A duality theorem for a four dimensional Willmore energy

Abstract

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 $\Phi$ from a four dimensional compact manifold without boundary $\Sigma$ into $\mathbb{R}^5$, the energy $\mathcal{E}_{GR}(\Phi)$ 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 $\Sigma$ 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 $\Sigma^4$, $\mathcal{E}_{GR}$ is never bounded from below on the set of immersions from $\Sigma$ 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.