The smallest Laplace eigenvalue of homogeneous 3-spheres

TL;DR

This paper derives an explicit formula for the smallest non-zero Laplace eigenvalue on homogeneous 3-spheres, providing new spectral estimates and showing the spectrum uniquely determines the metric up to isometry.

Contribution

It provides a complete spectral characterization of homogeneous 3-spheres and Berger spheres, including explicit eigenvalue formulas and metric distinguishability results.

Findings

Explicit eigenvalue formula for all homogeneous metrics on 3-spheres

Improved eigenvalue estimates in terms of diameter

Spectrum uniquely determines left-invariant metrics on SU(2)

Abstract

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.