The Willmore problem for surfaces with symmetry
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917114685685760 |
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| author | Kusner, Rob Lü, Ying Wang, Peng |
| author_facet | Kusner, Rob Lü, Ying Wang, Peng |
| contents | The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [22], but the general conjecture [12] that Lawson's [18] minimal surface $ξ_{g,1}\subset\mathbb{S}^3$ minimizes $W$ among surfaces of genus $g>1$ remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces $M\subset\mathbb{S}^3$ share the ambient symmetries $\widehat{G}_{g,1}$ of $ξ_{g,1}$. In fact, we show each Lawson surface $ξ_{m,k}$ satisfies the corresponding $W$-minimizing property under a smaller symmetry group $\widetilde{G}_{m,k}=\widehat{G}_{m,k}\cap SO(4)$. We also describe a genus 2 example where known methods do not ensure the existence of a $W$-minimizer among surfaces with its symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Willmore problem for surfaces with symmetry Kusner, Rob Lü, Ying Wang, Peng Differential Geometry The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [22], but the general conjecture [12] that Lawson's [18] minimal surface $ξ_{g,1}\subset\mathbb{S}^3$ minimizes $W$ among surfaces of genus $g>1$ remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces $M\subset\mathbb{S}^3$ share the ambient symmetries $\widehat{G}_{g,1}$ of $ξ_{g,1}$. In fact, we show each Lawson surface $ξ_{m,k}$ satisfies the corresponding $W$-minimizing property under a smaller symmetry group $\widetilde{G}_{m,k}=\widehat{G}_{m,k}\cap SO(4)$. We also describe a genus 2 example where known methods do not ensure the existence of a $W$-minimizer among surfaces with its symmetry. |
| title | The Willmore problem for surfaces with symmetry |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.12582 |