What is an example of harmonic form on a compact manifold?

Question

I'm doing some self-study on Hodge theory and elliptic operators right now. I'm trying to come up with an example of a harmonic $p$-form $\omega$ on a compact manifold, i.e. a form such that $d\omega =0$ and $d^* \omega=0$ where $d^*=(-1)^{np+n+1}\star d\star$ and $\star$ is the Hodge star. But I can't seem to find an example of one in any texts or through my own imagination.

Can anyone give me a (nontrivial) example of a harmonic form on a compact manifold?

Answer 1

If you start with a Kahler manifold $X$ of dimension $n$ (that is, a complex manifold equipped with a Hermitian metric $h$ whose associated $(1,1)$-form $\omega := -\operatorname{Im}h$ is $d$-closed), then $\omega^k$ is harmonic for any $k$. This is because $$ * \frac{\omega^k}{k!} = \frac{\omega^{n-k}}{(n-k)!}. $$ For cheap examples of such manifolds, take any hypersurface in a complex projective space defined by a homogeneous polynomial, equipped with the restriction of the Fubini-Study metric.

Alternatively, consider a torus $X = \mathbb{R}^n / \mathbb{Z}^n$ equipped with the flat metric induced by the Euclidean one. The tangent bundle of the torus is trivial, so one gets isomorphisms $$ C^{\infty}\Bigl(X, \bigwedge{}^k\, T_X^*\Bigr) = C^{\infty}(X) \otimes \bigwedge{}^k\, \mathbb{R}. $$ That is, the smooth $k$-forms on $X$ can be viewed as forms $$ u(x) = \sum_{J} u_J(x) dx_{j_1} \wedge \cdots \wedge dx_{j_k}, $$ where $J = (j_1, \ldots, j_k)$ is a multiindex and $u_J(x)$ are smooth functions on the torus (which can be pulled back to smooth functions on $\mathbb{R}^n$). It is illuminating to prove that such a form $u$ is harmonic if and only if all the $u_J$ are constant.

For really fun times, note that the definition of the Hodge $*$ operator doesn't depend on the metric being positive-definite, but only on it being nondegenerate. Consider then the torus $X = \mathbb{R}^2 / \mathbb{Z}^2$ equipped with the metric induced by $$ \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} $$ on $\mathbb{R}^2$ and compute what the harmonic forms on it are. Extra points for noticing what about this makes geometers focus on positive-definite metrics.

Answer 2

Not precisely an example, but take any (smooth enough) $p$-form for any $p$, and apply heat equation to infinity. You will end up with an harmonic $p$-form, which in general is nontrivial. Take a look at Riemannian Geometry and Geometric Analysis of Jürgen Jost if you want to know more about it.

Eigenfunctions of the laplacian will vanish at exponential rate (with rate given by the eigenvalue), whilst the harmonic part will remain unchanged during the process.