arXiv research

Flat tori found non-isometric pairs with identical Laplace eigenvalues.

We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces.…

Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are i…

Compactness proven for isospectral Birkhoff billiard tables.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

Orbifold local orientability can be detected by heat invariants.

John Conway created pairs of domains that sound the same for a special kind of music.

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

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; an…

Revisiting a construction due to Vigneras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.

Study eigenvalues of CROSS spaces with various metrics.

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.

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (…

Paper proves non-compact inaudibility of symmetry and commutativity.

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 als…

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 th…

Researchers find isospectral but non-diffeomorphic nilmanifolds.

Paper shows spectra can't distinguish naturally reductive manifolds.

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 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}…

New findings show fundamental group is not audible in spherical space forms.

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γis a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the…

We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To pro…

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.

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as …

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the numbe…

We give a systematic way to construct almost conjugate pairs of finite subgroups of $Spin(2n+1)$ and $Pin(n)$ for $n\in \mathbb{N}$ sufficiently large. As a geometric application, we give an infinite family of pairs $M_1^{d_n}$ and $M_2^{d_n}$ of nearly Kähler manifolds that are isospectral for the Dirac and Laplace op…

Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace-Beltram…

Mathematicians decode geometric properties from eigenvalues over 112 years.

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…

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…

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is n…

We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group $Z_2^k$. For any n>3 (resp. n>5) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M_1, M_2, non homeomorphic to each other, that are Laplace isos…

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collecti…

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds $M$ and $M'$ which are isospectral for the Laplace operator on functions and such that $M$ has completely integrable geodesic flow in the sense of Liouville, while $M'$ has not. Moreover, for both manifolds we analyze the structure of t…

Recent developments link Steklov eigenvalues to manifold geometry.

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $τ$ we compare the notions of $τ$-representation equivalence with $τ$-isospectrality. We exhibit infinitely many $K$-types $τ$ so that, for arbitrary discrete subgroups $Γ$ and $Γ'$ of $G$, if the multiplicities of $…

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…

Generalizes isomonodromic-isospectral correspondence for twisted connections.

Develops new methods for isospectral orbifolds and regulator quotients.

We construct several new classes of isospectral manifolds with different local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the first examples of isospectral four-dimensional manifold…

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …

Infinite-genus surfaces have many isospectral hyperbolic structures.

We present a new construction for obtaining pairs of higher-step isospectral Riemannian nilmanifolds and compare several resulting new examples. In particular, we present new examples of manifolds that are isospectral on functions, but not isospectral on one-forms.