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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.26258 |
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| _version_ | 1866916046707884032 |
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| author | Katsuta, Shuhei |
| author_facet | Katsuta, Shuhei |
| contents | H. Fujimoto showed that for a complete minimal surface in $\mathbb{R}^m$, if the Gauss map is non-degenerate, then it omits at most $\frac{m(m + 1)}{2}$ hyperplanes in the complex projective space $\mathbb{P}^{m - 1}$ in general position, and that the number $\frac{m(m + 1)}{2}$ is best possible for all odd integers $m \geq 3$ and for even integers with $4 \leq m \leq 16$. In this paper, we prove that the number $\frac{m(m + 1)}{2}$ is also best possible for all even integers $m \geq 4$, as conjectured by Fujimoto. The main tool is a special planar network $(Γ_0, ω)$ in the theory of positive matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26258 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Fujimoto Conjecture via Total Positivity Katsuta, Shuhei Differential Geometry Combinatorics 53A10, 30D35, 15B48 H. Fujimoto showed that for a complete minimal surface in $\mathbb{R}^m$, if the Gauss map is non-degenerate, then it omits at most $\frac{m(m + 1)}{2}$ hyperplanes in the complex projective space $\mathbb{P}^{m - 1}$ in general position, and that the number $\frac{m(m + 1)}{2}$ is best possible for all odd integers $m \geq 3$ and for even integers with $4 \leq m \leq 16$. In this paper, we prove that the number $\frac{m(m + 1)}{2}$ is also best possible for all even integers $m \geq 4$, as conjectured by Fujimoto. The main tool is a special planar network $(Γ_0, ω)$ in the theory of positive matrices. |
| title | The Fujimoto Conjecture via Total Positivity |
| topic | Differential Geometry Combinatorics 53A10, 30D35, 15B48 |
| url | https://arxiv.org/abs/2605.26258 |