# First eigenvalue of the Laplacian on Berger spheres

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.

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Hi, Renato! It can be done probably this way. The Berger sphere $G/K=S^{2n+1}=SU(n+1)/SU(n)$ is a normal homogeneous space. Peter-Weyl decomposes $L^2(G)$ wrt left regular representation and this induces a decomposition of $L^2(G/K)$ into $G$-irreducible representations. The Laplacian acting on $L^2(G/K)$ is the same as the Casimir element of $\mathfrak g$ acting on those representations, and it acts as a scalar, for which there is an explicit and easy formula. The details are in Nolan Wallach's book on Harmonic Analysis (if I rememeber well), I think you can figure out. –  Claudio Gorodski Jun 13 '11 at 1:34
In fact, the one-parameter family of metrics do not consist of normal homogeneous metrics, except for the basic one. It is still possible to adapt the method nontheless. –  Claudio Gorodski Jun 13 '11 at 13:02
Hi Claudio, I believe you are refering to Lemma 5.6.4 on p. 124 of Wallach's book (the 1973 print), where it is shown that the Laplacian on a compact Lie group $G$ acts on the factors $\Gamma_\gamma E$ of the left regular representation decomposition as the scalar $c(\gamma)=\langle \Lambda_\gamma+\rho,\Lambda_\gamma+\rho\rangle-\langle\rho,\rho\rangle$, where $\gamma$ is the irreducible representation in question, $\Lambda_\gamma$ is the highest weight of $\gamma$ and $\rho$ is half the sum of the positive roots of $G$. However I still don't see how to apply this on any homogeneous space... –  Renato G Bettiol Jun 17 '11 at 16:32
If this is indeed the result you were mentioning, to obtain the Berger metrics on $SU(n+1)/SU(n)$ we consider a 1-parameter family of left-invariant metrics on $SU(n+1)$ (and the same should hold in the other homogeneous spaces $Sp(n+1)/Sp(n)$ and $Spin(9)/Spin(7)$). Even if we take, in the above formula for $c(\gamma)$, the inner product on $\mathfrak h^*$ (the Lie algebra of the maximal torus) to be the one induced by this 1-parameter family, why should $c(\gamma)$ be the eigenvalues of $\Delta_t$, even for the $\gamma$'s that have a non-zero vector fixed by $SU(n)$ (or $Sp(n),Spin(7)$)? –  Renato G Bettiol Jun 17 '11 at 16:39
No, the Laplacian of the normal homogeneous metric corresponds to the Casimir element and has the eigenavalues you described, but for the other metrics in the 1-parameter family you need to modify the Casimir element accordingly and correct the numerical value of the eigenvalues. Construct the Casimir element using a Killing orthonormal basis adapted to the submersion $SU(n+1)\to SU(n+1)/SU(n)$ and renormalize in one direction. For the normal homogeneous metric, Helgason's "Groups and geometric analysis" is also a good reference. –  Claudio Gorodski Jun 17 '11 at 16:59

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.