处收敛于零，则Laplacian的本质谱包含区间$[(n-1)^2K/4, \infty)$.
The essential spectrum of the Laplacian
本文证明了希尔伯特空间上自伴算子的本质谱的Weyl判别法的一个推广。然后将此判据应用于开流形上函数的拉普拉斯算子，得到了关于其本质谱的新结果。
We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.
In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.
A Sharp Lower Bound for the Spectrum of the Hodge Laplacian on Kähler Hyperbolic Manifolds and its Applications
本文研究了Kähler超曲面Hodge拉普拉斯算子谱的精确下界，并应用于有界对称域的谱分析。
The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I
本文的关注点是澄清无穷远曲率与拉普拉斯算子谱结构之间的关系。特别地，本文讨论了在本质谱中是否存在拉普拉斯算子的本征值的问题。这个问题的径向曲率的边界行为将被确定：我们将假设一个端部的径向曲率$K_{\rm rad.}$在无穷远处收敛到一个常数-1，衰减阶数为$K_{\rm rad.} + 1 = o(r^{-1})$，并证明没有嵌入在本质谱中的本征值。此外，为了证明这种衰变阶$K_{\rm rad.} + 1 = o(r^{-1})$是尖锐的，我们将构造一个流形，该流形具有径向曲率衰变$K_{\rm rad.} + 1 = O(r^{-1})$和嵌入在拉普拉斯算子的本质谱$[ \frac{(n-1)^2}{4}, \infty)$中的本征值$\frac{(n-1)^2}{4} + 1$。
Infimum of the exponential volume growth and the bottom of the essential spectrum of the Laplacian
本文的目的是指出Brooks的两个不等式中的“上确界”应改为“下确界”。Higuci教授已经知道了这篇论文的结果。因此，我想从这个预印本服务器上删除这篇论文。
We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.
Discretization of vector bundles and rough Laplacian
我们建立了一个统一的比较粗糙拉普拉斯算子的频谱（作用于复秩1或调和曲率的向量丛的部分）与离散算子的频谱（广义的离散磁拉普拉斯算子加上潜在的）作用于来自离散化过程的有限维空间。作为应用，我们得到了粗糙拉普拉斯算子的第一正特征值与完整性的比较。
The lower bound of the Ricci curvature that yields the infinite number of the discrete spectrum of the Laplacian
本文讨论Laplace-Beltrami算子的离散谱是无限还是有限的问题。这个问题的曲率的边界行为将被完全确定。
Joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds
本文研究三维Heisenberg Bieberbach流形上次拉普拉斯算子与Reeb向量场的联合谱，利用theta函数和有限群特征理论明确计算了其重数。
Discretization of Riemannian manifolds applied to the Hodge Laplacian
一个改编版本的证明（由于A。Weil）的著名de Rham定理允许我们一致地比较作用于微分形式（在紧致黎曼流形上）的Hodge Laplacian的谱和作用于与开覆盖（由足够小半径的球组成）相关的上链的组合Laplacian的谱。然后，我们展示了一个下界的第一个正特征值的组合拉普拉斯算子和推导出一个下界的第一个正特征值的霍奇拉普拉斯算子。
In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the $k$-form essential spectrum over a complete manifold with vanishing curvature at infinity or asymptotically nonnegative Ricci curvature. In addition, we will apply the generalized Weyl criterion to study the variation of the spectrum of a self-adjoint operator under continuous perturbations of the operator. In the particular case of the Laplacia…
Spectrum of the Laplacian on the Page metric
本文通过将问题约化为一维Sturm-Liouville问题并结合微扰分析与伪谱方法，数值构造了Page度量上拉普拉斯算子及Lichnerowicz拉普拉斯算子的谱。
We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.
The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds
我们考虑具有边界的紧致带边流形内部的开流形，并且黎曼度量在边界附近渐近于共形圆柱度量。我们表明，在磁场的存在下，不定义一个完整的相对上同调类的功能的拉普拉斯算子的本质谱消失。由此可见，即使在强磁场的作用下，本质谱也不稳定。我们也处理磁场经营者扰动电场。在同一背景下，我们描述的本质谱的$k$形式的拉普拉斯算子。当边界的$k$和$k-1$ de Rham上同调群消失时，这一点恰好消失。在所有纯点谱情形下，我们给出了特征值计数函数的Weyl型渐近性。在其他情况下，我们描述本质谱。
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