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.
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.
Let (E,h) be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.
The paper studies spectral convergence of connections on vector and principal bundles.
problem Continuity of eigenvalues of connection Laplacians on vector bundles.
method Introducing a new topology on metric measure spaces with isometric G-actions to analyze convergence of G-connections. result Established spectral convergence of connections on vector and principal bundles.
We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…
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…
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.
The paper equates the index of a vector bundle to the manifold's index and introduces an analytic torsion.
problem Equating the index of a vector bundle to the manifold's index.
method Adiabatic limit and Witten Laplacian.
result The Quillen metric of the analytic torsion on the vector bundle coincides with the manifold's metric.
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 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.
Proves stability of geometric flows on compact manifolds.
problem Stability of geometric flows of closed sections.
method General result about stability of geometric flows.
result Proves stability of modified Laplacian coflow and balanced flow.
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…
Unified approach to data processing using gauge theory.
problem Data representation and analysis with consistent symmetry.
method Geometric gauge theory for discrete vector bundles.
result Unified understanding of heat kernel properties and data transformation.
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.
The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.
problem Studying spectral properties of the horizontal Laplacian
method Interpreting the horizontal Laplacian as a twisted Laplacian acting on a flat vector bundle
result The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold
The paper proves heat kernel asymptotics for high power line bundles on complex manifolds.
problem Proving heat kernel asymptotics for Kodaira Laplacians of high power line bundles.
method Scaling technique applied to both compact and non-compact manifolds.
result Direct proof of holomorphic Morse inequalities and generalization to vector bundles.
Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geom…
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.
The paper studies harmonic symmetric bilinear forms on Riemannian manifolds and proves properties of the Bourguignon Laplacian.
problem Analyzing harmonic symmetric bilinear forms on Riemannian manifolds.
method Developed the theory of harmonic symmetric bilinear forms and proved properties of the Bourguignon Laplacian.
result The kernel of the Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold.
We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, w…
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…
The paper proves estimates for Hodge Laplacians on Lie groups.
problem Estimating Hodge Laplacians on semisimple Lie groups.
method Proves Schwartz estimates for Hodge Laplacian and Dirac operators.
result Generalizes results on symmetric spaces to Lie groups.
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.
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…
We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat in…
Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.
problem Asymptotic expansion of graph Laplacian on discretized surfaces.
method Relate spanning trees and cycle-rooted spanning forests to zeta-regularized determinants.
result Explicit formula for limit of cycle-rooted spanning forest probability and topological observables.
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal s…
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
Characterizes lamination spaces of graphs on a pair of pants.
problem Understanding lamination spaces of graphs on a pair of pants.
method Identifying lamination spaces as lattice polytopes and using graph exploration technique.
result Characterizes the polytopes that arise as lamination spaces of graphs on a pair of pants.
LA-VDM accelerates VDM using landmarks to improve data analysis.
problem Efficiently analyzing complex datasets with nonuniform sampling densities.
method Landmark-constrained two-stage normalization to accelerate VDM.
result LA-VDM accurately recovers parallel transport and converges to the connection Laplacian.
Develops global pseudo-differential calculus on homogeneous vector bundles.
problem Global theory of subelliptic pseudo-differential operators on homogeneous vector bundles.
method Global symbolic calculus, complex functional calculus, Hörmander system of vector-fields.
result Global pseudo-differential calculus on homogeneous vector bundles.
The study generalizes Bochner Laplacian results to Riemann surfaces.
problem Analyzing curvature vanishing line bundles on Riemann surfaces.
method Exploiting the relation of Bochner Laplacian on tensor powers with sR Laplacian.
result Bergman kernel expansion for semi-positive line bundles.
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
Conservation of heat in manifolds with boundary under mixed conditions.
problem Conservation of heat in manifolds with boundary and mixed conditions.
method Uniform lower bounds on the zero order piece of the Dirac Laplacian and on the endomorphism defining the mixed boundary condition.
result Conservation principle holds under suitable geometric control.
Unified framework for Sobolev spaces on vector bundles, including explicit integration by parts.
problem Developing a comprehensive theory for Sobolev spaces on vector bundles.
method Explicit higher-order geometric integration by parts formula on arbitrary Riemannian manifolds.
result Direct proofs of classical theorems in Sobolev spaces on vector bundles.
Study f-Laplace bounds on gradient Ricci shrinkers, applying to Betti numbers.
problem Bounding eigenvalues of f-Laplacian on gradient Ricci shrinkers. method Upper and lower bounds established using volume growth rate; extends to vector bundles.
result Explicit upper bounds for Betti numbers derived.
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of…
We consider the problem of identifying a unitary Yang-Mills connection ∇ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian ∇∗∇ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the cas…
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
In this paper we study the first eigenvalue of the Laplacian on a compact Kaehler manifold using stable bundles and balanced bases.
We study the behaviour of Laplace-type operators H on a complex vector bundle E → M in the adiabatic limit of the base space. This space is a fibre bundle M → B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/ε. Under a gap condi…
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
Study connects spectral properties to frame flows on curved manifolds.
problem Spectral properties and frame flows on curved manifolds.
method Link between spectral properties, frame flows, and polynomial maps between spheres.
result Ergodicity of frame flows on low-rank bundles.
Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.
problem Analyzing self-adjoint extensions of Dolbeault Laplacians on Riemann surfaces.
method Defined ζ-regularized determinants, introduced Robin mass, derived comparison formulas. result Explicit expressions for Robin mass in spinor bundles and scalar cases.
Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
problem Understanding gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
method Analyzes high tensor powers of Hermitian line bundles with non degenerate curvature, proving Riemann-Roch numbers for eigenvalue clusters and describing spectral projectors.
result Clusters and gaps in eigenvalues are described by Riemann-Roch numbers and have pointwise kernel descriptions.