arXiv research

New inequality for special forms on manifolds.

Study spectral gaps in hyperbolic rational homology spheres.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.

We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact $1$-forms $λ_1^*$ on rational homology spheres…

Researchers prove Seifert-Weber dodecahedral space is an L-space.

Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\leq V$. If $μ_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that $μ_1(M)\geq \frac c{d^3e^{2kd}}$ and $μ_{k+1}(M)\geq \frac c{d^2}$, where $c>0$ depends only on $V$, and $k$ is the number of co…

Optimizes metrics for the first curl eigenvalue on 3-manifolds.

The paper studies Dirac operators on large spectral three-manifolds.

Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.

Let $M^n$ be a n-dimensional compact manifold, with $n\geq3$. For any conformal class C of riemannian metrics on M, we set $μ_k^c(M,C)=\inf_{g\in C}μ_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where $μ_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<μ_k^c(M,C)\leqμ_k^…

Topological complexity for closed 1-forms

Study on Riemannian Poisson warped product spaces and their properties.

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

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.

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.

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.

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 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.…

We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in te…

New examples of Z/2 harmonic 1-forms and their branching sets are explored.

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$…

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…

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, $J^1π$. 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.

No nontrivial harmonic 1-forms on certain gradient Ricci solitons.

In this paper, we define conservative semibasic vector $1-$forms on the tangent bundle of a Finsler manifold. Using these vector $1-$forms, we characterize conservative $L-$Ehresmann connections with respect to the energy function. Then we find a correspondence between torsion-free semibasic vector $1-$forms and the su…

This paper extends Lusternik-Schnirelmann category to non-compact manifolds.

Classifies vector fields in the kernel of a 1-form, up to equivalence.

In this paper, the Douglas curvature of (α,β)-metrics, a special class of Finsler metrics defined by a Riemannian metric αand a 1-form β, is studied. These metrics with vanishing Douglas curvature in dimension n\geq3 are classified by using a new class of metrical deformations called β-deformations. The result shows th…

In this paper I suggest an alternative approach (using generic flat bundles and higher Massey products) to a Lusternik-Schnirelman type theory for closed 1-forms (cf. also math.DG/9811113)