# Explicit Eigenvalues of the Laplacian

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?

Besse (1978, p.202) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $$\mathrm S^n$$ due apparently to Heine (1863, §19; 1878, §128), 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))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.)

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:

Secondly, some cases yield to other methods:

• Lens spaces $$\mathrm S^{2n-1}\,/\,\mathbf Z_p$$ by Sakai (1976, p.256).

• Hopf manifolds $$M_\alpha$$ by Bedford & Suwa (1976, p.261).

• 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).

• 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) and Nilsson & Pope (1983, p.68).

• Riemannian two-step nilmanifolds $$G\,/\,\Gamma$$ by Pesce (1993).

1 Note added: Rogawski–Varadarajan (2012, p. 690) attribute the formula for $$c(\lambda)$$ to Casimir–van der Waerden (1935; note the reviewer). However, I’m not sure I can find it there...?

• What is meant by $\sum_{\alpha>0}\alpha$? – YCor Jul 21 '16 at 22:53
• @YCor : The sum of the positive roots. (So $\rho$ is half that, as usual I think?) – Francois Ziegler Jul 24 '16 at 16:48
• The Casimir method is used for the case of U(n) (although not semi-simple) in a paper by Voiculescu, where the Riemannian metric comes from the Hilbert-Schmidt norm on the Lie algebra – Guillaume Aubrun Aug 25 '16 at 11:13

You can compute the eigenvalues explicitly for flat tori $\mathbb{R}^n/\Gamma$, where $\mathbb{R}^n$ has the standard Euclidean metric and $\Gamma$ is a lattice. The eigenvectors all have the form $e^{2 \pi i \langle v, w \rangle}$ where $v \in \mathbb{R}^n$ and $w \in \Gamma^{\vee}$ lies in the dual lattice. The corresponding eigenvalue is $-4 \pi^2 \| w \|^2$.

Among other things, this allows you to reduce the problem of finding two nonisometric isospectral manifolds to the problem of finding two nonisomorphic lattices with the same theta function.

• @Piotr: why? Everything I’ve said is a straightforward exercise. Also I don’t have a reference, because I learned all of this as an exercise that was assigned to me once. – Qiaochu Yuan Jun 25 '18 at 6:57
• You are right. I did not realize that your argument is self-contained. I was too quick. – Piotr Hajlasz Jun 25 '18 at 12:38

Jeffrey Weeks has computed the spectra of homogeneous elliptic manifolds.

For arithmetic hyperbolic manifolds, the spectrum is in principle computable in the sense that one may define a Selberg zeta function arithmetically, whose roots give the spectrum.

Certain other homogeneous Heisenberg manifolds have their spectra computed.

In principle, one can compute the spectrum of any homogeneous compact Riemannian manifold because in this case the problem is essentially representation theoretic. However, performing this computation concretely takes a bit of skill.

If $(M_k,g_k)$, $k=1,2$ are Riemannian manifolds whose spectra you can compute, then, using the separations of variables trick, one can compute the spectrum of the product $(M_1\times M_2, g_1\times g_2)$. (The example of the flat torus is a special case of this principle.)

• Do you mean normal homogeneous spaces? And if not, how does one go about computing eigenvalues of left-invariant (not bi-invariant) metric on a compact Lie group? – Chris Judge Oct 13 '16 at 19:10
• I meant bi-invariant, and I was honestly thinking of of symmetric spaces. – Liviu Nicolaescu Oct 13 '16 at 23:14

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: