Proves Kato manifolds satisfy Hodge decomposition.
problem Proving Hodge decomposition for Kato manifolds.
method Relating cohomology to modification data and studying Bott-Chern and Aeppli cohomology.
result Kato manifolds satisfy Hodge decomposition.
New method preserves topology in Hodge decomposition for scalar and vector fields.
problem Topology-preserving Hodge decomposition on manifolds with boundaries.
method Comprehensive 5-component decomposition in Eulerian representation.
result Effective numerical experiments validate the method's accuracy and orthogonality.
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and L2 forms. We further extend the Hodge decomposition to the Sobolev space H1 for general k-forms on non-compact manifolds of nonpositive constant sectional curvature. As a res…
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.
A generalized complex manifold which satisfies the ∂∂-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we …
Study on Lp cohomology and Hodge decomposition for ALE manifolds.
problem Understanding Lp cohomology dimensions and harmonic forms in ALE manifolds. method Relating dimensions of Lp cohomology spaces to decaying harmonic forms, proving independence and jumps in dimensions, and providing Hodge decompositions. result Dimension of Lp reduced cohomology spaces in degree k is independent of p for k not equal to 1 or n-1, and jumps by a factor N-1 for k equal to 1 or n-1. The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.
problem Developing a Lp-Hodge decomposition on sub-Riemannian contact manifolds. method Using a Sobolev approach and recent results from [4] and [6].
result Established an Lp-Hodge decomposition theorem for Rumin's forms on sub-Riemannian contact manifolds. Let $G=\C^{n}\ltimes_φ \C^{m}$ with a semi-simple action $φ: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose G has a lattice Γ. Then we show that in some conditions on G and Γ, G/Γ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By t…
The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.
problem Generalizing Hodge theory to semisimple local systems.
method Establishing a canonical isomorphism and proving a global invariant cycle theorem.
result A new geometric proof of the Decomposition theorem for semisimple local systems.
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.
problem Analyzing harmonic forms and Hodge decomposition on almost Kähler manifolds.
method Using Hodge decomposition and the Hard Lefschetz Condition to study almost Kähler manifolds.
result The spaces of harmonic forms have the Hodge decomposition and the Hard Lefschetz Condition is satisfied.
Paper proves existence of Hadamard states for Maxwell equations.
problem Proving existence of Hadamard states for Maxwell equations on spacetime.
method Introducing Cauchy radiation gauge and new Hodge decomposition.
result Existence of Hadamard states for Maxwell equations proven.
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.
We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomol…
In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the ddc-lemma of Kähler geometry.
We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that t…
Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
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.
Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.
problem Understanding cohomology groups on foliated manifolds.
method Defined and studied Morse-Novikov cohomology relative to a foliation, proving homotopy invariance and extending to more general forms.
result Proved Hodge theorem and Poincaré duality for reduced leafwise Morse-Novikov cohomology groups on Riemannian foliations.
We study solutions for the Hodge laplace equation Δu=ω on p forms with Lr estimates for r>1. Our main hypothesis is that Δ has a spectral gap in L2. We use this to get non classical Lr Hodge decomposition theorems. An interesting feature is …
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.
The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of gov…
Decomposes financial networks to reveal cause-effect hierarchies during crises.
problem Complex financial networks are hard to interpret due to Granger causality.
method Helmholtz-Hodge-Kodaira decomposition to separate networks into rotational and gradient components.
result Precious metals and pharmaceutical products are identified as causal drivers during crises.
New GPs model edge functions on complex networks, capturing divergence and curl.
problem Modeling flow data on networks with independent learning of Hodge components.
method Developed Hodge-compositional edge GPs using Hodge decomposition.
result Hodge-compositional edge GPs can represent any edge function and capture flow relevance.
The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be L2-orthogonally decomposed into a curl-free, divergence-free, and a harmonic field. The harmonic field itself can be further decomposed into three component…
For a certain class of complexes of pre-Hilbert A-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert A-modules and derive the Hodge decomposition for them. We call these complexes self-adjoint parametrix possessing. We show that A-elliptic complexes o…
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
We study rank 1 flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.
Study of Ricci-Yamabe solitons on Walker 3-manifolds.
problem Characterizing Ricci-Yamabe solitons on Walker 3-manifolds.
method Using Hodge decomposition of De-Rham, the soliton field is found from the potential function.
result Classification of all Ricci-Yamabe and gradient Ricci-Yamabe solitons in a Walker 3-manifold.
A new geometric framework resolves singularities in anomalous transport.
problem Mathematical singularities in quantum Berry connections.
method Hodge-de Rham decomposition of the Brillouin zone.
result A smooth geometric proxy potential for anomalous transport.
This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.
problem Understanding the interplay between vortices and harmonic flows on compact surfaces.
method Hodge decomposition of Euler's equations, focusing on point vortices on compact Riemann surfaces.
result The harmonic part of the flow is constant on flat tori but not on non-flat tori.
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…
The paper introduces a new class of manifolds based on the Hodge decomposition and spectral sequences.
problem Characterizing and understanding new classes of compact complex manifolds.
method Introducing a new class of page-r-∂∂ˉ-manifolds and using spectral sequences. result Characterized and provided examples of the new class of manifolds.
Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Develops Hodge theory on ALG∗ manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG∗ manifolds. method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG∗ manifolds. result Non-existence of ALG∗ manifolds with non-negative Ricci curvature at infinity. We show that the eigenspaces of the Laplacian Δk on k-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of Δk lies in the spectrum of Δk+1.
A graph theory approach defines curl and decomposes vector fields.
problem Defining curl for vector fields on graphs and decomposing them.
method Definition of curl as orthogonal complement of circulation-free fields, proving analogues of vector field theorems.
result Helmholtz-Hodge decomposition on graphs: gradient, curl, and harmonic fields.
Study cohomology of Bigolin complex on complex manifolds.
problem Characterize cohomology of Bigolin complex on compact complex manifolds.
method Analyze the decomposition of the double complex into squares and zigzags, focusing on the zigzags contributing to cohomology.
result In complex dimension 3, multiplicities of zigzags are characterized by Betti, Hodge, Aeppli numbers plus Bigolin numbers.
The paper develops L2-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.
problem Proving the Hopf conjecture for almost Kähler manifolds.
method Developed L2-Hodge theory identities and applied them to prove vanishing theorems and refine estimates. result Proved the Hopf conjecture for compact almost Kähler manifolds with negative sectional curvature.
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.
We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holon…
Novel method CHPCA simplifies complex market dynamics.
problem Quantifying interactions in rapidly evolving consumer goods markets.
method Complex Hilbert Principal Component Analysis (CHPCA) and Hodge decomposition.
result Revealed comovements and customer heterogeneity in consumer choice process.
In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic behaviour of spaces of harmonic forms by means of a certain Čech-de Rham complex…
Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition o…
Study on G2-structures using Laplacian coflow and solitons.
problem Characterizing and understanding G2-structures and their solitons. method Using the irreducible G2-decomposition of the Hodge Laplacian and Lie derivative, characterizing infinitesimal symmetries and soliton conditions. result Proof of the absence of compact shrinking solitons for the Laplacian coflow.
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.