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

Abstract

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 $\Sigma$.