On a sharp form of curvature conjecture for minimal graphs
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912551294468096 |
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| author | Kalaj, David |
| author_facet | Kalaj, David |
| contents | Recently, the author and Melentijević resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{π^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant \( c_0 \) is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant \( c_1 \). In addition, we show that for any prescribed unit normal vector \( \mathbf{n} \), there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is \( (0,0,1) \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a sharp form of curvature conjecture for minimal graphs Kalaj, David Differential Geometry Complex Variables Recently, the author and Melentijević resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{π^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant \( c_0 \) is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant \( c_1 \). In addition, we show that for any prescribed unit normal vector \( \mathbf{n} \), there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is \( (0,0,1) \). |
| title | On a sharp form of curvature conjecture for minimal graphs |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2508.16928 |