Two classification results for stationary surfaces of the least moment of inertia
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909811995574272 |
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| author | López, Rafael |
| author_facet | López, Rafael |
| contents | A surface in Euclidean space $\r^3$ is said to be an $α$-stationary surface if it is a critical point of the energy $\int_Σ|p|^α$, where $α\in\r$. We prove that all ruled $α$-stationary surfaces are vector planes (for all $α$) and a type of elongated helicoids (for $α=1$). The second result of classification asserts that if $α\not=-2,-4$, any $α$-stationary surface foliated by circles must be a rotational surface. If $α=-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $α=-2$, we find many non-spherical cyclic $(-2)$-stationary surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two classification results for stationary surfaces of the least moment of inertia López, Rafael Differential Geometry 53A10, Secondary: 53C42 A surface in Euclidean space $\r^3$ is said to be an $α$-stationary surface if it is a critical point of the energy $\int_Σ|p|^α$, where $α\in\r$. We prove that all ruled $α$-stationary surfaces are vector planes (for all $α$) and a type of elongated helicoids (for $α=1$). The second result of classification asserts that if $α\not=-2,-4$, any $α$-stationary surface foliated by circles must be a rotational surface. If $α=-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $α=-2$, we find many non-spherical cyclic $(-2)$-stationary surfaces. |
| title | Two classification results for stationary surfaces of the least moment of inertia |
| topic | Differential Geometry 53A10, Secondary: 53C42 |
| url | https://arxiv.org/abs/2507.12398 |