Searcharxiv⌕ Search

arXiv subjects

Juan Pablo Rossetti

Publications and source records attributed to Juan Pablo Rossetti.

13 recordsLinked to original sources
arXiv2024

Do the Hodge spectra distinguish orbifolds from manifolds? Part 2

In \cite{GGKM-SSS} we examined the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We showed that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$ Here we give both positive and negative inverse spectral results for the individual $p$-spectra considered separately. For example, we give conditions on the codimension of the singular set which guarantee that the volume of the singular set is determined, and in many cases we show by providing counterexamples that the conditions are sharp.

math.DG↗
arXiv2023

Isospectral CR manifolds with respect to the Kohn Laplacian

We prove that the spectrum of the Kohn Laplacian does not determine the equivalence classes of CR manifolds. We construct pairs of odd-dimensional elliptic manifolds that are not equivalent as CR manifolds but whose Kohn Laplacians have the same spectrum. These manifolds are endowed with the CR structures inherited from the canonical CR structure on the sphere of the same dimension. We provide three different constructions among lens spaces and an additional one among elliptic manifolds with non-cyclic fundamental groups.

math.DG↗
arXiv2019

Recent results on the spectra of lens spaces

In this paper we report on recent results by several authors, on the spectral theory of lens spaces and orbifolds and similar locally symmetric spaces of rank one. Most of these results are related to those obtained by the authors in [IMRN (2016), 1054--1089], where the spectra of lens spaces were described in terms of the one-norm spectrum of a naturally associated congruence lattice. As a consequence, the first examples of Riemannian manifolds isospectral on $p$-forms for all $p$ but not strongly isospectral were constructed. We also give a new elementary proof in the case of the spectrum on functions. In this proof, representation theory of compact Lie groups is avoided and replaced by the use of Molien's formula and a manipulation of the one-norm generating function associated to a congruence lattice. In the last four sections we present several recent results, open problems and conjectures on the subject.

math.DG↗
arXiv2015

Non-strongly isospectral spherical space forms

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 isospectral lens spaces, via a thorough study of the associated generating functions.

math.DG↗
arXiv2015

Spectra of lens spaces from 1-norm spectra of congruence lattices

To every $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, with $n=2m-1$ and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on $L$ with the number of lattice elements of a given $\|\cdot\|_1$-length in $\mathcal L$. As a consequence, we show that two lens spaces are isospectral on functions (resp.\ isospectral on $p$-forms for every $p$) if and only if the associated congruence lattices are $\|\cdot\|_1$-isospectral (resp.\ $\|\cdot\|_1$-isospectral plus a geometric condition). Using this fact, we give, for every dimension $n\ge 5$, infinitely many examples of Riemannian manifolds that are isospectral on every level $p$ and are not strongly isospectral.

math.DG↗
arXiv2014

Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent

Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; and the number of primitive closed geodesics of each length and orientability class, counting a geodesic running along the boundary as half orientation-preserving and half orientation-reversing, and discounting imprimitive geodesics appropriately. This implies that Laplace-isospectral hyperbolic 2-orbifolds determine equivalent linear representations of Isom(H^2), and are isospectral for any natural operator.

math.DG↗
arXiv2012

Representation equivalence and p-Spectrum of constant curvature space forms

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 $\bigwedge^p(\mathbb{R}^n)_\mathbb{C}$. We give an expression of the multiplicity $d_λ(p,Γ)$ of the eigenvalues of the $p$-Hodge-Laplace operator in terms of multiplicities $n_Γ(π)$ of specific irreducible unitary representations of $G$. As a consequence, we extend results of Pesce for the spectrum on functions to the $p$-spectrum of the Hodge-Laplace operator on $p$-forms of $Γ\backslash X$, and we compare $p$-isospectrality with $τ_p$-equivalence for $0\leq p\leq n$. For spherical space forms, we show that $τ$-isospectrality implies $τ$-equivalence for a class of $τ$'s that includes the case $τ=τ_p$. Furthermore we prove that $p-1$ and $p+1$-isospectral implies $p$-isospectral. For nonpositive curvature space forms, we give examples showing that $p$-isospectrality is far from implying $τ_p$-equivalence, but a variant of Pesce's result remains true. Namely, for each fixed $p$, $q$-isospectrality for every $0\leq q\leq p$ implies $τ_q$-equivalence for every $0\leq q\leq p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with $τ_p$-equivalence.

math.SP↗
arXiv2011

Strongly isospectral manifolds with nonisomorphic cohomology rings

For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is Kähler while $M'$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ there is a family of eight compact flat manifolds (four of them Kähler) having very different cohomology rings. In particular, the cardinalities of the sets of primitive forms are different for all manifolds.

math.DG↗
arXiv2009

Isospectral hyperbolic surfaces have matching geodesics

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.

math.DG↗
arXiv2008

Isospectral orbifolds with different maximal isotropy orders

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on $p$-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean $\R^3$ and are shown to be isospectral on functions using dimension formulas for the eigenspaces. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties.

math.DG↗
arXiv2006

Tetra and Didi, the cosmic spectral twins

We introduce a pair of isospectral but non-isometric compact flat 3-manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, Didi has four half-twisting geodesics. Nevertheless, these spaces are isospectral. This isospectrality can be proven directly by matching eigenfunctions having the same eigenvalue. However, the real interest of this pair--and what led us to discover it--is the way isospectrality emerges from the Selberg trace formula, as the result of a delicate interplay between the lengths and twists of closed geodesics.

math.DG↗
arXiv2005

Generalized Hantzsche-Wendt Flat Manifolds

We study the family of closed Riemannian n-manifolds with holonomy group isomorphic to $Z_2^{n-1}$, which we call generalized Hantzsche-Wendt manifolds. We prove results on their structures, compute some invariants, and find relations between them, illustrated in graph connecting the family.

math.GT↗