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Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$

Jun Matsumoto

TL;DR

The paper advances Osserman's problem by examining the total ramification weight $\nu_g$ of the Gauss map for complete minimal surfaces of finite total curvature. It introduces a constructive framework to produce meromorphic functions on punctured spheres with $\nu_g>2$, and applies order-weight estimates to constrain possible Gauss maps. Within the topology of the known $3$- and $4$-punctured examples, it proves the Miyaoka--Sato case is unique for the $3$-punctured sphere and fully classifies the $4$-punctured case with $D_g=2$, revealing new examples beyond Kawakami--Watanabe, while also producing a new $4$-punctured example with $D_g=1$ and $\nu_g=2.5$. The results provide a systematic method for constructing complete minimal surfaces with finite total curvature and $\nu_g>2$, and open new directions for understanding the limits of $\nu_g$ and the structure of the Gauss map in this setting.

Abstract

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 $ν_g = 2.5$.

Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$

TL;DR

The paper advances Osserman's problem by examining the total ramification weight of the Gauss map for complete minimal surfaces of finite total curvature. It introduces a constructive framework to produce meromorphic functions on punctured spheres with , and applies order-weight estimates to constrain possible Gauss maps. Within the topology of the known - and -punctured examples, it proves the Miyaoka--Sato case is unique for the -punctured sphere and fully classifies the -punctured case with , revealing new examples beyond Kawakami--Watanabe, while also producing a new -punctured example with and . The results provide a systematic method for constructing complete minimal surfaces with finite total curvature and , and open new directions for understanding the limits of and the structure of the Gauss map in this setting.

Abstract

Motivated by Osserman's problem on the number of omitted values of the Gauss map of a complete minimal surface with finite total curvature in , its totally ramified value number (referred to in this paper as the \emph{total weight of totally ramified values}) has attracted significant interest. The value of 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 and . Subsequently, in 2024, Kawakami and Watanabe gave another minimal surface defined on the four-punctured Riemann sphere that also satisfies and . To date, these remain the only two known examples of such surfaces satisfying . In this paper, we provide a systematic construction of meromorphic functions on punctured Riemann spheres that satisfy . As a consequence, we obtain the following results for complete minimal surfaces of finite total curvature with 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 and , which include examples other than Kawakami--Watanabe's one. (3) Furthermore, we construct a new example on the four-punctured sphere satisfying and .
Paper Structure (8 sections, 16 theorems, 126 equations, 2 figures)

This paper contains 8 sections, 16 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complete minimal immersions of finite total curvature into R3 and their Gauss maps
  4. Osserman's problem and its relationship with the total weight of totally ramified values of the Gauss map
  5. Some Estimates for orders and total weight of totally ramified values of meromorphic functions on punctured Riemann surfaces
  6. Complete minimal surfaces on four-punctured Riemann spheres with $C(\varSigma) = -16\pi$ and $\nu_g = 2.5$
  7. The case where $D_g = 2$
  8. The case where $D_g \leq 1$

Key Result

Proposition 3.1

Let $\overline{\varSigma}$ be a compact Riemann surface, and $p_1, \dots, p_n \in \overline{\varSigma}\ (n \in \boldsymbol{Z}_{\geq 0})$. Let $g : \overline{\varSigma} \setminus \{p_1, \dots, p_n\} \to \overline \boldsymbol{C}$ be a meromorphic function. Assume that the function $g$ extends meromorp where $e_p$ is the multiplicity of $g$ at $p$, $\nu(b_j)$ is the order of a totally ramified value

Figures (2)

  • Figure 1: Complete minimal surface with $C(\varSigma) = - 16 \pi, D_g = 2$ and $\nu_g = 2.5$ in Case 1 for each $\omega$ of (1), (2), and (5).
  • Figure 2: Complete minimal surface with $C(\varSigma) = - 16 \pi, D_g = 1, R_g = 2$ and $\nu_g = 2.5$

Theorems & Definitions (32)

  • Definition 1.1: cf. Nev70, Kaw06
  • Remark 1.2
  • Example 2.5: MS94, Kaw06
  • Example 2.7: KW24
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['thm:estimate_order']}
  • ...and 22 more