The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.
problem Formulating Lichnerowicz type formulas and Kastler-Kalau-Walze theorems for statistical de Rham Hodge operators.
method Developed Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
result Proved Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.
problem Analyzing perturbations of de Rham Hodge operators on manifolds with boundaries.
method Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
result Proves Kastler-Kalau-Walze type theorems for perturbations of de Rham Hodge operators on 4D and 6D manifolds with or without boundaries.
Discretizes Hodge-Dirac operators on a torus.
problem Capturing geometric aspects of continuum Hodge theory in discrete settings.
method Discrete exterior calculus framework, Hodge-Dirac and Laplace operators.
result Proves discrete Hodge decomposition theorem on combinatorial torus.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on t…
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.
problem Simplicity of Hodge Laplacian and curl operator eigenvalues along metric families.
method Generalized Teytel's method to compute meagre codimension of metrics with specific eigenvalue multiplicities.
result Simplicity of Hodge Laplacian and curl operator is not a meagre codimension 2 property.
Proves the Hodge conjecture for complex projective manifolds.
problem Proving the Hodge conjecture for complex projective manifolds.
method Utilizing the Dirac-Dolbeault operator and Nash-Moser generalized inverse function theorem.
result Existence of complex submanifolds whose fundamental classes span rational Hodge classes.
Study the boundary operator property on simplicial complexes, proving essential properties for Hodge theory.
problem Characterize the boundary operator property ∂∂=0 on simplicial complexes. method Characterization in ℓ2 terms of recurrence of links, defining relative cohomology, and proving harmonic eigenforms. result Essential properties for Hodge theory, including weak decomposition and existence of harmonic eigenforms.
In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.
Let (X,h) be a compact and irreducible Hermitian complex space of complex dimension v>1. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…
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 consider generalized Hodge-Laplace operators αdδ+βδd for α,β>0 on p-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.
Eigenvalue estimates for weighted manifolds with applications.
problem Eigenvalue estimates for weighted Riemannian manifolds.
method Derivation of various eigenvalue estimates for the Hodge Laplacian acting on differential forms.
result Derivation of an inequality relating eigenvalues of the Jacobi operator and the spectrum of the Hodge Laplacian.
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.
problem Ambiguity in viscous operator choice for Navier-Stokes equations on Riemannian manifolds.
method Kinematic construction of strain rate from Lie-dragged vectors, excluding Hodge Laplacian due to antisymmetric part.
result Kinematic selection uniquely identifies the deformation Laplacian, resolving analytical obstructions.
We prove R-bisectoriality and boundedness of the H∞-functional calculus in Lp for all 1<p<∞ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on k-forms.
We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.
We treat two quite different problems related to changes of complex structures on Kähler manifolds by using global geometric method. First, by using operators from Hodge theory on compact Kähler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As …
Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact …
An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calderón-Zygmund and singular integral operators in Lp(C). We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We wi…
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
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…
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.
Reformulates elasticity complex with new differential and Hodge star operators.
problem Elasticity complex and compatibility condition reformulation.
method Generalized differential complex of Dubois-Violette-Henneaux.
result Integrating formula to recover displacement from strain.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…
We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering X over a compact manifold M of dimension n+1. Let Σ be a hypersurface in M which does not disconnect M and such that M−Σ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (Σ)$ is trivial, we can con…
The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This…
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms f…
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the stand…
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
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.
The study bounds Riesz transforms on manifolds with controlled curvature.
problem Bounding Riesz transforms on manifolds with controlled curvature.
method Established Lp-boundedness of local covariant Riesz transforms for differential forms. result Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
The paper develops a new Laplacian for manifold learning from data.
problem Learning manifold structures from point cloud data.
method Constructs deformed Hodge Laplacians and proves their spectral convergence.
result Empirical operators converge to the classical Hodge Laplacian.
The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.