arXiv research

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

Study coclosed G2-structures on SU(2)²-invariant manifolds.

We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $\mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $\mathfrak g$ to a six-dimensional Lie algebra $\mathfrak h$, with kernel contained in the center of $…

We prove that Einstein coclosed G_2-structures are nearly parallel.

The study characterizes G₂-structures on 2-step nilpotent Lie groups.

Classifies nilpotent Lie groups with specific $G_2$-structures.

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic …

Completes classification of G2-structures on specific nilpotent Lie groups.

This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.

Study on $G_2$-structures using Laplacian coflow and solitons.

Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.

Study conformal Killing forms on specific nilpotent Lie groups.

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an…

We prove a refined Kato inequality for closed and coclosed differential $(p,q)$ forms on a Kahler manifold.

Flow solves $G_2$ system, proving existence of torsion-free metrics.

We introduce coG_2-vector fields, coRochesterian 2-forms and coRochesterian vector fields on manifolds with a coclosed G_2-structure as a continuous of work from [15], and we show that the spaces of coG_2-vector fields and of coRochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the …

Study $G_2$-flows reducing to complex geometry flows, focusing on $G_2$-anomaly and $G_2$-Laplacian coflow.

Study heterotic G2-system on 2-step nilmanifolds with torus bundles.

If M is a riemannian manifold, then the inclusion of the complex of coclosed harmonic forms into the de Rham complex induces a linear isomorphism in cohomology. If M has at most countably many connected components, this linear isomorphism is a Frechet isomorphism.

A flow from hypersymplectic to hyperkähler structures is described.

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} ψ= - Δ_d ψ$ where $ψ$ is the dual 4-form of a $G_2$-structure $φ$ and $Δ_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($dψ=0$). We study this flow for two explicit examples of coclosed $…

Topological complexity for closed 1-forms

Study on Riemannian Poisson warped product spaces and their properties.

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…

New insights into cohomology of closed 1-forms.

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

Indices of vector fields and 1-forms studied for singular varieties and actions.

The variational problem for the functional $F=\frac12\|φ^*ω\|_{L^2}^2$ is considered, where $φ:(M,g)\to (N,ω)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the D…

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

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.

New metrics produce discrete zero sets for nondegenerate harmonic forms.

We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…

The study resolves a conjecture about harmonic forms on compact manifolds.

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

Study topological properties of foliations induced by closed 1-forms on orbifolds.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

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.

Due to spectral obstructions, a scattering theory in the Lax-Phillips sense for the wave equation for differential p-forms on H^{n+1} cannot be developed. As a consequence, Huygens' principle for the wave equation in this context does not hold. If we restrict the class of forms and we consider the case of coclosed p-fo…

Constructs deformations of Vaisman manifolds preserving foliations.

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.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.

We prove in this article that given a linearly concave domain $D$ in the projective space $\Bbb{CP}^{n}$, 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 $\Bbb{CP}^{n}$ 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.