A duality theorem for a four dimensional Willmore energy

We prove an analog of Bryant’s duality theorem for a four dimensional Willmore energy ${\mathcal{E}}_{GR}$ obtained by Graham-Reichert and Zhang. We show that for an immersion $\mathrm{\Phi }$ from a four dimensional compact manifold without boundary $\mathrm{\Sigma }$ into ${\mathbb{R}}^5$, the energy ${\mathcal{E}}_{GR}(\mathrm{\Phi })$ is equal to two energies on its conformal Gauss map $Y$. One defined only in terms of the image of $Y$, which is the analog of the area functional for Willmore surfaces, and an other one defined on maps from $\mathrm{\Sigma }$ into the De Sitter space ${\mathbb{S}}^{5,1}$, which is the analog of the Dirichlet energy for Willmore surfaces. We prove that even when restricted to immersions of a given topological manifold ${\mathrm{\Sigma }}^4$, ${\mathcal{E}}_{GR}$ is never bounded from below on the set of immersions from $\mathrm{\Sigma }$ into ${\mathbb{R}}^5$. We exhibit a second conformally invariant energy ${\mathcal{E}}_P$ which is bounded from below and whose construction is closer to the two dimensional Willmore energy.