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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2602.07551 |
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| _version_ | 1866908819344326656 |
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| author | Matsumoto, Jun |
| author_facet | Matsumoto, Jun |
| contents | Motivated by Osserman's problem on the number $D_g$ of omitted values
of the Gauss map of a complete minimal surface with finite total curvature in $\boldsymbol{R}^3$,
its totally ramified value number $ν_g$ (referred to in this paper as the
\emph{total weight of totally ramified values})
has attracted significant interest.
The value of $ν_g$ provides more detailed information than the number of omitted values alone.
In 2006, Kawakami first found that a minimal surface defined on the three-punctured Riemann sphere,
originally constructed by Miyaoka and Sato, satisfies $D_g = 2$ and $ν_g = 2.5 > 2$.
Subsequently, in 2024, Kawakami and Watanabe gave another minimal surface
defined on the four-punctured Riemann sphere that also satisfies $D_g = 2$ and $ν_g = 2.5$.
To date, these remain the only two known examples of such surfaces satisfying $ν_g > 2$.
In this paper, we provide a systematic construction of meromorphic functions
on punctured Riemann spheres that satisfy $ν_g > 2$.
As a consequence, we obtain the following results for complete minimal surfaces of finite total curvature with $ν_g = 2.5$ within the topological types of the known examples: (1) For the three-punctured sphere, we prove the uniqueness of Miyaoka--Sato's example. (2) For the four-punctured sphere, we completely determine the surfaces with $D_g=2$ and $ν_g = 2.5$, which include examples other than Kawakami--Watanabe's one. (3) Furthermore, we construct a new example on the four-punctured sphere satisfying $D_g=1$ and $ν_g = 2.5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07551 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$ Matsumoto, Jun Differential Geometry Complex Variables Motivated by Osserman's problem on the number $D_g$ of omitted values of the Gauss map of a complete minimal surface with finite total curvature in $\boldsymbol{R}^3$, its totally ramified value number $ν_g$ (referred to in this paper as the \emph{total weight of totally ramified values}) has attracted significant interest. The value of $ν_g$ provides more detailed information than the number of omitted values alone. In 2006, Kawakami first found that a minimal surface defined on the three-punctured Riemann sphere, originally constructed by Miyaoka and Sato, satisfies $D_g = 2$ and $ν_g = 2.5 > 2$. Subsequently, in 2024, Kawakami and Watanabe gave another minimal surface defined on the four-punctured Riemann sphere that also satisfies $D_g = 2$ and $ν_g = 2.5$. To date, these remain the only two known examples of such surfaces satisfying $ν_g > 2$. In this paper, we provide a systematic construction of meromorphic functions on punctured Riemann spheres that satisfy $ν_g > 2$. As a consequence, we obtain the following results for complete minimal surfaces of finite total curvature with $ν_g = 2.5$ within the topological types of the known examples: (1) For the three-punctured sphere, we prove the uniqueness of Miyaoka--Sato's example. (2) For the four-punctured sphere, we completely determine the surfaces with $D_g=2$ and $ν_g = 2.5$, which include examples other than Kawakami--Watanabe's one. (3) Furthermore, we construct a new example on the four-punctured sphere satisfying $D_g=1$ and $ν_g = 2.5$. |
| title | Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$ |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2602.07551 |