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



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



Explicit eigenvalues of the Laplacian
[Besse (1978, p.202)](http://ams.org/mathscinet-getitem?mr=496885) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine ([1863](https://zbmath.org/?q=an:062.1614cj), [§19](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002151928&physid=phys139); [1878](https://emis.de/cgi-bin/JFM-item?10.0332.01), [§128](https://archive.org/stream/handbuchderkugel01heinuoft#page/461)), this gives $\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$ and $\mathbf{OP}^2$.


**Edit:** Also, for $M$ a compact semisimple Lie group it is well known (due apparently to [Freudenthal (1954)](https://books.google.com/books?id=WiNXgJfbF5wC&pg=PA428))1 that the Laplacian (= Casimir) acts on the $\lambda$-subspace in the Peter-Weyl decomposition $L^2(M)=\smash{\bigoplus\_\lambda V\_\lambda^{\phantom\*}\!\otimes V\_\lambda^\*}$ by the scalar $c(\lambda):=\smash{\|\lambda + \rho\|^2-\|\rho\|^2}$; so these are its eigenvalues. $(\lambda$: dominant weight; $\rho=\frac12\!\sum\limits\_{\alpha > 0}\alpha$; $\|\cdot\|$: Killing norm.) 




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**Further edit:** The literature contains quite a few more cases than the answers so far. As no single source or search word easily returns them, I list here what I found (others’ answers not repeated):


First, the Casimir method above extends to give the spectrum of the normal metric on $G/H$ ($G$ compact semisimple, $H$ closed). In fact, by Frobenius reciprocity, $V\_\lambda$ occurs in $L^2(G/H) = \operatorname{Ind}\_H^G1$ with multiplicity equal to the dimension of $V\_\lambda{}^H=\{H$-fixed vectors in $V\_\lambda\}$. So the eigenvalues are exactly all $c(\lambda)$ for $\lambda$ such that $V\_\lambda{}^H\ne0$. After spheres, this method was applied to:


* Stiefel manifolds $\mathrm{SO}\_n\,/\,\mathrm{SO}\_{n-m}$ by [Levine (1969, p.519)](http://ams.org/mathscinet-getitem?mr=412743), [Gelbart (1974)](http://ams.org/mathscinet-getitem?mr=425519), [Strichartz (1975)](http://ams.org/mathscinet-getitem?mr=380277).
* CROSSes ($\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$, $\mathbf{OP}^2$) by [Berger & al. (1971, pp.159-173)](http://ams.org/mathscinet-getitem?mr=282313), [Cahn & Wolf (1976)](http://ams.org/mathscinet-getitem?mr=397801).
* Flag manifolds $G\,/\,T$ ($T$: maximal torus) by [Yamaguchi (1979, p.110)](http://ams.org/mathscinet-getitem?mr=530024).
* Grassmannians $\mathrm{Gr}\_2(\mathbf R^n)$ by [Strese (1980, p.78)](http://ams.org/mathscinet-getitem?mr=623695) and [Tsukamoto (1981)](http://ams.org/mathscinet-getitem?mr=628842).
* Aloff-Wallach spaces $\mathrm{SU}\_3\,/\,\mathrm S^1$ by [Urakawa (1984, p.984)](http://ams.org/mathscinet-getitem?mr=755857) and [Joe et al. (2001, p.417)](http://ams.org/mathscinet-getitem?mr=1827691).
* Symmetric spaces $\mathrm{SU}\_n\,/\,\mathrm{SO}\_n$ by [Gurarie (1992,](//ams.org/mathscinet-getitem?mr=1174965) [p.253)](//books.google.com/books?id=tRBqyIFmde8C&ots=wLY3Sdxlei&pg=PA253).
* Grassmannians $\mathrm{Gr}\_n(\mathbf C^{n+m})$ and $\mathrm{SU}\_{n+m}\,/\,\mathrm{SU}\_n\times\mathrm{SU}\_m$ by [Ben Halima (2007, pp.546, 549)](http://ams.org/mathscinet-getitem?mr=2371957).


Secondly, some cases yield to other methods:


* Lens spaces $\mathrm S^{2n-1}\,/\,\mathbf Z\_p$ by [Sakai (1976, p.256)](http://ams.org/mathscinet-getitem?mr=397807).
* Hopf manifolds $M\_\alpha$ by [Bedford & Suwa (1976, p.261)](http://ams.org/mathscinet-getitem?mr=418172).
* Berger spheres (total spaces of the Hopf fibration $\mathrm S^1\to\mathrm S^{2n+1}\to \mathbf{CP}^n$ with rescaled fiber) by [Tanno (1979, p.184)](http://ams.org/mathscinet-getitem?mr=538918).
* Jensen spheres (total spaces of the Hopf fibration $\mathrm S^3\to\mathrm S^{4n+3}\to \mathbf{HP}^n$ with rescaled fiber) by [Tanno (1980, p.103)](http://ams.org/mathscinet-getitem?mr=597687) and [Nilsson & Pope (1983, p.68)](http://ams.org/mathscinet-getitem?mr=727944).
* Grassmannians $\mathrm{Gr}\_2(\mathbf C^n)$ by [Sumitomo & Tandai (1985, p.153)](//ams.org/mathscinet-getitem?mr=86i:58136).
* Riemannian two-step nilmanifolds $G\,/\,\Gamma$ by [Pesce (1993)](http://ams.org/mathscinet-getitem?mr=1145731).




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1 **Note added:** Rogawski–Varadarajan ([2012](//ams.org/mathscinet-getitem?mr=3001808), p. 690) attribute the formula for $c(\lambda)$ to Casimir–van der Waerden ([1935](https://zbmath.org/?q=an:61.0475.02); note the reviewer). However, I’m not sure I can find it there...?


