Electrostatic skeletons and condition of strict descent
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910103189323776 |
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| author | Huang, Linhang |
| author_facet | Huang, Linhang |
| contents | Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Electrostatic skeletons and condition of strict descent Huang, Linhang Complex Variables Mathematical Physics Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons. |
| title | Electrostatic skeletons and condition of strict descent |
| topic | Complex Variables Mathematical Physics |
| url | https://arxiv.org/abs/2604.03861 |