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The first eigenvalue of a homogeneous CROSS. (English) Zbl 1493.53076

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 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:
(i)
A 1-parameter family \(\mathbf{g}(t)\) of \(\operatorname{SU}(n + 1)\)-invariant metrics on \(S^{2n+1}\);
(ii)
A 3-parameter family \(\mathbf{h}(t_1, t_2, t_3)\) of \(\operatorname{Sp}(n + 1)\)-invariant metrics on \(S^{4n+3}\);
(iii)
A 1-parameter family \(\mathbf{k}(t)\) of \(\operatorname{Spin}(9)\)-invariant metrics on \(S^{15}\);
(iv)
A 1-parameter family \(\mathbf{h}(t)\) of \(\operatorname{Sp}(n + 1)\)-invariant metrics on \(\mathbf{C}P^{2n+1}\),
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.

MSC:

53C35 Differential geometry of symmetric spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J53 Isospectrality
35P15 Estimates of eigenvalues in context of PDEs
35B35 Stability in context of PDEs
58J55 Bifurcation theory for PDEs on manifolds
53C18 Conformal structures on manifolds

Citations:

Zbl 0469.53043
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Full Text: DOI arXiv

References:

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