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### First eigenvalue for strictly convex domains

Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
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### When Spectrum of Laplacian in a non-compact manifold is infinite and discrete?

We know that the spectrum of Laplacian in compact smooth manifolds are discrete and infinite. There is a question about spectrum of Laplacian in non-compact case in mathoverflow, Spectrum of ...
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### non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...
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### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3$ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
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### Laplace eigenvalue and floer theory

I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow. Any ...
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### Limited expansion of mean curvature of geodesic spheres

I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...
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### Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
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### Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...
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### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...
If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem \begin{cases} -\Delta u=\lambda u & \mbox{in }\Omega\\ u=0 & \mbox{on }\partial\... 1answer 190 views ### Does the Laplacian commutes with elements of the basis of the Lie algebra? Let G be a compact Lie group with Lie algebra \mathfrak{g}. I know that if g is semi-simple then the Laplace-Beltrami operator on G agrees with the Casimir element and therefore commutes with ... 2answers 379 views ### First eigenvalue of the Laplacian on a regular polygon Consider the Laplacian eigenvalue problem -\Delta u = \lambda u on \Omega with Dirichlet boundary conditions. Let \lambda_1 denote the first eigenvalue. The following theorem is well known: (... 2answers 415 views ### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold It has been proved by Li-Yau and Zhong-Yang that if M is a closed Riemannian manifold of dimension n with nonnegative Ricci curvature, then the first nonzero eigenvalue \lambda_1(M) of the (... 2answers 108 views ### Resolvent operator of fractional Laplacian For 0<\alpha<2, we define the fractional Laplacian with Fourier transform \begin{align} \widehat{(-\Delta)^{\frac{\alpha}{2}} u}(\xi) = |\xi|^\alpha \widehat u(\xi). \end{align} Consider the ... 1answer 387 views ### Stochastic interpretation of heat kernel on fiber bundle I'm looking for a stochastic interpretation of the heat equation for vector valued function. The classical set up is the following : If (M,g) is a riemannian manifold then we could consider the ... 1answer 174 views ### On fundamental solutions to Poisson equation on 3-dimensional manifolds I am interesting in solutions to Poisson equation\triangle \varphi = 4 \pi \rho \qquad (1) defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and $\... 1answer 52 views ### Analytical value for the first eigenvalue of a certain spherical triangle I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ... 1answer 84 views ### Does the green kernel converge as a series of functions? Let$(M,g)$be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a$L^2$operator acting on functions. Moreover we have the convergence of the following series,... 1answer 143 views ### Functional equations for spectral zeta function Functional equation in the theory of zeta functions is one of the important components of this theory. I am interested to know whether the similar property, having functional equation, for the ... 3answers 325 views ### Decompose the Laplacian Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator$A$and its adjoint$A^*$? I am interested in finding such a decomposition, but I could not get one by ... 0answers 158 views ### Solving gradient of an especial heat equation In my research I came up with a gradient of heat equation on a edge-weighted graph as: \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where ... 1answer 349 views ### Fractional Laplacian and stereographic projection The ordinary Laplacian on$\mathbb{R}^N$behaves nicely under a stereographic projection onto$\mathbb{S}^N\setminus\{P\}$. (Here$P$is either the north or south pole of the unit sphere$\mathbb{S}^N$... 0answers 211 views ### Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix I'm trying to find upper boundaries on the smallest Eigenvalue$\lambda_1$of$L + E$, where$L$is a standard Laplacian of an unweighted digraph, with$\lambda_1(L) = 0$and$E \in \{0,1\}^{n \times ...
I seem to see three different things that are being called the Laplacian of a graph, One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...