The asymptotic behavior of quasi-harmonic functions and eigenfunctions of the drift Laplacian

Critical set of eigenfunctions of the Laplacian

We give an upper bound for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.

Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type $L_p$ moment estimates of eigenfunctions.

Some inequalities between Laplacian eigenvalues on Riemannian manifolds

In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships between weighted $p$-Laplacian first eigenvalues. As an immediate application, we also obtain some eigenvalue comparison results between the first Dirichlet eigenvalue of the weighted Laplacian, the first clamped plate eigenvalue and the first buckling eigenvalue.

Topological aspects of $\mathbb{Z}/2\mathbb{Z}$ eigenfunctions for the Laplacian on $S^2$

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of particular interest is how these eigenvalues and eigenvectors change when viewed as functions on the configuration spaces of points.

On the essential spectrum of the Laplacian and the drifted Laplacian

This paper concerns the $L^2$ essential spectrum of the Laplacian $Δ$ and the drift Laplacian $Δ_f$ on complete Riemannian manifolds endowed with a weighted measure $e^{-f}d\;vol_g$. We prove that the essential spectrum of the drift Laplacian $Δ_f$ is $[0,+\infty) $ provided the Bakry-Émery curvature tensor $Ric_f$ is nonnegative and $f$ has sublinear growth . When $Ric_f \geq 1/2 g$ and $|\nabla f|^2 \leq f$, we show that the essential spectrum of the Laplacian is also $[0,+\infty)$. During the proofs of these results, the $f$-volume growth estimate plays an important role and may be of indep…

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.

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。

Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.

A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces

In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-Hölder inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical "Chiti Comparison Theorem". We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds.

Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

In a Hadamard manifold $M$, it is proved that if $u$ is a $λ$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $λ$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,λ$ and $H.$

Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order

In this paper, we investigate the buckling problem of the drifting Laplacian of arbitrary order on a bounded connected domain in complete smooth metric measure spaces (SMMSs) supporting a special function, and successfully get a general inequality for its eigenvalues. By applying this general inequality, if the complete SMMSs considered satisfy some curvature constraints, we can obtain a universal inequalities for eigenvalues of this buckling problem.

The heat trace for the drifting Laplacian and Schrödinger operators on manifolds

We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. In the case of a smooth weight function, we determine the full asymptotic expan…

拉普拉斯算子特征值的若干普适不等式及特征函数$L_{\infty}$范数的上界

Some universal inequalities of eigenvalues and upper bounds for the $L_{\infty}$ norm of eigenfunctions of the Laplacian

该短篇综述推导非负Ricci曲率闭黎曼流形上拉普拉斯算子特征值的Weyl型普适不等式与特征函数$L_{\infty}$范数上界，指出结果可由Milman与Cheng-Li的工作结合得到，相关详细证明此前未见文献报道。

区域态与磁拉普拉斯算子在双曲面上的改进 $L^\infty$ 绝对值界

Zonal states and improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces

本文研究了双曲面上磁拉普拉斯算子特征函数的改进 $L^\infty$ 绝对值界，并提出了磁区域态的概念，这些态在临界能量下饱和霍尔曼德界。

The Drift Laplacian and Hermitian Geometry

Let $(M^n, h)$ be a compact Hermitian manifold. Suppose $λ$ is the lowest eigenvalue of the complex Laplacian on $M$. We prove that $λ\geq C$ where $C$ depends only on the dimension $n$, the diameter $d$, the Ricci curvature of the Levi-Civita connection on $M$, and a norm, expressed in curvature, that determines how much $M$ fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng. We combine these results to obtain the main estimate.

构造具有预设节点集的黎曼度量以满足拉普拉斯特征函数

Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions

该研究构造了具有预设节点集的黎曼度量，使得拉普拉斯特征函数的零集与给定配置一致，并证明了特征值与体积的关系。

Dirichlet p-Laplacian eigenvalues and Cheeger constants on symmetric graphs

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.

若干模型空间上的双调和函数与双本征函数

Biharmonic functions and bi-eigenfunctions on some model spaces

该研究针对球面上的双拉普拉斯算子给出简便公式与本征相关结果，推导了一类模型空间上双调和方程径向解的积分形式，证明其存在真双调和函数，给出空间形式上的显式例子，并完成穿孔欧氏空间上正拉普拉斯真双调和函数的分类。

High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology

Let $Σ$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $Σ$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $ψ$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $Δψ=-λψ$ (with $λ=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $ψ$ given by~$Σ$, up to an ambient diffeomorphism.

The embedding dimension of Laplacian eigenfunction maps

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$. We call the smallest such $m$ the maximal embedding dimension of $M$. We show that the maximal embedding dimension of $M$ is bounded from above by a constant depending only on the dimension of $M$, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in $\mathbb{R}^3$, showing that the maximal embedding dimension only depends on bounds for the …

Exponential Decay of Eigenfunctions and Accumulation of Eigenvalues on Manifolds with Axial Analytic Asymptotically Cylindrical Ends

In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of the Laplacian on functions. In the case of a manifold with (non-compact) boundary it is either the Dirichlet Laplacian or the Neumann Laplacian. We show that the rate of exponential decay of an eigenfunction is prescribed by the distance from the corresponding eigenvalue to the ne…

On Sullivan's construction of eigenfunctions via exit times of Brownian motion

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $λ_1 = λ_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $τ= τ_D$ be the first exit time of Brownian motion from $D$. For each $λ\in \mathbb{C}$ …

Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to gen…

The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the con…

双曲曲面上磁 Laplace 算子的半经典缺陷测度

Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

本文研究闭合双曲曲面上恒定磁场中磁 Laplace 算子的半经典缺陷测度，并按低能、临界和高能三种能级分别刻画其不变测度、量子唯一遍历性及等分布性质。

带Wentzell型边界条件的漂移拉普拉斯算子的Cheeger型不等式

A Cheeger-type inequality for the drift Laplacian with Wentzell-type boundary condition

该研究针对带Wentzell型边界条件的流形上漂移拉普拉斯算子，结合体-边界相互作用的Cheeger型常数，证明了其第一非平凡特征值的下界，结果遵循Cheeger经典不等式的思路。