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

TL;DR

This paper establishes a connection between Hamiltonian minimal submanifolds in the space of oriented geodesics and critical points of certain curvature functionals on hypersurfaces in real space forms, with specific results for surfaces in 3D.

Contribution

It demonstrates that deformations of hypersurfaces induce Hamiltonian variations in geodesic spaces and characterizes Hamiltonian minimal submanifolds as normal congruences of critical hypersurfaces.

Findings

Deformations induce Hamiltonian variations in geodesic spaces.

Hamiltonian minimal submanifolds correspond to critical points of specific curvature functionals.

Results include characterizations for surfaces in 3D 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 $\Sigma$.