A duality theorem for a four dimensional Willmore energy

Abstract

We prove an analogue of the energy identities underlying Bryant's duality theorem for a four-dimensional Willmore energy ${\mathcal{E}}_{\mathrm{GR}}$ obtained by Graham--Reichert and Zhang in codimension one. We show that, for an immersion $\Phi$ of a compact four-dimensional manifold without boundary $\Sigma$ into ${\mathbb{R}}^5$, the energy ${\mathcal{E}}_{\mathrm{GR}}(\Phi )$ is equal to two energies associated with its conformal Gauss map $Y$: one defined only in terms of the image of $Y$, which is the analogue of the area functional for Willmore surfaces, and another defined on maps from $\Sigma$ into the de Sitter space ${\mathbb{S}}^{5,1}$, which is the analogue of the Dirichlet energy for Willmore surfaces. We prove that, even when restricted to immersions of a given topological manifold ${\Sigma }^4$, ${\mathcal{E}}_{\mathrm{GR}}$ is not bounded below on the set of immersions from $\Sigma$ into ${\mathbb{R}}^5$. We exhibit a second conformally invariant energy ${\mathcal{E}}_{\mathrm{P}}$ that is bounded below and whose construction is closer to that of the two-dimensional Willmore energy.

arXiv comment38pages, v2 : reference added and some typos corrected, v3 : typos corrected, corrections in the proof of Proposition I. v4: presentation improved, link between conformal Gauss map and normal tractor added

References in corpus (5)

• Positivity, negativity, and entanglement
• Conformally invariant bending energy for hypersurfaces
• Energy Quantization of Willmore surfaces at the boundary of the Moduli Space
• Global Conformal Invariants of Submanifolds
• Classification of branched Willmore spheres