the periodic curve $\gamma_1$, so that the natural projection $\widetilde{\Lambda}_{\mathfrak h}(\Sigma)\to \Lambda_{\mathfrak h}(\Sigma)$, $[\gamma_r]\mapsto \gamma_1$ is a covering map.

We pick a closed two-form ${\mathrm{\omega}}_0\in\Xi^2_{c_0}(M_0)$ and choose $\Omega_0:=\mathfrak p_0^*{\mathrm{\omega}}_0\in\Xi^2_C(\Sigma)$ as our auxiliary element for the computation of the volume. One sees that ${{\mathfrak{Vol}}}:\Xi^2_C(\Sigma)\to{{\mathbb R}}$ depends only on $(\mathfrak p_0,c_0)$ and not on ${\mathrm{\omega}}_0$. We associate to $\Omega=\Omega_0+{{\mathrm d}}\alpha$ the **action functional** $$\widetilde {\mathcal A}_{\Omega}:\widetilde\Lambda_{\mathfrak h}(\Sigma)\to{{\mathbb R}},\qquad\widetilde{\mathcal A}_{\Omega}\big([\gamma_r]\big):=\int_{[0,1]\times S^1}\Gamma^*\Omega + \int_{S^1}\gamma_0^*\alpha,$$ where $\alpha$ is chosen to be normalised if $C^n\neq0$, and $\Gamma:[0,1]\times S^1$ is the cylinder traced by the path $\{\gamma_r\}$. Like the volume, the functional $\widetilde{\mathcal A}_{\Omega}$ depends only on $(\mathfrak p_0,c_0)$ and not on ${\mathrm{\omega}}_0$. If $[\gamma_r]$ is a critical point of $\widetilde{\mathcal A}_{\Omega}$, then $\gamma_1\in\mathcal X(\Omega)$, provided $\gamma_1$ is embedded. Furthermore, the action is invariant under isotopies $\{\Psi_r:\Sigma\to\Sigma\}$ starting at the identity (Proposition \[prp:action\_inv\_isotopy\]): $$\widetilde {\mathcal A}_{\Psi_1^*\Omega}\big([\Psi^{-1}_r\circ\gamma_r]\big)=\widetilde{\mathcal A}_{\Omega}\big([\gamma_r]\big),\qquad\forall\,[\gamma_r]\in\widetilde\Lambda_{\mathfrak h}(\Sigma).$$ In general, $\widetilde{\mathcal  A}_\Omega$ does not descend to a functional on $\Lambda_{\mathfrak h}(\Sigma)$. More precisely, by Lemma \[l:exact\] this happens if and only if $c_0|_{\pi_2(M_0)}=a e_0|_{\pi_2(M_0)}$ for some $a\in{{\mathbb R}}$. However, as we see now, we can define an action on the set of Zoll forms $\Omega$ with $(\mathfrak p_\Omega,[{\mathrm{\omega}}_\Omega])\in\mathfrak Z_C^0(\Sigma)$. In this case, the volume of $\Omega$ can 