arXiv subjects

Emilio A. Lauret

Publications and source records attributed to Emilio A. Lauret.

At least 19 recordsLinked to original sources

arXiv2026

The fundamental group of a spherical space form is not audible

We revisit the problem of isospectral spherical space forms with non-cyclic fundamental groups after the works by Ikeda, Gilkey and Wolf. We find the first pair of spherical space forms with non-isomorphic fundamental groups and the same Laplace spectrum. This shows that the isomorphism class of the fundamental group is not audible among spherical space forms. We also found several instances where one can hear the fundamental group of a spherical space form (among spherical space forms).

math.DG↗

arXiv2026

Linear stability of Perelman's $ν$-entropy of standard Einstein manifolds

Paul Schwahn recently exhibited 112 non-symmetric, connected, simply connected, compact Einstein manifolds that are stable with respect to the total scalar curvature functional restricted to the space of Riemannian metrics with constant scalar curvature and fixed volume. This stability follows from the inequality $λ_L > 2E$, where $λ_L$ denotes the smallest eigenvalue of the Lichnerowicz Laplacian on TT-tensors and $E$ is the corresponding Einstein factor. In this paper, we estimate the smallest positive eigenvalue $λ_1$ of the Laplace-Beltrami operator for connected, simply connected, non-symmetric standard Einstein manifolds $(G/H,g_{\operatorname{st}})$ with $G$ a compact and connected simple Lie group. We obtain that $λ_1>2E$ for all of them excepting $7$ spaces. As a consequence of our estimates, we establish that all stable Einstein manifolds found by Schwahn are in fact linearly stable with respect to Perelman's $ν$-entropy.

math.DG↗

arXiv2026

Hodge Laplacian on $1$-forms of homogeneous $3$-spheres

We study the spectrum of the Hodge-Laplacian on $1$-forms for left-invariant metrics on the Lie group $\operatorname{SU}(2) \cong S^3$ and its quotient $\operatorname{SO}(3)\cong P^3(\mathbb{R})$. To the best of our knowledge, we provide the first explicit computation of the full spectrum of the Hodge-Laplacian for a canonical variation by determining the eigenvalues of Berger 3-spheres and analyzing their resulting splitting behavior. Furthermore, we propose and rigorously prove an explicit formula for the first eigenvalue of general homogeneous metrics on $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$. The formal proof of this result was autonomously discovered by an advanced AI model, providing a notable case study for AI-driven mathematical research. Finally, leveraging this explicit formula, we apply these spectral results to the inverse problem, showing that the spectrum on $1$-forms determines the metric up to isometry. The source code for the symbolic computations, visualizations, and a Monte Carlo stress test is provided in the electronic supplementary material [He26].

math.DG↗

arXiv2025

First Laplace eigenvalue of strongly isotropy irreducible spaces

We study the smallest positive eigenvalue $λ_1$ of the Laplace-Beltrami operator associated with any compact strongly isotropy irreducible space. We provide an explicit expression for all simply connected cases. Furthermore, every strongly isotropy irreducible space is automatically an Einstein manifold, and we prove for each of them that $E<λ_1\leq 16E$, where $E$ denotes the corresponding Einstein constant.

math.DG↗

arXiv2025

Spectrally distinguishing symmetric spaces II

The action of the subgroup $\operatorname{G}_2$ of $\operatorname{SO}(7)$ (resp.\ $\operatorname{Spin}(7)$ of $\operatorname{SO}(8)$) on the Grassmannian space $M=\frac{\operatorname{SO}(7)}{\operatorname{SO}(5)\times\operatorname{SO}(2)}$ (resp.\ $M=\frac{\operatorname{SO}(8)}{\operatorname{SO}(5)\times\operatorname{SO}(3)}$) is still transitive. We prove that the spectrum (i.e.\ the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric $g_0$ on $M$ coincides with the spectrum of a $\operatorname{G}_2$-invariant (resp.\ $\operatorname{Spin}(7)$-invariant) metric $g$ on $M$ only if $g_0$ and $g$ are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

math.DG↗

arXiv2025

The smallest Laplace eigenvalue of homogeneous 3-spheres

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of 3-dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non-zero eigenvalue in terms of the diameter for homogeneous 3-spheres. Another application shows that the spectrum of the Laplace--Beltrami operator distinguishes up to isometry any left-invariant metric on SU(2). It is also proved the non-existence of constant scalar curvature metrics conformal and arbitrarily close to any non-round homogeneous metric on the 3-sphere. All of the above results are extended to left-invariant metrics on SO(3), that is, homogeneous metrics on the 3-dimensional real projective space.

math.DG↗

arXiv2025

Spectrally distinguishing symmetric spaces I

We prove that the irreducible symmetric space of complex structures on $\mathbb R^{2n}$ (resp.\ quaternionic structures on $\mathbb C^{2n}$) is spectrally unique within a $2$-parameter (resp.\ $3$-parameter) family of homogeneous metrics on the underlying differentiable manifold. Such families are strong candidates to contain all homogeneous metrics admitted on the corresponding manifolds. The main tool in the proof is an explicit expression for the smallest positive eigenvalue of the Laplace-Beltrami operator associated to each homogeneous metric involved. As a second consequence of this expression, we prove that any non-symmetric Einstein metric in the homogeneous families mentioned above is $ν$-unstable.

