The first eigenvalue of a homogeneous CROSS (Q2074710)

The underlying manifolds of compact symmetric spaces often admit homogeneous Riemannian metrics other than their symmetric space metric. Homogeneous metrics on compact rank-one Riemannian symmetric spaces were classified by \textit{W. Ziller} [Math. Ann. 259, 351--358 (1982; Zbl 0469.53043)]. Up to homotheties, in addition to the canonical symmetric space metrics, that is, the round metric of constant sectional curvature 1 on \(S^n\) and \(\mathbb{R}P^n\), and the Fubini-Study metrics \(g_{FS}\) on the projective spaces \(\mathbb{C}P^n\), \(\mathbb{H}P^n\), and \(\operatorname{Ca}P^2\), they are as follows: \begin{itemize} \item[(i)] A 1-parameter family \(\mathbf{g}(t)\) of \(\operatorname{SU}(n + 1)\)-invariant metrics on \(S^{2n+1}\); \item[(ii)] A 3-parameter family \(\mathbf{h}(t_1, t_2, t_3)\) of \(\operatorname{Sp}(n + 1)\)-invariant metrics on \(S^{4n+3}\); \item[(iii)] A 1-parameter family \(\mathbf{k}(t)\) of \(\operatorname{Spin}(9)\)-invariant metrics on \(S^{15}\); \item[(iv)] A 1-parameter family \(\mathbf{h}(t)\) of \(\operatorname{Sp}(n + 1)\)-invariant metrics on \(\mathbf{C}P^{2n+1}\), \end{itemize} where \(t\) and \(t_i\) denote positive real numbers. All metrics in (i), (ii), and (iii) descend to homogeneous metrics invariant under the same groups on \(\mathbb{R}P^{2n+1}\), \(\mathbb{R}P^{4n+3}\), and \(\mathbb{R}P^{15}\), respectively. In this paper, the authors compute the first eigenvalue of the Laplacian \(\lambda_1(M,g)\) for every homogeneous metric \(g\) on the underlying manifold of a compact rank-one symmetric space \(M\), completing the results previously available. For example: Theorem ~B. The first eigenvalue of the Laplacian on \((\mathbb{C}P^{2n+1},\mathbf{h}(t))\) is given by \[\lambda_1(\mathbb{C}P^{2n+1}, \mathbf{h}(t))=\min\{8n+8/t^2, 8(n+1)\}.\] As a first application, the authors prove that two compact rank-one Riemannian symmetric spaces, endowed with homogeneous metrics, are isospectral if and only if they are isometric. As a second application, they finalize the classification of homogeneous metrics on a compact rank-one Riemannian symmetric space that are stable solutions of the Yamabe problem.