arXiv research

This short note aims at (re)proving that the symmetrically normalized graph Laplacian $L=\Id - D^{-1/2}WD^{-1/2}$ (from a graph defined from a Gaussian weighting kernel on a sampled smooth manifold) converges towards the continuous Manifold Laplacian when the sampling become infinitely dense. The convergence rate with …

Study a modified Laplacian equation in spacetime.

Graph Laplacians converge under symmetric divergence conditions.

Proposes a symmetric graph autoencoder for unsupervised learning.

The paper studies eigenvalues and Cheeger constants on symmetric graphs.

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

Enhances GCNs using VAT for better node classification.

Formula for twisted orbital integrals using hypoelliptic Laplacian.

Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^n$. The spaces of symmetric eigentensors are explicitly given.

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimens…

The study examines the geometry of Lichnerowicz Laplacian's kernel on various spaces.

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of complex symmetric covariant tensor fields on the complex projective space $P^n(\comp)$. The spaces of symmetric eigentensors are explicitly given.

The paper studies harmonic symmetric bilinear forms on Riemannian manifolds and proves properties of the Bourguignon Laplacian.

New method for mixed memberships using symmetrized Laplacian inverse matrix.

We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

Study eigenvalues of drift Laplacian on symmetric self-shrinkers in R^3.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

The paper studies Laplacians on smooth distributions and proves they are multipliers in $C^*$-algebras.

More than forty years ago J. H. Samson has defined the Laplacian $Δ_{sym}$ acting on the space of symmetric covariant $p$-tensors on an $n$-dimensional Riemannian manifold $(M, g)$. This operator is an analogue of the well known Hodge-de Rham Laplacian $Δ$ which acts on the space of exterior differential $p$-forms ($1 …

In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

We prove the existence of a smooth family of non-compact domains $Omega_s \subset R^{n+1}$ bifurcating from the straight cylinder $B^n \times R$ for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary. The domains $Omega_s$ are rotationally s…

New method of symmetrization applied to PDEs on spheres.

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

New method to bound Laplacian eigenvalues of geodesic balls.

Let $(M,g)$ be a globally symmetric space of noncompact type, of arbitrary rank, and $Δ$ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted fr…

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an e…

Paper studies minimax optimal regression using Laplacian smoothing over graphs.

The paper compares eigenvalues of Laplacians on fibred manifolds using symmetrization techniques.

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

Extends graph theory to hypergraphs with manifold-valued nodes.

Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the ce…

Stability of Einstein metrics under Ricci iteration studied.

In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distributi…

We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.

The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.

Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians $L_m$ are defined for each positive integer $m$, which generalize the ordinary Laplace-Beltrami operator. We show …

Inverse problem solved for rotationally symmetric manifolds using eigenvalues and resonances.

The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.

Enhances privacy in federated learning with Laplacian smoothing.

New nodal domain theorems for symmetric matrices via signed graphs.