arXiv research

It is shown that an HKT-space with closed parallel potential 1-form has $D(2,1;-1)$-symmetry. Every locally conformally hyperkähler manifold generates this type of geometry. The HKT-spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic conditio…

Study existence of harmonic 1-forms on Calabi-Yau manifolds.

Upper and lower bounds for magnetic Laplacian eigenvalues on manifolds.

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…

Eigenvalues of 1-form Laplacian on hyperbolic manifolds relate to geodesic cycles.

In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a …

The paper bounds eigenvalues of magnetic Schroedinger operators on compact manifolds.

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the…

Boundary rigidity proven for non-reversible Finsler metrics.

Proves existence of global positive LCK potential on certain manifolds.

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

New theorem on Lee classes for LCK manifolds with potential.

Study properties of 3D almost η-Ricci solitons with diagonal metrics.

Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.

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…

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

New insights into cohomology of closed 1-forms.

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

Recovering matrix valued potentials from wave equation data on stationary spacetimes.

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…

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence …

New metrics produce discrete zero sets for nondegenerate harmonic forms.

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.

The paper defines vector 1-forms on Finsler manifolds and constructs connections.

Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.

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…

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

New connections found between curvature and Euler characteristic using Schrödinger operators.

Study Euler obstruction of 1-forms on determinantal singularities.

Vanishing theorem for $L^2$-harmonic forms on Riemannian manifolds with parallel 1-form.

Modified dynamical systems retain Turing universality.

The paper studies local normal forms of singular contact forms and primitive 1-forms.

Constructs deformations of Vaisman manifolds preserving foliations.

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

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.…

The study classifies rational 1-forms on the Riemann sphere with simple poles.

Research shows growth in Higgs field strength for certain equations.

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.