In short: Are harmonic forms critical points of some generalized Dirichlet functional?
Let $(M,g)$ be a smooth Riemannian manifold, denote by $\Omega^k(M)$ the space of $k$-valued forms on $M$. Let $d:\Omega^k(M) \to \Omega^{k+1}(M),\delta:\Omega^{k+1}(M) \to \Omega^k(M)$ be the exterior derivative and its adjoint, respectively.
A $k$-form $\sigma$ on $M$ is called harmonic if $d\sigma=\delta \sigma=0$ (or equivalently if $\Delta \sigma=0$ where $\Delta=d\delta +\delta d$).
$0$-harmonic forms (also known as harmonic functions) are critical points of the Dirichlet integral: $E(\sigma)=\int_M |d\sigma|_{g}^2 \operatorname{Vol}_g$. Is there an analogous variational realization for forms of higher degree?
(Perhaps $E(\sigma)=\int_M |d\sigma|_{g}^2 + |\delta \sigma|_{g}^2 \operatorname{Vol}_g$ is the right choice?)
Also, what about vector-valued forms?
Any reference for a variational treatment would be appreciated.