Classifies rational differential forms on the Riemann sphere based on their isotropy group.
problem Classifying rational differential forms on the Riemann sphere based on their isotropy group.
method Analyzing the isotropy group of rational differential forms with simple poles and zeros.
result Finite subgroups of PSL(2, C) are realizable as isotropy groups for rational 1-forms on the Riemann sphere.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
The study classifies rational 1-forms on the Riemann sphere with simple poles.
problem Classifying rational 1-forms on the Riemann sphere with specified pole conditions.
method Recognized three equivalent atlases, proved submanifold properties, and used PSL(2,C) action.
result Quotients of isochronous 1-forms admit stratified orbit types.
Algorithm constructs integrals of hyperexponential 1-forms.
problem Computing integrals of hyperexponential 1-forms.
method Using Schanuel's conjecture, decomposes Hω into f(F(x))+H(x)R(x). result Algorithm constructs a basis of cohomology of differential 1-forms.
The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.
problem Understanding the spectral gap for coclosed 1-forms in hyperbolic 3-manifolds.
method Constructing sequences of manifolds and analyzing their spectral properties and homology growth.
result Sequences of hyperbolic manifolds can have uniform spectral gaps for coclosed 1-forms but unbounded torsion homology growth.
We consider the closed orbit structure of generic gradient flows of Morse closed 1-forms. The torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. We extend this result to arbitrary Morse closed 1-form…
New metrics produce discrete zero sets for nondegenerate harmonic forms.
problem Creating metrics to produce discrete zero sets for nondegenerate harmonic forms.
method Metric perturbation to produce new nondegenerate harmonic forms with discrete zero sets.
result Existence of metrics producing discrete zero sets for nondegenerate harmonic forms.
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. New examples of hyperbolic 3-manifolds where Seiberg-Witten equations fail.
problem Finding hyperbolic 3-manifolds without irreducible Seiberg-Witten solutions.
method Combining hyperbolic geometry, upper bound for eigenvalues, Selberg trace formula, and precise numerical bounds.
result First examples of hyperbolic 3-manifolds where Seiberg-Witten equations do not admit irreducible solutions.
The paper describes spectra of operators on rational homogeneous varieties.
problem Spectrum of invariant (1,1)-forms and Weitzenböck remainder. method Detailed analysis using index theory and Lie-theoretic data.
result Explicit formula for smallest eigenvalue and new lower bounds.
Study Fano fibrations and Kähler-Einstein metrics on their bases.
problem Understand geometric structures and metrics on Fano fibrations.
method Construct (1,1)-forms and solve twisted Kähler-Einstein equations. result Singular Kähler metrics on Fano fibrations satisfy twisted Kähler-Einstein equations.
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformat…
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
Study on Riemannian Poisson warped product spaces and their properties.
problem Characterizing and understanding Riemannian Poisson warped product spaces.
method Formal treatment of Killing and 2-Killing 1-forms on Riemannian Poisson manifolds, including Bochner type results.
result Characterization of 2-Killing 1-form on (R2,g,Π) and Bochner type results on compact spaces. This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lu…
The paper explores parallel 1-forms on special Finsler manifolds and their properties.
problem Investigating parallel 1-forms on specific Finsler manifolds.
method Analyzing Landsberg manifolds, metrizability freedom, and specific Finsler metrics.
result Landsberg surfaces with parallel 1-forms are necessarily Berwaldian, and the metrizability freedom is at least 2.
New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
In this paper we continue the study of generic properties of the Novikov complex, began in the work "The incidence coefficients in the Novikov complex are generically rational functions" ( dg-ga/9603006). For a Morse map f:M→S1 there is a refined version of Novikov complex, defined over the Novikov completion of …
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…
S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. The paper defines vector 1-forms on Finsler manifolds and constructs connections.
problem Characterizing conservative connections on Finsler manifolds.
method Defining conservative semibasic vector 1-forms and constructing connections.
result A correspondence between torsion-free semibasic vector 1-forms and vertical vector fields.
A conformal metric g with constant curvature one and finite conical singularities on a compact Riemann surface Σ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function f on Σ\{singularities}, called the {\it developing …
Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.
problem Modifying Z/2 harmonic 1-forms under weak regularity assumptions.
method Calabi surgery method, involving cutting and pasting closed 2-valued 1-forms.
result Flexible construction and modification of Z/2 harmonic 1-forms.
The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.
problem The vanishing and finiteness of p-harmonic 1-forms on submanifolds.
method Using BiRic curvature conditions to prove theorems.
result Theorems on vanishing and finiteness of p-harmonic 1-forms.
Study topological properties of foliations induced by closed 1-forms on orbifolds.
problem Characterize the topology of foliation leaves induced by closed 1-forms on orbifolds.
method Establish criteria for the compactness of foliation leaves and extend a topological result to orbifolds.
result Criteria for the compactness and coexistence of foliation leaves are established.
We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel…
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
Vanishing theorem for L2-harmonic forms on Riemannian manifolds with parallel 1-form.
problem Proving vanishing of L2-harmonic forms on Riemannian manifolds with a parallel 1-form. method Using L2 Morse-Novikov cohomology and a vanishing theorem. result The L2-harmonic forms on the manifold are identically zero. The paper studies local normal forms of singular contact forms and primitive 1-forms.
problem Local normal forms of singular contact forms and primitive 1-forms.
method Combines classical normalization techniques and toric approach.
result Extends and improves previous results on first-order contact forms and primitive 1-forms.
Constructs deformations of Vaisman manifolds preserving foliations.
problem Deforming Vaisman manifolds while maintaining their canonical foliations.
method Uses a basic 1-form with specific properties to construct transverse deformations.
result Basic 1-forms exist in abundance for constructing deformations.
We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.
We prove in this article that given a linearly concave domain D in the projective space CPn, a 1-dimensional comlex analytic set V in D, and a meromorphic 1-form φ on V, V is a subset of an algebraic variety of CPn and φ is the restriction to V of an algebraic 1-form on $\Bbb{CP}^{…
Some years ago Moshé Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathem…
The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with a closed 1-form to a traditional problem with a Morse function.…
New examples of Z/2 harmonic 1-forms and their branching sets are explored.
problem Exploring the properties and examples of Z/2 harmonic 1-forms and their branching sets.
method Elementary constructions and families of Z2 harmonic 1-forms. result The branching set Σ of a Z2 harmonic 1-form can exhibit various features including non-trivial links, multiple covers, and immersed structures. Paper explores connection cochain in abelian extensions and its relation to connection forms.
problem Understanding the connection cochain in abelian extensions.
method Apply Moriyoshi's connection cochain concept to abelian extensions and relate it to connection 1-forms.
result Established the relationship between connection cochain and connection 1-forms in abelian extensions.
We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, J1π. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian i…
The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
No nontrivial harmonic 1-forms on certain gradient Ricci solitons.
problem Existence of nontrivial harmonic 1-forms on gradient Ricci solitons.
method Proving Liouville theorems for harmonic 1-forms.
result No nontrivial L2-integrable harmonic 1-forms on specified solitons. This paper extends Lusternik-Schnirelmann category to non-compact manifolds.
problem Extending Lusternik-Schnirelmann category to non-compact manifolds.
method Explanation and extension of Farber's results to non-compact manifolds.
result Farber's results hold equally well on non-compact manifolds, and new phenomena occur in gradient flows.
Classifies vector fields in the kernel of a 1-form, up to equivalence.
problem Classifying vector fields in the kernel of a 1-form.
method Equivalence relation, local models, transversal unfoldings.
result Provides a list of local models and transversal unfoldings for vector fields.