三球面的Laplacian本征模

Pollicott-Ruelle spectrum and Witten Laplacians

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family o…

Maximization of the second Laplacian Eigenvalue on the Sphere

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on $S^m$. For $S^{2}$, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides' work and recent developments on the hyperbolic center of mass and provides also a simpler proof for $S^2$.

通过Belyi函数和拉普拉斯算子行列式研究奇异常曲率球面的三角剖分

Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

本文通过Belyi函数和拉普拉斯行列式研究奇异常曲率球面的三角剖分，并计算了不同正多面体的行列式。

球面商上 Kohn Laplacian 的特征值分布

Distribution of Eigenvalues of the Kohn Laplacian on Sphere Quotients

本文证明了球面商上 Kohn Laplacian 的 Weyl 型定理，指出在三维情况下可从其谱确定基本群，并给出了复 Green 算子的 Sobolev 估计。

2-球面上带三个锥形奇点的常曲率度量的拉普拉斯算子行列式

Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

该研究推导了带三个锥形奇点的2-球面常曲率度量下Friedrichs拉普拉斯算子zeta正则化谱行列式的显式公式，证明了对称奇点对应行列式驻点（小表面积时为极小值），并建立了其与刘维尔场论中刘维尔作用量的关联。

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.

关于$1$-形式的Hodge拉普拉斯算子在同构于$S^3$的齐性3球面上的研究

Hodge Laplacian on $1$-forms of homogeneous $3$-spheres

本文研究了SU(2)和SO(3)上的左不变度量下Hodge-Laplacian的谱，首次明确计算了Berger 3球面的特征值并提出并证明了齐性度量下第一个特征值的显式公式，展示了1形式谱决定度量的全等性。

$L^p$ Norms of Eigenfunctions on Regular Graphs and on the Sphere

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the $2$-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.

球面 $\mathbb{S}^{2}$ 和 $\mathbb{S}^{3}$ 上拉普拉斯算子点扰动的高能本征函数

High-energy eigenfunctions of point perturbations of the Laplacian on the spheres $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$

研究球面上点散射子扰动拉普拉斯算子的本征函数序列的半经典测度，证明所有不变测度都是半经典测度，且当散射子包含一对对径点时，可构造非不变测度的本征函数序列。

Zeta Determinant for Laplace Operators on Riemann Caps

The goal of this paper is to compute the zeta function determinant for the massive Laplacian on Riemann caps (or spherical suspensions). These manifolds are defined as compact and boundaryless $D-$dimensional manifolds deformed by a singular Riemannian structure. The deformed spheres, considered previously in the literature, belong to this class. After presenting the geometry and discussing the spectrum of the Laplacian, we illustrate a method to compute its zeta regularized determinant. The special case of the deformed sphere is recovered as a limit of our general formulas.

Calabi-Yau超曲面上线丛的Laplacian数值谱

Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

本文首次数值计算了Calabi-Yau三重流形上取值于线丛的Dolbeault Laplacian谱，并通过三个例子验证了算法的准确性，为弦紧致化中Yukawa耦合计算提供了必要工具。

子流形上拉普拉斯和薛定谔算子特征值的泛不等式，

Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds,

我们在欧几里得空间、球面、实、复和四元数射影空间的紧致子流形（可能有非空边界）上建立了薛定谔算子的特征值不等式，这些不等式与Payne、Pólya、Weinberger和Yang关于欧几里得整环上拉普拉斯算子的不等式有关，但它们以显式的方式依赖于中曲率。在后面的章节中，我们证明了齐次黎曼空间上的薛定谔算子的类似结果，更一般地，证明了任何允许特征映射进入球面的黎曼流形上的薛定谔算子的类似结果，以及海森堡群子域上的科恩拉普拉斯算子的类似结果。在这个分析的后果是一个扩展的赖利不等式，界定任何特征值的拉普拉斯算子的中曲率，和频谱标准的浸入式流形在齐性空间。