Andrey A. Yakovlev
We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.
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The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.
Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on ${\bf R}^n$ endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to $[0, +\infty)$.
We explicitely compute the essential spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $dσ^2= y^{2a}dy^2 + y^{2b}dθ_{\partial M}^2$, where $y$ is a boundary defining function on a compact manifold with boundary $M$.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are obtained.
Capacities-to-masses for Laplace-Beltrami operators on complete Riemannian manifolds
本文提出创新方法研究完备黎曼流形上拉普拉斯-贝尔特拉米算子的容量与质量，解决了文献中关于极限变分容量的公开问题。
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtain…
We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.
In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.
We study on a new kind of surface covered by translation and factorable (TF-type) surfaces in the three dimensional Euclidean space. We consider I and III Laplace-Beltrami operator surfaces of a TF-type surface. Then we obtain degrees and classes of algebraic surfaces of the surfaces using eliminate methods on software programme.
The purpose of this note is to provide a short cut presentation of a Mayer-Vietoris formula due to Burghelea-Friedlander-Kappeler for the regularized determinant in the case of elliptic operators of Laplace Beltrami type in the form typically needed in applications to torsion.
We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.
We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $ϕ_0$ of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the point where $ϕ_0$ attains its maximum. If $ϕ_0$ is a constant function and $n \geq 4$, these domains are close to the complement of geodesic balls of small radiu…
An explicit formula for the Laplace-Beltrami operator on the Stiefel manifold
本文基于约束流形的一般框架，推导了正交Stiefel流形上Laplace-Beltrami算子用环境欧氏坐标表示的显式公式，并将球面与特殊正交群作为特例。
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Émery Ricci curvature condition and applications
本文在 Bakry-Émery Ricci 曲率有下界的完备黎曼流形上建立了 Laplace-Beltrami 算子热核的高斯上界，并据此证明了非负次调和函数的 L^1-Liouville 性质及特征值的下界与底谱上界。
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite dimensional. Using Jucys-Murphy elements of the Hecke algebra the eigenvalues of the Laplace-Beltrami operator for the Hopf algebra O(SLq(N)) are computed.
In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.
We generalize the result of [Matveev-Topalov 2001] to all signatures: we show that in all signatures the Killing tensors constructed by projectively equivalent metrics correspond to commuting differential operators
Laplace-Beltrami operator on the orthogonal group in Cartesian coordinates
本研究借助嵌入梯度向量场方法推导约束流形上Laplace-Beltrami算子的环境坐标通用公式，进一步给出正交群上该算子的显式表达式，并完成相关函数的应用验证。
We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the $Γ_2$ operator may not be bounded from below but the horizontal Bakry-Emery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin-…
Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2…
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessella…
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a half-helicoid than on any other screw surface in $\mathbb{R}^2 \times \mathbb{S}^1$ with the same boundary.
The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. As a result, we will obtain the absolute continuity of the Laplace-Beltrami operator.
Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $λ_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfac…
It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $δ>0$ and for an arbitrarily large $L>0$ …
On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator
本文研究连通流形上黎曼度量集合中使得 Laplace–Beltrami 算子具有给定重数特征值的子集的结构，利用 Besson 的几何刻画证明除无穷余维集合外所有度量满足强 Arnold 假设，并得到任意重数特征值对应度量集合的精确余维。
We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $O\Big(\big(\frac{\log n}{n}\big)^\frac{1}{2m}\Big)$ to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of $M$. No information on the submanifold $M$ is needed in the construction of the graph or the "out-of-sample extension" of the eigenvectors. Of …
Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces
本文扩展共轭函数法至黎曼曲面，利用hp自适应有限元求解Laplace--Beltrami方程，实现含奇异性和尖点复杂边界域的高精度共形映射计算。
The smallest Laplace eigenvalue of homogeneous 3-spheres
本文推导齐性3维球面（即带左不变度量的SU(2)）上Laplace–Beltrami算子最小非零特征值的显式公式，完整刻画Berger球面谱，并推广至SO(3)。
The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces
本文研究了紧致自然红uctive齐性空间上拉普拉斯-贝叶斯算子的谱，通过广义Casimir算子的特征值和球形表示推导出公式，并应用于正常齐性度量的变体，首次分析了谱对度量变形参数的依赖性。
Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions
本文研究 Laplace-Beltrami 特征函数与高阶 Cheeger 常数之间的关系，通过特征函数上水平集和软阈值构造给出高阶 Cheeger 比率的上界，并将其推广到动态 Laplacian 以揭示动力系统中的低动态 Cheeger 比率集合。
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary.
Volume Growth and Recurrence of Fractional Powers of the Laplace--Beltrami Operator
该研究针对黎曼流形上由α/2稳定次支配子从属布朗运动得到的过程X^(α)，证明了积分条件∫^∞ dt/V(o,t^(1/α))=∞可推出X^(α)回归，解决了Grigor'yan问题26的充分性部分。
We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term $E\log E$. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are als…
We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.
First Laplace eigenvalue of strongly isotropy irreducible spaces
该研究针对紧强迷向不可约空间的Laplace-Beltrami算子最小正特征值，给出单连通情形显式表达式，并证明其与Einstein常数满足$E<λ_1\leq 16E$的不等式关系。
On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin…
The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an uppe…
Convergence of the Point Integral method for Poisson equation on point cloud
本文研究了点积分方法在带有Neumann边界条件的子流形上的泊松方程收敛性，分析了该方法在点云上的离散化性能。
Self-adjointness criteria and self-adjoint extensions of the Laplace-Beltrami operator on $α$-Grushin manifolds
本文针对α-Grushin流形上Laplace-Beltrami算子的自伴性问题，引入适应奇异几何的伪微分算子演算，给出曲率Laplacian本质自伴性判据，并在非本质自伴情形下构造自然自伴延拓，结果推广至该演算下椭圆的一大类微分算子。
Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$ Δ_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M $$ where ${K(u):={\frac{1}{(4π)^{d/2}}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and ${h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the …
Conformal Spherical Splines for the Laplace--Beltrami Operator on Genus-Zero Surfaces: Construction, Algorithm, and Experiments
本文用共形球样条求解零亏格曲面上Laplace--Beltrami算子的Poisson方程和特征值问题，仅需共形因子描述几何，实验验证了最优收敛阶并优于参数曲面有限元。
The classical and quantum particle on a flag manifold
针对旗流形上的测地线与Laplace–Beltrami算子谱问题，本文给出一族可显式求解的不变度量，并用自旋链型有限维空间逼近函数Hilbert空间以确定谱。
We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided th…
This is an expository article on the question of whether zero lies in the spectrum of the Laplace-Beltrami operator acting on differential forms on a manifold.
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperboli…
A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \cite{RoeOpen}. It allows a description `a la Novikov-Shubin \cite{NS2} of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale ``Weyl asymptotics'' relation. Moreover, in analogy with the Connes-Wodzicki result \cite{CoCMP,Co,Wo}, the asymptotic dimension d measures the singular traceability (at 0) of …
Finsler Geometry, Graph Neural Networks, and You
针对图拉普拉斯只能近似各向同性算子的局限，提出基于芬斯勒拉普拉斯的图神经网络层，证明其收敛性并恢复非线性扩散方程的几何结构。
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