A duality theorem for a four dimensional Willmore energy
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916207494430720 |
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| author | Martino, Dorian |
| author_facet | Martino, Dorian |
| contents | 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 $Φ$ from a four dimensional compact manifold without boundary $Σ$ into $\mathbb{R}^5$, the energy $\mathcal{E}_{GR}(Φ)$ 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 $Σ$ 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 $Σ^4$, $\mathcal{E}_{GR}$ is never bounded from below on the set of immersions from $Σ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11433 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A duality theorem for a four dimensional Willmore energy Martino, Dorian Differential Geometry 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 $Φ$ from a four dimensional compact manifold without boundary $Σ$ into $\mathbb{R}^5$, the energy $\mathcal{E}_{GR}(Φ)$ 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 $Σ$ 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 $Σ^4$, $\mathcal{E}_{GR}$ is never bounded from below on the set of immersions from $Σ$ 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. |
| title | A duality theorem for a four dimensional Willmore energy |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2308.11433 |