Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916305854005248 |
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| author | Hirsch, Jonas Kusner, Rob Mäder-Baumdicker, Elena |
| author_facet | Hirsch, Jonas Kusner, Rob Mäder-Baumdicker, Elena |
| contents | We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $\mathcal{W}: =\frac{1}{4} \int|\vec H|^2$. In codimension one, we prove that the $\mathcal{W}$-Morse index for any inverted minimal sphere or real projective plane with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4π}-3$. We also consider several geometric properties -- for example, the property that all $m$ asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the $\mathcal{W}$-Morse index of their inverted surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_14367 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions Hirsch, Jonas Kusner, Rob Mäder-Baumdicker, Elena Differential Geometry Analysis of PDEs We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $\mathcal{W}: =\frac{1}{4} \int|\vec H|^2$. In codimension one, we prove that the $\mathcal{W}$-Morse index for any inverted minimal sphere or real projective plane with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4π}-3$. We also consider several geometric properties -- for example, the property that all $m$ asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the $\mathcal{W}$-Morse index of their inverted surfaces. |
| title | Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2110.14367 |