On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb{S}}_{p,1}^{n+1}$ induce a Hamiltonian variation of the normal congruence in the space $\mathbb{L}({\mathbb{S}}_{p,1}^{n+1})$ of oriented geodesics. As an application, we show that every Hamiltonian minimal sumbanifold in $\mathbb{L}({\mathbb{S}}^{n+1})$ (resp. ${\mathbb L}({\mathbb H}^{n+1})$) with respect to the (para-) Kaehler Einstein structure is locally the normal congruence of a hypersurface $\mathrm{\Sigma }$ in ${\mathbb{S}}^{n+1}$ (resp. ${\mathbb{H}}^{n+1}$) that is a critical point of the functional ${\cal W}(\Sigma)=\int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}$,where$k_i$denotetheprincipalcurvaturesof$\Sigma$and$\epsilon\in{-1,1}$. In addition, for $n=2$, we prove that every Hamiltonian minimal surface in $\mathbb{L}({\mathbb{S}}^3)$ (resp. $\mathbb{L}({\mathbb{H}}^3)$) with respect to the (para-) Kaehler conformally flat structure is locally the normal congruence of a surface in ${\mathbb{S}}^3$ (resp. ${\mathbb{H}}^3$) that is a critical point of the functional ${\cal W}‘(\Sigma)=\int_\Sigma\sqrt{H^2-K+1}$(resp\mathrm{.}${\cal W}’(\Sigma)=\int_\Sigma\sqrt{H^2-K-1}; $),where$H$and$K$ denote, respectively, the mean and Gaussian curvature of $\mathrm{\Sigma }$.