The smallest Laplace eigenvalue of homogeneous 3-spheres

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 three‐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 three‐dimensional real projective space.

Left-invariant geometries on SU(2) are uniformly doubling

“…Remark 8.7. In a very recent preprint [31], E. A. Lauret has given an exact expression for the smallest eigenvalue λ g of SU(2) in terms of the parameters of the metric, which in our notation reads as follows: Indeed, (8.10) is consistent with (8.9). In earlier work, as part of a more general construction, H. Urakawa [55] computed λ g for a particular one-parameter family of metrics g(t) on SU(2), which in our notation is [55,Theorem 5].…”

The first eigenvalue of a homogeneous CROSS

“…That is no longer the case on S 4n+3 , h(t 1 , t 2 , t 3 ) when not all t i 's are the same, and these metrics are also not normal homogeneous, which renders the computation of their first eigenvalue substantially more challenging. This was recently achieved in [Lau19a] in the lowest dimensional case S 3 , h(t 1 , t 2 , t 3 ) , i.e., that of left-invariant metrics on SU(2) ∼ = S 3 , laying the groundwork for the cases n ≥ 1, which are settled in our first main result.…”

“…These are instances where optimality in a geometric variational problem is not necessarily achieved with the most symmetries, since a global minimizer exists in every conformal class, and a conformal class contains at most one homogeneous metric (up to homotheties). Stable homogeneous spheres among canonical variations of the round metric were classified in [BP13a], and among S 3 , h(t 1 , t 2 , t 3 ) in [Lau19a]. Thus, the only families left to consider are CP 2n+1 , ȟ(t) , for which the stability classification follows easily from Theorem B, see Remark 6.3, and S 4n+3 , h(t 1 , t 2 , t 3 ) , which is settled in our next main result.…”

“…For the convenience of the reader, the complete list of homogeneous metrics on CROSSes that are stable solutions to the Yamabe problem is provided in Table 2, in Appendix A, combining Theorem D and Remark 6.3 with [BP13a,Lau19a].…”

Spectral Uniqueness of Bi-Invariant Metrics on Symplectic Groups

“…To the author's knowledge, Question 1.1 has been answered only in the cases SU(2) ≃ S 3 by Tanno [Ta73] and SO(3) by Schmidt and Sutton [SS14] 1 . Actually, Schmidt and Sutton proved that any left-invariant metric on any of these two groups is uniquely determined by its spectrum in M G (see [La18] for a recent alternative proof). It was shown that the first four heat invariants cannot coincide for non-isometric pairs.…”