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

TL;DR

This paper computes the spectrum of the Hodge-Laplacian on 1-forms for homogeneous 3-spheres, providing explicit formulas, analyzing eigenvalue splitting, and demonstrating spectral determination of the metric up to isometry.

Contribution

It offers the first explicit spectrum computation for a canonical variation of Berger 3-spheres and introduces an AI-discovered proof of the first eigenvalue formula for homogeneous metrics.

Findings

Explicit eigenvalues for Berger 3-spheres are determined.

An explicit formula for the first eigenvalue of homogeneous metrics is proven.

The spectrum on 1-forms uniquely determines the metric up to isometry.

Abstract

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].