Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.
problem Geodesics and Laplace-Beltrami spectrum on flag manifolds.
method Invariant metrics and finite-dimensional approximations.
result Explicit solutions for geodesics and spectrum.
The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.
problem Analyzing heat-type equations on manifolds with specific boundary conditions.
method Proving Schauder estimates for the Laplace-Beltrami operator on manifolds with fibered boundaries and a Φ-metric.
result The proof of parabolic Schauder estimates for the Laplace-Beltrami operator.
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 o…
Lower bound found for eigenvalue of hypersurface in Riemannian manifold.
problem Finding bounds for eigenvalues of hypersurfaces in Riemannian manifolds.
method Used minimally embedded hypersurface and Ricci curvature constraints.
result Provided a lower bound for the first eigenvalue.
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.
The study bounds heat kernel for manifolds with specific curvature conditions.
problem Estimating heat kernel for manifolds with Bakry-Émery Ricci curvature.
method Gaussian upper bound for heat kernel, proving L^1-Liouville property, deriving eigenvalue bounds.
result Established Gaussian upper bound for heat kernel, derived eigenvalue bounds.
Anisotropic metric on manifolds uniquely determined by boundary data.
problem Determining Riemannian metrics on compact manifolds from boundary measurements.
method Analysis of the Dirichlet-to-Neumann map for the Laplace-Beltrami operator.
result Riemannian metrics can be uniquely determined up to isometry.
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 …
We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension n≥2, with volume close to the volume of the manifold. If the first (positive) eigenfunction φ0 of the Laplace-Beltrami operator over the manifold is a nonconst…
Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.
problem Understanding the spectrum and geodesics of sigma models on flag manifolds.
method Connecting SU(n) spin chains to sigma models and calculating spectra and geodesics.
result Calculated the spectrum of the Laplace-Beltrami operator and geodesics for CP1 and F3. 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 t…
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 eig…
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. …
New criterion for wave operators on Kato-Ricci manifolds.
problem Existence and completeness of wave operators for Laplace-Beltrami operators.
method Proves L1 criterion using heat semigroup estimates for Kato-Ricci manifolds. result Establishes new conditions for wave operators on Kato-Ricci manifolds.
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 manifold…
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
We explicitely compute the essential spectrum of the Laplace-Beltrami operator for p-forms for the class of warped product metrics dσ2=y2ady2+y2bdθ∂M2, where y is a boundary defining function on a compact manifold with boundary M.
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
Study shows algebraic nature of manifold submetries on compact spaces.
problem Understanding manifold submetries on compact homogeneous spaces.
method Analyzes singular Riemannian foliations and manifold submetries on compact normal homogeneous spaces.
result Establishes a one-to-one correspondence between algebras of preserved functions and manifold submetries.
Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.
problem Calculating the spectrum of the Laplace-Beltrami operator on homogeneous spaces.
method Formula derivation based on eigenvalues of a generalized Casimir operator and spherical representations.
result First detailed computation and investigation of the spectrum for a family of metrics on the Aloff-Wallach manifold.
The paper bounds Cheeger ratios of eigenfunctions and their level sets.
problem Understanding geometric features of Riemannian manifolds through eigenfunctions.
method Constructive upper bounds on Cheeger constants using eigenvalues and eigenfunctions.
result Upper bounds on Cheeger ratios of eigenfunction level sets and their superlevel sets.
Study on 2-ruled hypersurfaces in a Walker 4-manifold.
problem Characterize and analyze 2-ruled hypersurfaces in a Walker 4-manifold.
method Define and analyze three types of 2-ruled hypersurfaces, compute Gaussian and mean curvatures, and study Laplace-Beltrami operators.
result Characterizations and properties of 2-ruled hypersurfaces in a Walker 4-manifold.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
Leibniz cohomology reveals connections on manifolds.
problem Understanding connections on Riemannian manifolds using Leibniz cohomology.
method Expressing Levi-Civita connection as a cochain in Leibniz cohomology of vector fields.
result Vanishing of Leibniz coboundary implies eigenfunctions of the Laplacian.
Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.
problem Understanding the spectral properties of Laplace-Beltrami operators on Riemannian symmetric spaces.
method Using symplectic geometry and geometric quantization, associating flag manifolds to symmetric spaces and relating their Satake diagrams.
result Harmonic polynomials on flag manifolds induce all eigenfunctions on symmetric spaces.
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 eige…
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…
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.
Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.
problem Spectral uniqueness of complex/quaternionic structures on manifolds.
method Explicit expression for smallest positive eigenvalue of Laplace-Beltrami operator.
result Irreducible symmetric spaces are spectrally unique within families of homogeneous metrics.
The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.
problem Bounding eigenvalues of geometrically finite hyperbolic manifolds of infinite volume.
method Provided a lower bound on the kth eigenvalue of the Laplace-Beltrami operator by the kth eigenvalue of a neighborhood of the thick part of the convex core.
result Recovered a theorem bounding the bottom eigenvalue from below by a specific formula involving the volume of the 1-neighborhood of the convex core.
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from (ε,ρ)-proximity graphs. result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.
problem Classical and quantum problems for 1D sigma model with specific target space.
method Mapping to Gaudin model, solving polynomial equations.
result Explicit description of geodesics and spectrum found.
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
The paper bounds eigenvalues and integrals of eigenfunctions on hyperbolic manifolds.
problem Eigenvalues and integrals of eigenfunctions on compact hyperbolic manifolds.
method Spectral decompositions and consistency conditions derived from quadruple overlap integrals.
result Upper bounds on Laplacian eigenvalues and triple overlap integrals.
Study the geometry of gas giant planets to infer their internal structure.
problem Determine the interior structure of gas giant planets using boundary data.
method Geometric analysis of Riemannian manifolds with conformal blow-up at the boundary.
result The interior structure of a gas giant is uniquely determined by different types of boundary data.
We consider non-self-adjoint Schrödinger operators Δ+V where Δ is the Laplace-Beltrami operator on a Zoll manifold X and V∈C∞(X,C). We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on α-Grushin manifolds. method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.
problem Constraining the spectra of Laplace operators on hyperbolic manifolds and orbifolds.
method Linear programming and spectral identities derived from the conformal bootstrap and Selberg trace formula.
result Upper bounds on the first and second Laplacian eigenvalues, and spectral gaps of hyperbolic 3-manifolds and orbifolds.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
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 pr…