arXiv research

New inequality for special forms on manifolds.

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

Study spectral gaps in hyperbolic rational homology spheres.

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…

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^…

New method solves differential equations on manifolds, with applications in physics.

The spectral properties of p-forms on the fundamental domains of regular tesselations of the d-dimensional sphere are discussed. The degeneracies for all ranks, p, are organised into a double Poincare series which is explicitly determined. In the particular case of coexact forms of rank (d-1)/2, for odd d, it is shown …

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…

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…

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

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

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

Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacia…

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

New interpretation of complex hyperbolic form as Weil-Petersson form.

Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…

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…

The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…

The study examines parallel forms on manifolds, focusing on specific dimensions and forms.

The study shows that the second fundamental form is intrinsic under certain conditions in space forms.

New forms generalize Whitney forms with rational coefficients for numerical analysis.

This paper classifies quadratic form parameters over integers and computes their Witt groups.

Paper presents a new flat triangular form for systems.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

The paper finds a contact form on SL(2p) for p > 1.

Researchers solve conformal Killing forms on Kaehler manifolds.

Characterizes Whitney forms on simplices and proves their uniqueness.

Study of tautological forms on curve moduli spaces.

Analytic surgery and gluing formula for torsion forms in fiber bundles.

Paper presents a new triangular form for flat systems.

Special p-forms are forms which have components φ_{μ_1...μ_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form φ\in Λ^p R^d is called democratic if the set of nonzero components {φ_{μ_1...μ_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry …

Paper transforms torse-forming vector fields into simpler forms.

The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.

Study on immersions with flat normal bundle in curved spaces.

Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.

We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of $d ω$ and its Hodge dual $* dω$ where $ω$ is the fundamental 2-form of the nearly Kähler stru…

Diffeology explores $k$-forms and bundles with more information than traditional differential forms.

New conformally invariant forms help identify Einstein metrics.

In this paper, we first define the equivariant infinitesimal $η$-form, then we compare it with the equivariant $η$-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the equivariant $η$-form, modulo exact forms, as a function on the acting Lie group. This result…

Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…

New metrics produce discrete zero sets for nondegenerate harmonic forms.

Enhances power of covariance matrix tests for high-dimensional data.

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

New findings show fundamental group is not audible in spherical space forms.

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

The paper proves unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.