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 ${\mathbb{S}}_{p,1}^{n+1}$ 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 $\Sigma$ in ${\mathbb{S}}^{n+1}$ (resp. ${\mathbb{H}}^{n+1}$) that is a critical point of the functional $\mathcal{W}(\Sigma )={\int }_{\Sigma }{\left({\Pi }_{i=1}^n|\epsilon +k_i^2|\right)}^{1/2}$, 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 ${\mathcal{W}}^{\prime }(\Sigma )={\int }_{\Sigma }\sqrt{H^2-K+1}$ (resp. ${\mathcal{W}}^{\prime }(\Sigma )={\int }_{\Sigma }\sqrt{H^2-K-1}$), where $H$ and $K$ denote, respectively, the mean and Gaussian curvature of $\Sigma$.

10 pages

References in corpus (2)

• On the Geometry of Spaces of Oriented Geodesics
• Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface