arXiv research

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.

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differ…

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…

Introduces a new Hodge theory using vector fields on manifolds.

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…

Discretizes Hodge-Dirac operators on a torus.

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stre…

The study bounds Riesz transforms on manifolds with controlled curvature.

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error w…

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, $φ$-differential operators, that is, they live in the generalised $φ$-calculus studied by the authors i…

A manifold with fibered cusp metrics $X$ can be considered as a geometrical generalization of locally symmetric spaces of $\mathbb{Q}$-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology $H^p(X)$. Similar to the situ…

We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kaehler manifolds this leads to simple proofs of eigenvalue estima…

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 goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.

Study Hodge Laplacians for manifold data, improving error bounds.

In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

The paper proves inequalities for twisted differential forms on manifolds.

Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…

A general theory of quantum spinor structures on quantum spaces is presented, within the conceptual framework of the formalism of quantum principal bundles. Quantum analogs of all basic objects of the classical theory are constructed and analyzed. This includes Laplace and Dirac operators, quantum versions of Clifford …

New method generalizes eigenvalue inequality to surfaces with boundaries.

In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on $(1, 1)$-forms. The method is effective in proving an optimal result when $M$ has nonnegative bisectional curvature. It also provides …

Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.

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…

The article studies spinor and tensor fields on curved spaces, deriving formulas and spectra.

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general ge…

The concept of harmonic metallic structure on a metallic pseudo-Riemannian manifold is introduced. In the case of compact manifolds we prove that harmonicity of a metallic structure $J$, with $J^2=pJ+qI$ and $p^2+4q\neq 0$, is equivalent to $dJ=0$. Conditions for a harmonic metallic structure to be preserved by harmoni…

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

We study the $p$-spectrum of a locally symmetric space of constant curvature $Γ\backslash X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(Γ\backslash G)_{τ_p}$, where $τ_p$ is the complexified $p$-exterior representation of $\mathrm{O}(n)$ on $\bigwedge^p(\mathbb{R}…

Rust library solves complex equations on abstract simplicial complexes.

We study solutions for the Hodge laplace equation $Δu=ω$ on $p$ forms with $\displaystyle L^{r}$ estimates for $\displaystyle r>1.$ Our main hypothesis is that $Δ$ has a spectral gap in $\displaystyle L^{2}.$ We use this to get non classical $\displaystyle L^{r}$ Hodge decomposition theorems. An interesting feature is …

Introduces a new elliptic operator with positive eigenvalue.

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, $Δ_f$, acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z_2^k, with 0<k<n. This formula implies that any two compact flat manifolds with holonomy group Z…

We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\,F)$ of complementary regular holomorphic foliations on such a manifold. The main idea is to …

Smooth bundles with rough data maintain Hodge kernel isomorphism.

The paper extends the Cheeger-Müller theorem to spaces with conical singularities.

Promotes spectral functionals to noncommutative fields and proves a theorem.

New operators help focus on specific areas in complex math problems.

Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on t…

Study Chern number inequalities for negative curvature Kähler manifolds.

The study explores discrete versions of Riemannian geometry structures on manifolds.

Study examines metrics and functionals for Hodge-Dirac operator on manifolds.

Paper introduces a new multilinear functional for spectral triples and computes its properties.

Numerical experiments support conjecture about opers and nonabelian Hodge.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.