arXiv research

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can …

In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a smooth p-form is non-increasing along the Ricci flow. The L^{\infty} norm is showe…

Local holomorphic maps preserving (p,p) forms are shown to be isometries.

Study generalizes map properties between Hermitian manifolds preserving specific forms.

We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, for instance, pairs of flat manifolds o…

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…

We give a lower bound on the number of small positive eigenvalues of the p-form Laplacian in a certain type of collapse with curvature bounded below.

A special p-form is a p-form which, in some orthonormal basis {e_μ}, has components φ_{μ_1...μ_p} = φ(e_{μ_1},..., e_{μ_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.

Characterizes differential forms and vector fields with constant coefficients on manifolds.

Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.

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…

We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…

The paper explores families of almost complex structures and transverse (p,p)-forms.

We explicitely compute the essential spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $dσ^2= y^{2a}dy^2 + y^{2b}dθ_{\partial M}^2$, where $y$ is a boundary defining function on a compact manifold with boundary $M$.

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.

Formulas derived for operators on forms in anti-de Sitter spaces.

In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as $i_{\dot γ}dσ=0$, where $σ$ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where $σ$ is a differen…

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 …

Solves complex Monge-Ampère equation for $(p,p)$-forms on Kähler manifolds.

We give an explicit description of the spectrum of the Hodge--Laplace operator on $p$-forms of an arbitrary lens space for any $p$. We write the two generating functions encoding the $p$-spectrum as rational functions. As a consequence, we prove a geometric characterization of lens spaces that are $p$-isospectral for e…

In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of se…

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 …

Let $(M,I,J,K)$ be a hyperkaehler manifold, $\dim_\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping Dolbeault-closed $(2p,0)$-forms to closed $(n+p,n+p)$-forms, and positive $(2p,…

Let $D, Ω_1, ..., Ω_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $Ω_1 \times... Ω_m$ preserving the invariant $(p, p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank …

We present a gauge invariant generalization of Maxwell's equations and p-form electromagnetism to Kaehler spacetimes.

We show that every Killing p-form on a compact quaternion-Kähler manifold has to be parallel for p greater than 1.

Free differential algebras (FDA's) provide an algebraic setting for field theories with antisymmetric tensors. The "presentation" of FDA's generalizes the Cartan-Maurer equations of ordinary Lie algebras, by incorporating p-form potentials. An extended Lie derivative along antisymmetric tensor fields can be defined, an…

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is less than a positive constant depending only on $n$. Second, we show that the di…

In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar curvature, under curvature pinching conditions. Secondly, we prove vanishing results …

Eigenvalue bounds for forms on warped manifolds studied.

We make a computational study to know what kind of isospectralities among lens spaces and lens orbifolds exist considering the Hodge--Laplace operators acting on smooth $p$-forms. Several evidenced facts are proved and some others are conjectured.

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H, the Laplacian can be decomposed into operators in the conjugate of the generalised…

A Killing $p$-form on a Riemannian manifold is a $p$-form whose covariant derivative is totally anti-symmetric. In this paper we give the complete (local) description of 4-dimensional Riemannian manifolds (M,g) carrying non-parallel Killing 2-forms $\varphi$. If $M$ is connected and oriented, we show that there exists …

Paper develops techniques to solve complex PDEs involving higher cohomology forms.

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on $p$-forms, we determine all $p$-isos…

On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\leq p<\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues.

Let (M,g) be a compact Einstein manifold with smooth boundary. We consider the spectrum of the p form valued Laplacian with respect to a suitable boundary condition. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.

Survey on invariant conformal Killing forms on Lie groups.

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

On any compact manifold of dimension greater than 3, we exhibit a metric whose first positive eigenvalue for the Laplacian acting on p-form is of multiplicity 2. As a corollary, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian with multiplicity 1 or 2.

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere…

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $Ω$ in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bdΩ)/{\rm Vol}(Ω)$.

In this paper we describe recent results on explicit construction of lens spaces that are not strongly isospectral, yet they are isospectral on $p$-forms for every $p$. Such examples cannot be obtained by the Sunada method. We also discuss related results, emphasizing on significant classical work of Ikeda on isospectr…

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…

This paper is devoted to the horizontal (``characteristic'') cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov C-spectral sequence and the Krasil'shchik C-cohomology are obtained. As an appl…

The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms f…