Paper provides integral representation for fractional Laplace-Beltrami operators.
problem Fractional Laplace-Beltrami operators on general Riemannian manifolds.
method Two different proofs for the same result: compact manifolds with or without boundary, and riemannian manifolds without boundary with bounded Ricci curvature.
result Integral representation of fractional Laplace-Beltrami operators for general Riemannian manifolds.
Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic s…
The paper constructs Laplace-Beltrami operators on noncommutative tori.
problem Developing Laplace-Beltrami operators for noncommutative tori.
method Construction of Laplace-Beltrami operators with consideration of non-trivial modular automorphisms.
result Laplace-Beltrami operators on noncommutative tori have properties similar to those on ordinary Riemannian manifolds.
Improved bounds on Laplace-Beltrami operator eigenvalues on real projective plane.
problem Improving upper bounds for Laplace-Beltrami operator eigenvalues.
method Analyzing eigenvalues with even indexes and providing bounds for Dirichlet, Neumann, and Steklov eigenvalues.
result Enhanced upper bounds for Laplace-Beltrami operator eigenvalues on the real projective plane.
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.
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.
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.
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 studies the spectrum of Laplace-Beltrami operators on complex spaces.
problem Analyzing the spectrum of Laplace-Beltrami operators on compact complex spaces.
method Examined the Friedrichs extension of Laplace-Beltrami and Hodge-Kodaira Laplacians, providing estimates for eigenvalues and trace-class properties.
result Discrete spectrum and trace-class properties of Laplace-Beltrami operators on compact complex spaces.
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.
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.
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 differ…
Study eigenvalues of Laplace-Beltrami under Ricci flow on 3-manifolds.
problem Eigenvalue behavior under Ricci flow on 3-manifolds.
method Monotonic quantities and bounds constructed for the first eigenvalue.
result Eigenvalue tends to zero in converging cases after rescaling.
Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on Rn endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).
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 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.
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 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.
The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using hp-adaptive finite element methods. result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
Upper bounds on Laplace-Beltrami operator found via volume growth.
problem Finding upper bounds for the spectral function on homogeneous spaces.
method Spectral embeddings and volume growth of balls.
result Improved bounds on eigenvalues and heat kernel.
Graph Laplacian converges to Laplace-Beltrami operator with a specific rate.
problem Convergence of graph Laplacian to Laplace-Beltrami operator on random geometric graphs.
method Analysis of random geometric graphs and eigenvalue convergence rates.
result Eigenvalues and eigenvectors of graph Laplacian converge to Laplace-Beltrami operator with rate O((nlogn)2m1). 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 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.
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…
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.
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.
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.
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…
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…
Estimates on Dirac operator eigenvalues for reducible manifolds.
problem Estimating eigenvalues of the Dirac operator on specific manifolds.
method Using Laplace-Beltrami operator and scalar curvature for estimates.
result Sharp estimates on the first eigenvalue, reducing to Alexandrov's result.
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. …
Study eigenvalues of CROSS spaces with various metrics.
problem Eigenvalue of Laplace-Beltrami operator on CROSSes.
method Explicit formulae for the first eigenvalue of CROSSes with homogeneous metrics.
result Homogeneous metrics on CROSSes are isospectral if and only if they are isometric.
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…
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 Delau…
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. 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.
Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).
problem Finding uniform bounds for spectral gaps of Schottky subgroups.
method Establishes explicit lower bounds for the second eigenvalue of the Laplace-Beltrami operator.
result Uniform and explicit lower bounds for the second eigenvalue of congruence coverings.
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…
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
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…
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 geometric quantum confinement on special incomplete Riemannian manifolds.
problem Characterize quantum confinement on Grushin-type manifolds.
method Constant-fibre direct integral scheme combined with Weyl's analysis.
result Fully characterizes essential self-adjointness of Laplace-Beltrami operator.
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.
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.
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in R3 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 cylinde…
The paper proves eigenvalues are simple for specific operators on bundles.
problem Eigenvalue simplicity for connection Laplacian and G-simplicity on bundles. method Analyzes connections on vector bundles and principal bundles, proving eigenvalue simplicity for a residual set of connections.
result Eigenvalues of the connection Laplacian and Laplace-Beltrami operator are simple for specified conditions.