Minimal surfaces over the Pitot quadrilaterals
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918225291247616 |
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| author | Dragović, Vladimir Kalaj, David |
| author_facet | Dragović, Vladimir Kalaj, David |
| contents | We develop a fully explicit framework for constructing Scherk-type minimal graphs over the Pitot quadrilaterals (i.e. such that the two pairs of opposite sides have the same total length). For any Pitot quadrilateral \(Q\), we first produce a harmonic diffeomorphism of the unit disk onto \(Q\), whose dilatation is the square of a Möbius automorphism determined directly by the vertices of \(Q\). Using this map as the Weierstrass data, we obtain a minimal graph \(Σ\) whose Gauss map is a univalent Möbius transformation and whose height function exhibits alternating blow-up behavior along opposite sides of \(Q\), mirroring the classical Scherk surfaces. We further construct an associated canonical surface \(Σ^\diamond\), with the same boundary asymptotics, and prove a sharp curvature comparison theorem: at the harmonic center of \(Q\), among all bounded minimal graphs with matching normal direction and mixed derivative, \(Σ^\diamond\) uniquely maximizes the absolute Gaussian curvature. This provides a complete and constructive description of Scherk-type minimal graphs over all, both convex or concave, Pitot quadrilaterals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal surfaces over the Pitot quadrilaterals Dragović, Vladimir Kalaj, David Differential Geometry Complex Variables We develop a fully explicit framework for constructing Scherk-type minimal graphs over the Pitot quadrilaterals (i.e. such that the two pairs of opposite sides have the same total length). For any Pitot quadrilateral \(Q\), we first produce a harmonic diffeomorphism of the unit disk onto \(Q\), whose dilatation is the square of a Möbius automorphism determined directly by the vertices of \(Q\). Using this map as the Weierstrass data, we obtain a minimal graph \(Σ\) whose Gauss map is a univalent Möbius transformation and whose height function exhibits alternating blow-up behavior along opposite sides of \(Q\), mirroring the classical Scherk surfaces. We further construct an associated canonical surface \(Σ^\diamond\), with the same boundary asymptotics, and prove a sharp curvature comparison theorem: at the harmonic center of \(Q\), among all bounded minimal graphs with matching normal direction and mixed derivative, \(Σ^\diamond\) uniquely maximizes the absolute Gaussian curvature. This provides a complete and constructive description of Scherk-type minimal graphs over all, both convex or concave, Pitot quadrilaterals. |
| title | Minimal surfaces over the Pitot quadrilaterals |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2512.01029 |