The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
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.
To every Hermitian vector bundle with connection over a compact Riemannian manifold M one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of M and prove that their spectra converge, as…
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
Maps embed manifolds using heat kernels of connection Laplacian.
problem Embedding manifolds in Euclidean space.
method Using heat kernels of the connection Laplacian and truncated heat kernels.
result Maps can be made arbitrarily close to isometries.
The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian. Unlike the …
Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.
problem Determining geometric data from boundary symbol of connection Laplacian.
method Analyze symbol of Dirichlet-to-Neumann operator associated with connection Laplacian.
result Geometric data on boundary and normal derivatives are determined by symbol.
Uniqueness proof for Calderón's problem on real-analytic vector bundles.
problem Recovering topology and geometry from boundary measurements of vector bundles.
method Real-analyticity assumption for uniqueness proof.
result Topology and geometry of vector bundles can be recovered from boundary measurements.
The paper develops a method to sparsify magnetic Laplacians using multi-type spanning forests.
problem Sparsifying magnetic Laplacians for large and dense graphs.
method Sampling multi-type spanning forests using a determinantal point process.
result The method provides statistical guarantees for estimating the connection Laplacian.
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.
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…
We consider a connection ∇X on a complex line bundle over a Riemann surface with boundary M0, with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L:=∇X∗∇X+q, with q a complex valued potential, uniquely determines the…
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a…
Modified Laplacian connects to Yang-Mills instantons on manifolds.
problem Understanding instantons on 4D manifolds.
method Infinite dimensional Lévy Laplacian defined on manifolds, parameterized by curves in orthogonal rotations.
result Instantons on 4D manifolds are related to the modified Lévy Laplacian under specific curve conditions.
Improved matching for multiple objects using a novel reweighting method.
problem Current multi-object matching methods have limitations and are not robust.
method Proposes a novel iterative reweighting strategy using the graph connection Laplacian.
result Demonstrates superior performance over state-of-the-art methods.
Given a class of closed Riemannian manifolds with prescribed geometric conditions, we introduce an embedding of the manifolds into ℓ2 based on the heat kernel of the Connection Laplacian associated with the Levi-Civita connection on the tangent bundle. As a result, we can construct a distance in this class which …
This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.
problem Understanding the mathematics behind spectral clustering and its equivalence to PCA.
method Dividing spectral clustering into two categories based on graph connectivity and proving the equivalence to PCA.
result Spectral clustering and PCA are equivalent, with specific proofs for fully connected and multi-connected graphs.
Proves rotational symmetry for Serrin-type problems in doubly connected domains.
problem Proving symmetry in Serrin-type problems for doubly connected domains.
method Employing the technique from arXiv:2109.11255 and comparing with the classical moving plane method.
result Rotational symmetry results for Serrin-type problems in doubly connected domains.
In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry.
Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.
problem Modeling conic Laplacian on \(\mb P^1\) with specific boundary conditions.
method Fourier decomposition, Legendre equations, gluing map, Friedrichs spectrum, Weyl function.
result Explicit computation of eigenfunctions and \(S\)-matrix.
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.
We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.
The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.
problem Finding surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map.
method Investigated 2D submanifolds of R^2, classified surfaces of genus 0 or with k≥3 boundary components.
result One-parameter family of 2D submanifolds with commuting Laplacian and Dirichlet-to-Neumann map.
The paper finds upper bounds for eigenvalues of a weighted Laplacian.
problem Finding bounds for eigenvalues of a weighted Laplacian.
method Analyzing the operator \( L_h \) with Neumann boundary conditions.
result Existence of upper bounds for eigenvalues of the weighted Laplacian.
We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set the bandwidth by optimizing the Laplacian's ability to preser…
New derivation of Type IIA flow metrics.
problem Flow of metrics in Type IIA theory.
method Adapted to Laplacian flow, uses projected Levi-Civita connection.
result New derivation of flow equations.
We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boun…
We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact Kähler manifolds.
In this paper we consider the problem of identifying a connection ∇ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ∇∗∇ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of t…
Study spectral properties of graph Laplacian for manifold data.
problem Understanding spectral properties of graph Laplacian for manifold data.
method Non-asymptotic error bounds on spectral properties of empirical graph Laplacian.
result Eigenvalues and eigenspaces of empirical graph Laplacian are close to Laplace-Beltrami operator of manifold.
The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
Study resolvents of Bochner Laplacians on compact manifolds.
problem Analyzing the resolvents of Bochner Laplacians in the semiclassical limit.
method Introducing Heisenberg semiclassical pseudodifferential operators to study sections of line bundles.
result Resolvents and spectral projections of Bochner Laplacians are studied in the large power limit.
Formulae connect heat kernels on glued manifolds.
problem Connecting heat kernels on joined manifolds.
method Proved gluing formulae for Laplacian heat kernels.
result Formulae linking heat kernels on joined manifolds.
Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.
problem Analyzing spectra of magnetic Laplacians on non-simply connected manifolds.
method Definition of spectral varieties and construction of conformal invariants.
result New conformal invariants of immersions of surfaces into 3- and 4-dimensional spaces.
Paper proves conditions for estimating precision matrices with Laplacian constraints.
problem Estimating high-dimensional precision matrices with Laplacian constraints.
method Minimizing Stein's loss with conditions on graph connectivity and Laplacian constraints.
result High-dimensional consistency achieved with Laplacian constraints, independent of graph structure.
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …
The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
problem Finding lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
method Analyzing the spectrum of the Laplacian with magnetic Neumann boundary conditions, focusing on multiply connected domains with convex curves. Lower bounds are derived based on geometric invariants such as area, perimeter, diameter, and fluxes around inner holes.
result Sharp lower bounds for the first eigenvalue are derived for doubly connected domains and domains with an arbitrary number of holes, and a lower bound is obtained for Aharonov-Bohm operators with an arbitrary number of poles when holes shrink to points.
The paper studies eigenvalues of a special Laplacian system on compact manifolds.
problem Investigating the first eigenvalue of the (p,q)-Laplacian system on compact manifolds. method Analyzing the (p,q)-Laplacian system on compact Riemannian manifolds without boundary. result For large eigenvalues, there exists a conformal metric to the standard metric of Sm. New formula for Lichnerowicz Laplacian on homogeneous spaces.
problem Finding new Einstein metrics on homogeneous spaces.
method Using Casimir operators to derive a new formula for the Lichnerowicz Laplacian.
result Derives many new Einstein metrics stable in the Einstein-Hilbert sense.
In this article we prove that the spectrum of the Laplacian on k-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.
Study finds bounds for fundamental tone on special Riemannian manifolds.
problem Finding bounds for fundamental tones on specific Riemannian manifolds.
method Developed a general lower bound for the fundamental tone of the p-Laplacian.
result Results applied to negatively curved manifolds, warped products, and Riemannian submersions.
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components …
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…
We estimate the eigenvalues of connection Laplacians in terms of the non-triviality of the holonomy.
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.