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.

Appendix A. First Eigenvalue and Yamabe Stability in the Remaining Homogeneous CROSSes

For the convenience of the reader, we now provide formulae (with references) for the first eigenvalue $\lambda _1(M,\mathrm {g})$ of the Laplacian on all CROSSes M, endowed with a homogeneous ${\mathsf {G}}$-invariant metric $\mathrm {g}$, as presented in Table 1 below.

The (complete) spectrum of a CROSS, endowed with its canonical symmetric space metric, can be found in [8, p. 202]. Detailed spectral computations for $\mathbb {S}^n$, $\mathbb {R}P^n$, and $\mathbb {C}P^n$ are given in [7]; for $\mathbb {H}P^n$ and $\mathbb {C}\mathrm {a}P^2$, see [15]. Regarding the remaining homogeneous metrics, we have that:

• (i) The first eigenvalue of $\mathbf{g }(t)$ on $\mathbb {S}^{2n+1}$ is computed in [38], and an inspection of which eigenfunctions are $\mathbb {Z}_2$-invariant yields its first eigenvalue on $\mathbb {R}P^{2n+1}$;

• (ii) The first eigenvalue of $\mathbf{h }(t_1,t_2,t_3)$ on $\mathbb {S}^3$ and $\mathbb {R}P^3$ are computed in [24], and the special cases where two of $t_1,t_2,t_3$ coincide done previously in [40];

• (iii) The first eigenvalue of $\mathbf{h }(t_1,t_2,t_3)$ on $\mathbb {S}^{4n+3}$ and $\mathbb {R}P^{4n+3}$ are computed in Theorem A, and the special case $t_1=t_2=t_3$ done previously in [39];

• (iv) The first eigenvalue of $\mathbf{k }(t)$ on $\mathbb {S}^{15}$ is computed in [12, Proposition 7.3], and an inspection of which eigenfunctions are $\mathbb {Z}_2$-invariant yields its first eigenvalue on $\mathbb {R}P^{15}$;

• (v) The first eigenvalue of $\check{\mathbf{h }}(t)$ on $\mathbb {C}P^{2n+1}$ is computed in Theorem B.

As an alternative reference for (i) and the special case $t_1=t_2=t_3$ in (iii) one may use, respectively, [12, Propositions 5.3 and 6.3]. These homogeneous metrics, together with those in (iv), account for all isometry classes of distance spheres in rank one symmetric space. A unified and explicit description of their full spectrum was recently obtained by the authors [11, Theorem A].

The above computations are carried out in one of two possible ways. The first, and more general, is the Lie-theoretic approach described in Sect. 2, which is used in (ii) and (iii), and generalizes the classical approach developed for canonical symmetric space metrics (see e.g. [41, 43]). The second, which relies on the existence of Riemannian submersions with minimal fibers, is explained in detail in [5] and [10], building on the earlier works [38,39,40], and is used in (i), in the special case $t_1=t_2=t_3$ in (iii), as well as in (iv) and (v).

We also include in Table 1 formulae for the scalar curvature of these CROSSes. The computation for the symmetric space metric on $\mathbb {S}^n$, $\mathbb {R}P^n$, $\mathbb {C}P^n$, $\mathbb {H}P^n$, and $\mathbb {C}\mathrm {a}P^2$ follows from the computation of their Einstein constants, which are, respectively, $n-1$, $n-1$, $2(n+1)$, $4(n+2)$, and 36, under the normalization convention that these metrics have $\sec =1$ for $\mathbb {S}^n$ and $\mathbb {R}P^n$, and $1\le \sec \le 4$ in the remaining cases. The computation for the other homogeneous metrics uses the Gray–O’Neill formula [9, Proposition 9.70], see also (5.6) and [12, Proposition 4.2]. In Table 3, by solving the inequality

$$\begin{aligned} {\text {scal}}(M,\mathrm {g})<(\dim M-1)\lambda _1(M,\mathrm {g}), \end{aligned}$$

(A.1)

we present the range of parameters for which these metrics are stable solutions to the Yamabe problem. If equality holds in (A.1), $\mathrm {g}$ is labeled as degenerate stable.

Remark A.1

For the convenience of the reader, we also identify some small imprecisions and typos in the literature. First, the multiplicity of the kth eigenvalue of the round sphere, $\lambda _k(\mathbb {S}^d,\mathrm {g}_{\mathrm {round}})=k(k+d-1)$, is given by (3.34). Unfortunately, this formula appears with (the same) typos in [7, p. 162] and [16, p. 35].

Second, the computation of some heat invariants of $\mathbb {C}\mathrm {a}P^2$ carried out in [15] is incorrect. For instance, the ratio $a_1/a_0$ of the first two heat invariants, which is equal to $\frac{{\text {scal}}}{6}$, evaluates to a negative number according to the formulae in [15, §13]. The correct values for these invariants are given in [4, Theorem 2.1]. More precisely, in the notation of [15, §12], the values of $\eta _j$ are correct, except for $\eta _6=-{175}/{4}$, $\eta _3=2864323/256$, and $\eta _1=18445239/4096$. Furthermore, the second row of $\zeta _{P^2(\text {Cay})}$ in [15, p. 20] should be replaced with

$$\begin{aligned} \zeta _{P^2(\text {Cay})} (t) = \frac{3!}{7! 11!} e^{(121/72)t} \sum _{j=0}^7 \eta _j\, (-1)^j\, g^{(j)}(\tfrac{t}{18}) + O (1), \end{aligned}$$

which gives, for any $0\le m\le 7$,

$$\begin{aligned} a_m= \frac{3!}{7!11!} (4\pi )^8 \sum _{k=0}^m \left( \frac{121}{72}\right) ^{\! k} {\eta _{7-m+k}}\, \frac{(7-m+k)!}{k!} \, 18^{8+k-m}. \end{aligned}$$

Using the above, one obtains the correct value $a_1/a_0=4/3$, according to the normalization used in [15], for which the scalar curvature of $\mathbb {C}\mathrm {a}P^2$ is ${\text {scal}}=8$.

Table 1 First eigenvalue of the Laplacian, scalar curvature, and volume of homogeneous metrics on simply-connected CROSSes

Table 2 First eigenvalue of the Laplacian, scalar curvature, and volume of homogeneous metrics on the non-simply-connected CROSSes

Table 3 Classification of homogeneous metrics on CROSSes that are stable solutions to the Yamabe problem, with same conventions as in Table 1