TITLE: Explicit eigenvalues of the Laplacian
QUESTION [36 upvotes]: Let $(M,g)$ be a compact manifold without boundary.

Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?

An important example is the $n$-sphere with its standard metric. To find eigenvalues, we embed $S^n$ inside $\mathbb{R}^{n+1}-\{0\}$ in the usual way, consider a positive homogeneous function $f\in C^\infty(\mathbb{R}^{n+1}-\{0\})$ of degree $s$, and then take the restriction to the sphere of the Laplacian $\Delta$ on $\mathbb{R}^{n+1}-\{0\}$ applied to the function $|x|^{-s} f$. The result is that if $f$ is harmonic relative to the Laplacian on $\mathbb{R}^{n+1}-\{0\}$, then the restriction to $S^n$ of $\Delta(|x|^{-s} f)$ is a scalar multiple of the restriction of $f$ to $S^n$, with the scalar being $s(s+n-2)$.
One sees very quickly that for more complicated manifolds, such a method does not apply. Various authors comment that the spectrum of the Laplacian is not easy to determine explicitly, and much of the literature seems to be consumed only with estimates for certain eigenvalues of the Laplacian given various constraints on the geometry of $(M,g)$.
Are there other interesting manifolds for which the spectrum of the Laplacian is known? In particular, are they known for ellipsoids?

REPLY [7 votes]: Generalizing the case of flat tori, one can compute explicitely the spectrum of many compact flat manifolds. See for instance 

spectrum on $p$-forms Miatello and Rossetti or the survey on isospectral compact flat manifolds in Contemp. Math. 491 AMS, 83--113.

There are also more progress for lens spaces than the one by Sakai mentioned by Ziegler.
See for instance

the description of the associated generating functions in Thm. 3.2 given by Ikeda and Yamamoto,
the same for the spectrum on $p$-forms in Thm. 2.3 by Ikeda (``Riemannian manifolds $p$-isospectral but not $p+1$-isospectral'', in Geometry of manifolds, Perspect. Math. 8, 1989),
the multiplicity of each eigenvalue in Thm. 3.8 of this article,
the associated generating function written as a rational function in Thm. 3.6 here,
the Dirac spectrum in Thm. 4.3 here. 

Other contributions on compact homogeneous spaces:

spectra on $p$-forms on $S^n$ and $P^n(\mathbb{C})$ Ikeda & Taniguchi, 
Grassmann Manifolds Tsukamoto, Fida El Chami, Tsagas & Kalogeridis,
Quaternionic Grassmann manifolds Fida El Chami,
$Sp(n)/U(n)$ Hong & Chen, Tsagas & Kalogeridis.