Spectral Theory of the Laplacian

The spectrum of the Hodge Laplacian encodes geometric and topological information about the manifold. Spectral geometry asks: "Can one hear the shape of a drum?"

The Eigenvalue Problem

Definition10.5Spectrum of the Laplacian

On a compact Riemannian manifold $(M, g)$ (without boundary), the eigenvalue problem for the Hodge Laplacian on $k$ -forms is

\Delta \omega = \lambda \omega, \quad \omega \in \Omega^k(M).

The spectrum $\mathrm{Spec}_k(M, g)$ is the set of eigenvalues $0 \leq \lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \to \infty$ , counted with multiplicity.

Theorem10.4Spectral theorem for the Hodge Laplacian

On a compact Riemannian manifold:

The eigenvalues $0 \leq \lambda_0 \leq \lambda_1 \leq \cdots$ are non-negative and tend to $+\infty$ .
The eigenspaces $E_\lambda = \{\omega : \Delta\omega = \lambda\omega\}$ are finite-dimensional.
The eigenforms form a complete orthonormal basis for $L^2\Omega^k(M)$ .
The multiplicity of $\lambda = 0$ equals the $k$ -th Betti number $b_k$ .

Spectral Geometry

ExampleSpectra of model spaces

Circle $S^1$ of length $2\pi$ : On functions, $\Delta f = -f''$ , eigenvalues $\lambda_n = n^2$ ( $n \geq 0$ ), with eigenfunctions $\cos(n\theta)$ , $\sin(n\theta)$ .
Sphere $S^n$ : On functions, $\lambda_k = k(k+n-1)$ with multiplicity $\binom{k+n}{n} - \binom{k+n-2}{n}$ (spherical harmonics of degree $k$ ).
Flat torus $T^n = \mathbb{R}^n/\Lambda$ : On functions, $\lambda_v = 4\pi^2|v|^2$ for $v$ in the dual lattice $\Lambda^*$ .

Weyl's Law and the Heat Kernel

Theorem10.5Weyl's asymptotic law

Let $N_k(\lambda) = \#\{j : \lambda_j \leq \lambda\}$ be the eigenvalue counting function for the Laplacian on functions on a compact $n$ -manifold. Then

N_0(\lambda) \sim \frac{\omega_n \mathrm{Vol}(M)}{(2\pi)^n} \lambda^{n/2} \quad \text{as } \lambda \to \infty,

where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$ . This determines $\dim(M)$ and $\mathrm{Vol}(M)$ from the spectrum.

Definition10.6Heat kernel

The heat kernel $K(t, x, y)$ is the fundamental solution of the heat equation $(\partial_t + \Delta_x)K = 0$ with $K(0, x, y) = \delta_y(x)$ . It has the spectral expansion

K(t, x, y) = \sum_j e^{-\lambda_j t} \phi_j(x) \phi_j(y),

where $\{\phi_j\}$ are orthonormal eigenfunctions. The heat trace is $Z(t) = \int_M K(t, x, x) \, \mathrm{dvol} = \sum_j e^{-\lambda_j t}$ .

RemarkCan you hear the shape of a drum?

Milnor (1964) showed that two 16-dimensional flat tori can have the same Laplacian spectrum but not be isometric, answering Kac's question negatively. However, the spectrum determines many geometric invariants: dimension, volume, total scalar curvature (via the heat trace asymptotics $Z(t) \sim (4\pi t)^{-n/2}\sum_{k=0}^\infty a_k t^k$ , where $a_0 = \mathrm{Vol}(M)$ and $a_1 = \frac{1}{6}\int_M S \, \mathrm{dvol}$ ).

RemarkThe Atiyah-Singer index theorem

The index of an elliptic operator $D$ (the difference $\dim\ker D - \dim\ker D^*$ ) can be computed from topological data via the heat kernel: $\mathrm{ind}(D) = \mathrm{tr}(e^{-tD^*D}) - \mathrm{tr}(e^{-tDD^*})$ for any $t > 0$ . This underlies the Atiyah-Singer index theorem, one of the deepest results connecting analysis, geometry, and topology.