The smallest Laplace eigenvalue of homogeneous 3-spheres
Emilio A. Lauret
TL;DR
This work delivers an explicit formula for the smallest nonzero Laplace eigenvalue on all homogeneous 3-spheres, equivalently on SU(2) with left-invariant metrics, namely lambda1 = min{a^2+b^2+c^2, 4(b^2+c^2)} for a≥b≥c>0, together with precise multiplicities. It also fully describes the Berger-sphere spectra and extends the results to SO(3), enabling sharp diameter-based estimates and establishing global spectral rigidity: the left-invariant metric is determined by its spectrum within SU(2) or SO(3). The paper then connects these spectral facts to geometric analysis: improved eigenvalue-diameter bounds, a Yamabe-type local rigidity result for homogeneous metrics, and consequences for the conformal geometry of S^3. The methods rely on Lie-theoretic Casimir computations, representation theory of SU(2), and Gershgorin circle estimates, yielding a robust framework for analyzing homogeneous spectra and their geometric implications.
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.