Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the so-called Berger spheres), i.e., for each $t>0$, let $S^{2n+1}\_t$ (resp $S^{4n+3}\_t$) be the Riemannian manifold $S^{2n+1}$ (resp $S^{4n+3}$) endowed with the metric obtained by scaling the round metric by a factor of $t^2$ in the direction of the fibers $S^1$ (resp $S^3$). For each $t>0$, they are the total space of a Riemannian submersion with totally geodesic fibers isometric to $tS^1$ (resp $tS^3$). In this context, the Laplacian $\Delta\_t$ of the canonical variation can be related to the original Laplacian $\Delta\_1$ in terms of the *vertical Laplacian* $\Delta\_v$, by the formula $$\Delta\_t=\Delta\_1+(\tfrac{1}{t^2}-1)\Delta\_v,$$ see Berard-Bergery and Bourguignon [Illinois Journ of Math, 1982]. The operator $\Delta\_v$ acts on functions of the total space and is defined by $$(\Delta\_v f)(p)=\Delta\_{F\_p}(f\vert\_{F\_p})$$ where $\Delta\_{F\_p}$ is the Laplacian of the fiber $F\_p$.


Since the fibers are totally geodesic, the spectrum of $\Delta\_v$ coincides with the spectrum of the Laplacian of the fiber, and all the above operators commute. In particular, $L^2$ of the total space admits a decomposition in simultaneous eigenspaces of $\Delta\_t$ and $\Delta\_v$. As a consequence, we have the following inclusion of spectra: $$Spec(\Delta\_t) \subset Spec(\Delta\_1)+(\tfrac{1}{t^2}-1) Spec(\Delta\_F).$$ 


I am interested in computing $\lambda\_1(t)$, the first non-zero eigenvalue of $\Delta\_t$, for all $t$. By the above, for all $t>0$, we know $\lambda\_1(t)$ is of the form $\mu\_k+(\tfrac{1}{t^2}-1)\phi\_j$ where $\mu\_k\in Spec(\Delta\_1)$ is an eigenvalue of the Laplacian on the original total space and $\phi\_j\in Spec(\Delta\_F)$ is an eigenvalue of the Laplacian of the fiber. The problem is that not all combinations of $\mu\_k$'s and $\phi\_j$'s give an eigenvalue of $\Delta\_t$ and the first non-zero eigenvalue of $\Delta\_t$ might be given by different combinations of $\mu\_k$'s and $\phi\_j$'s for each $t$. Recall that since total space and fibers are spheres, both $\mu\_k$'s and $\phi\_j$'s are easy to compute, namely the $k$th eigenvalue of the $m$-sphere is $k(k+m-1)$.


In the case of the first family with 1-dim fibers, this computation follows from a paper of Tanno [Tohoku Math Journ, 1979]. The trick is to look at a trajectory of the vector tangent to the $S^1$ fiber (a great circle) and solve the eigenvalue equation there (which becomes an ODE). Using this he proves that the only combinations of $\mu\_k$'s and $\phi\_j$'s permitted are when $0\leq j\leq k$ and $k-j$ is even. He then also finds explicit eigenfunctions when $k=j$; $j=1$ and $k$ odd; $j=2$ or $j=0$ and $k$ even. With these, it is very easy to compute $\lambda\_1(t)$ for this family where the fibers are 1-dim.


However, I do not know any way of extending the result to the case of 3-dim fibers, or also the case of 7-dim fibers, $S^7\to S^{15}\to S^8$. It seems that Tanno's technique uses strongly the fact that the fibers are 1-dim. Any suggestion on how to compute this other first eigenvalues would be greatly appreciated.



First eigenvalue of the Laplacian on Berger spheres
Tanno, the same author of the paper above mentioned in the case of 1-dim fibers, has another paper one year later [Tanno, Shûkichi. Some metrics on a $(4r+3)$-sphere and spectra.
Tsukuba J. Math. 4 (1980), no. 1, 99–105. ] in which he addresses exactly my original question, regarding the case of 3-dim fibers. As a consequence, one has an explicit formula for the first eigenvalue of the Laplacian $\Delta\_t$ of $S^{4n+3}\_t$ for all $t>0$. The technique is indeed related to Claudio's suggestion of using representation theory, since the Laplacian acts as the Casimir element. The case of $S^7\to S^{15}\to S^8$ can also be treated similarly, obtaining the first eigenvalue explicitly.


