Weighted $\infty$-Willmore Spheres
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910375280115712 |
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| author | Gallagher, Ed Moser, Roger |
| author_facet | Gallagher, Ed Moser, Roger |
| contents | On the two-sphere $Σ$, we consider the problem of minimising among suitable immersions $f \,\colon Σ\rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $ξ$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H \in \{ \pm \vert \vert ξH \vert \vert_{L^\infty} \}$ away from the nodal set of the PDE system, and $H = 0$ on the nodal set (if it is non-empty). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07468 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weighted $\infty$-Willmore Spheres Gallagher, Ed Moser, Roger Differential Geometry Analysis of PDEs 49Q10 (Primary), 35A15, 53A05, 53C42, 58E12 (Secondary) On the two-sphere $Σ$, we consider the problem of minimising among suitable immersions $f \,\colon Σ\rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $ξ$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H \in \{ \pm \vert \vert ξH \vert \vert_{L^\infty} \}$ away from the nodal set of the PDE system, and $H = 0$ on the nodal set (if it is non-empty). |
| title | Weighted $\infty$-Willmore Spheres |
| topic | Differential Geometry Analysis of PDEs 49Q10 (Primary), 35A15, 53A05, 53C42, 58E12 (Secondary) |
| url | https://arxiv.org/abs/2211.07468 |