arXiv research

Lower bounds for Hodge-Laplacian spectrum on orbifolds.

Study on the spectrum of drift Laplacian on Ricci expanders.

The paper extends Laplacian spectra approximations to vector bundles.

This paper concerns the $L^2$ essential spectrum of the Laplacian $Δ$ and the drift Laplacian $Δ_f$ on complete Riemannian manifolds endowed with a weighted measure $e^{-f}d\;vol_g$. We prove that the essential spectrum of the drift Laplacian $Δ_f$ is $[0,+\infty)$ provided the Bakry-Émery curvature tensor $Ric_f$ is …

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.

New outlier detection method using graph Laplacian spectrum boosts performance.

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

The paper extends decay estimates to graphs with positive spectrum.

Study the spectrum of Page's metric on complex projective spaces.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

The study extends $L^p$-spectrum analysis to warped products and Kleinian groups.

In this article we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.

New CR manifolds found with same Kohn Laplacian spectra.

We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…

Study on Kohn Laplacian spectrum on sphere quotients.

In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4λ_1 > h^2$, where $λ_1$ is the bottom of the $L^2$-spectrum and $h$ the…

We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

The study examines spectral properties of the Laplacian on forms for open Riemannian manifolds.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of bal…

Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).

We present some recent results on the behavior of the spectrum of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit).

Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.

In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.

Sharp bounds on scalar curvature spectrum and rigidity theorems.

We consider a complete noncompact smooth Riemannian manifold $M$ with a weighted measure and the associated drifting Laplacian. We demonstrate that whenever the $q$-Bakry-Émery Ricci tensor on $M$ is bounded below, then we can obtain an upper bound estimate for the heat kernel of the drifting Laplacian from the upper b…

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $Δ_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<Δu, u>_{L^2}\geq \frac14||u||^2_{L^2}$…

In this paper, we establish a kind of splitting theorem for the eigenvalues of a specific family of operators on the base of a warped product. As a consequence, we prove a density theorem for a set of warping functions that makes the spectrum of the Laplacian a warped-simple spectrum. This is then used to study the gen…

We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we …

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is th…

Study on spectral stability of Riemannian coverings.

The Hodge spectra help distinguish orbifolds from manifolds with singularities.

New mathematical tools for studying knots and links.

In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the $k$-form essential spectrum over a complete manifold with vanishing…

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant $(n-1)K$ converges to ze…

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions w…

Given a sequence of regular finite coverings of complete Riemannian manifolds, we consider the covering solenoid associated with the sequence. We study the leaf-wise Laplacian on the covering solenoid. The main result is that the spectrum of the Laplacian on the covering solenoid equals the closure of the union of the …

The $L^2$-$\partial\overline\partial$-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…

We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a cor…

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…

We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients.

Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.

Paper bounds the lowest spectrum of manifolds with curvature constraints.

Polygon $p$-widths are found via billiard trajectories.

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curv…

To every Hermitian vector bundle with connection over a compact Riemannian manifold $M$ one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of $M$ and prove that their spectra converge, as…