The first eigenvalue of a homogeneous CROSS

Abstract

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.

Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

“…Remark 6.4. Concerning G/K = Sp(n + 1)/ Sp(n) ≃ S 4n+3 (n ≥ 1) as at the end of the previous proof, it has been recently determined in [BLP20] an explicit expression for λ 1 (S 4n+3 , g) for every G-invariant metric g on S 4n+3 . This might help to obtain explicit constants C 1 , C 2 > 0 such that C 1 ≤ λ 1 (S 4n+3 , g) diam(S 4n+3 , g) 2 ≤ C 2 for all g ∈ M(G, K), analogous as those obtained in [La19] for S 3 .…”

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

“…3.3] replacing (p, q) with (p + q, p), and setting a = b = c = r/ √ 2, which implies that ν (q) j (a, b, c) = 1 2 r 2 q(q + 2) for all 1 ≤ j ≤ q + 1. Accordingly, the multiplicity (4.4) is q + 1 times that in [BLP,(3.11)], since ν (q) j does not depend on j in this case.…”

“…Such eigenvalues are called basic, and are independent of t. Although it has been known for a long time that all eigenvalues are sums of basic eigenvalues with certain Laplace eigenvalues of the fiber [BBB82,BB90], determining exactly which sums of eigenvalues from KP n and S 2d−1 indeed appear in the spectrum of the total space can be somewhat impractical. We circumvent this with an alternative Lie-theoretic approach based on [MU80], recently used in [Lau19,BLP] and expanded in Section 2 below, which yields our first main result: Theorem A. The spectrum of the Laplace-Beltrami operator on the homogeneous sphere S N −1 , g(t) , N = 2d(n + 1), as in (1.3), consists of the eigenvalues (1.4) λ (p,q) (t) = 4p p + q + d(n + 1) − 1 + 2dnq + q(q + 2d − 2) 1 t 2 , p, q ∈ N 0 , which are basic if q = 0, and have multiplicity…”

“…The Lie-theoretic method. In this section, we describe the Lie-theoretic procedure to compute the Laplace-Beltrami spectrum of G/H, g (r,s) , which relies on knowledge of representation branching rules involving G, K, and H. The discussion below is based on [MU80] and our earlier work [BLP,§2], and provides a computationally efficient alternative to more classical methods in [BBB82,BB90].…”