Separable surfaces that are critical points of the Dirichlet energy
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918499472900096 |
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| author | López, Rafael |
| author_facet | López, Rafael |
| contents | In this paper, we study surfaces $z=φ(x,y)$ in Euclidean space that satisfy the equation $φ_{xx}+φ_{yy}=\fracΛ{2}$ where $Λ\in\r$ is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth functions of one variable. If $Λ=0$, we find a large family of surfaces with interesting symmetry properties. However, if $Λ\not=0$, we show that the surfaces must be either surfaces of revolution or of the type $z=f(x)+g(y)$; furthermore, explicit parametrizations of these surfaces are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13033 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Separable surfaces that are critical points of the Dirichlet energy López, Rafael Differential Geometry Primary 53A10, Secondary 53C42 In this paper, we study surfaces $z=φ(x,y)$ in Euclidean space that satisfy the equation $φ_{xx}+φ_{yy}=\fracΛ{2}$ where $Λ\in\r$ is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth functions of one variable. If $Λ=0$, we find a large family of surfaces with interesting symmetry properties. However, if $Λ\not=0$, we show that the surfaces must be either surfaces of revolution or of the type $z=f(x)+g(y)$; furthermore, explicit parametrizations of these surfaces are obtained. |
| title | Separable surfaces that are critical points of the Dirichlet energy |
| topic | Differential Geometry Primary 53A10, Secondary 53C42 |
| url | https://arxiv.org/abs/2605.13033 |