math.DG↗

arXiv2025

Isospectral spherical space forms and orbifolds of highest volume

We prove that $\operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\geq5$. Furthermore, we show that $\operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\geq11$ and $n\equiv 1\pmod 5$, or $n\geq7$ and $n\equiv 2\pmod 5$, or $n\geq3$ and $n\equiv 3\pmod 5$.

math.DG↗

arXiv2024

The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems

The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of hyperbolic manifolds and discuss the extent to which analogous results hold in the setting of spherical manifolds. We conclude with a number of open problems.

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↗

arXiv2023

Diameter and displacement of sphere involutions

We show that spheres in all dimensions $\geq3$ can be deformed to have diameter larger than the distance between any pair of antipodal points. This answers a question of Yurii Nikonorov.

math.DG↗

arXiv2022

The stability of standard homogeneous Einstein manifolds

Back in 1985, Wang and Ziller obtained a complete classification of all homogeneous spaces of compact simple Lie groups on which the standard or Killing metric is Einstein. The list consists, beyond isotropy irreducible spaces, of 12 infinite families (two of them are actually conceptual constructions) and 22 isolated examples. We study in this paper the nature of each of these Einstein metrics as a critical point of the scalar curvature functional.

math.DG↗

arXiv2022

Full Laplace spectrum of distance spheres in symmetric spaces of rank one

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.

math.DG↗

arXiv2021

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on $G$ in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets $\mathcal S$ of the space of left-invariant metrics $\mathcal M$ on $G$ such that there exists a positive real number $C$ depending on $G$ and $\mathcal S$ such that $λ_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all $g\in\mathcal S$. The existence of the constant $C$ for $\mathcal S=\mathcal M$ is the original conjecture.

math.DG↗

arXiv2021

The first eigenvalue of a homogeneous CROSS

We provide explicit formulae for the first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that homogeneous metrics on CROSSes are isospectral if and only if they are isometric, and also discuss their stability (or lack thereof) as solutions to the Yamabe problem.

math.DG↗

arXiv2021

Strong representation equivalence for compact symmetric spaces of real rank one

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 $λ$ in the spectra of the Laplace operators acting on sections of the induced $τ$-vector bundles over $Γ\backslash G/K$ and $Γ'\backslash G/K$ agree for all but finitely many $λ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent in $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_π, L^2(Γ\backslash G))=\dim \operatorname{Hom}_G(V_π, L^2(Γ'\backslash G))$ for all $π\in \widehat G$ satisfying $\operatorname{Hom}_K(V_τ,V_π)\neq0$). In particular $Γ\backslash G/K$ and $Γ'\backslash G/K$ are $τ$-isospectral (i.e.\ the multiplicities agree for all $λ$). We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $τ$ of the representation $τ_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_{\mathbb C}^*M$. We show that for such $τ$, in most cases $τ$-isospectrality implies $τ$-representation equivalence. We construct an explicit counter-example for $G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1}$.

math.DG↗

arXiv2021

Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto λ_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant metrics $g$ on $G/K$. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when $K$ is trivial; the only particular cases known so far are when $G$ is abelian, $\operatorname{SU}(2)$, and $\operatorname{SO}(3)$. In this article we prove the existence of the mentioned upper bound for every compact homogeneous space $G/K$ having multiplicity-free isotropy representation.

math.DG↗

arXiv2020

Strong multiplicity one theorems for locally homogeneous spaces of compact type

Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $Γ$ be a finite subgroup of $G$, and let $τ$ be a finite-dimensional representation of $K$. For $π$ in the unitary dual $\widehat G$ of $G$, denote by $n_Γ(π)$ its multiplicity in $L^2(Γ\backslash G)$. We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_Γ(π)$ for $π$ in the set $\widehat G_τ$ of irreducible $τ$-spherical representations of $G$. More precisely, for $Γ$ and $Γ'$ finite subgroups of $G$, we prove that if $n_Γ(π)= n_{Γ'}(π)$ for all but finitely many $π\in \widehat G_τ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent, that is, $n_Γ(π)=n_{Γ'}(π)$ for all $π\in \widehat G_τ$. Moreover, when $\widehat G_τ$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset $\widehat {F}_τ$ of $\widehat G_τ$ verifying some mild conditions, the values of the $n_Γ(π)$ for $π\in\widehat F_τ$ determine the $n_Γ(π)$'s for all $π\in \widehat G_τ$. In particular, for two finite subgroups $Γ$ and $Γ'$ of $G$, if $n_Γ(π) = n_{Γ'}(π)$ for all $π\in \widehat F_τ$ then the equality holds for every $π\in \widehat G_τ$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.

math.RT↗