Let $(M,g)$ be a compact manifold without boundary.

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

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?