We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${ \mathbb S}^{ n+1}_{ p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${ \mathbb L}({ \mathbb S}^{ n+1}_{ p,1})$ 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 $\Sigma$ 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 $k_i$ denote the principal curvatures of $\Sigma$ 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. ${ \cal W}'(\Sigma)=\int_\Sigma\sqrt{ H^2-K-1}\; $), where $H$ and $K$ denote, respectively, the mean and Gaussian curvature of $\Sigma$.
Categories: math.DG