Juan Pablo Rossetti

Flat manifolds isospectral on p-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, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that βp(M)<βp(M’), for 0<p<n and pairs isospectral on p-forms for every p odd, and having different holonomy groups, ℤ4 and ℤ22, respectively.

Length spectra andp-spectra of compact flat manifolds

AbstractWe compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.

Comparison of Twisted P-Form Spectra for Flat Manifolds with Diagonal Holonomy

We give an explicit formula for the multiplicities of the eigenvalues ofthe Laplacian acting on sections of natural vector bundles over acompact flat Riemannian manifold MΓ = Γ\ℝn,Γ a Bieberbach group. In the case of the Laplacian acting onp-forms, twisted by a unitary character of Γ, when Γ hasdiagonal holonomy group F ≃ ℤ2k, these multiplicities have acombinatorial expression in terms of integral values of Krawtchoukpolynomials and the so called Sunada numbers. If the Krawtchoukpolynomial Kpn(x)does not have an integral root, this expressioncan be inverted and conversely, the presence of such roots allows toproduce many examples of p-isospectral manifolds that are notisospectral on functions. We compare the notions of twistedp-isospectrality, twisted Sunada isospectrality and twisted finitep-isospectrality, a condition having to do with a finite part of thespectrum, proving several implications among them and getting a converseto Sunadas theorem in our context, for n ≤ 8. Furthermore, a finitepart of the spectrum determines the full spectrum. We give new pairs ofnonhomeomorphic flat manifolds satisfying some kind of isospectralityand not another. For instance: (a) manifolds which are isospectral onp-forms for only a few values of p ≠ 0, (b) manifolds which aretwisted isospectral for every χ, a nontrivial character of F, and(c) large twisted isospectral sets.

Spectra of Lens Spaces from 1-Norm Spectra of Congruence Lattices

Fil: Lauret, Emilio Agustin. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina

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.

Representation Equivalence and p-Spectrum of Constant Curvature Space Forms

We study the p-spectrum of a locally symmetric space of constant curvature Γ∖X, in connection with the right regular representation of the full isometry group G of X on

Non-strongly isospectral spherical space forms

L^{2}(\varGamma \backslash G)_{\tau_{p}}

Isospectral Hantzsche-Wendt manifolds

, where τp is the complexified p-exterior representation of

Describing the platycosms

\operatorname{O}(n)

Hearing the Platycosms

on