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

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 $ν_g = 2.5$.

💡 Research Summary

The paper addresses a refined version of Osserman’s classical problem concerning the number of omitted values D₍g₎ of the Gauss map g of a complete minimal surface in ℝ³ with finite total curvature. While Osserman proved D₍g₎ ≤ 3 (and conjectured that D₍g₎ = 3 may never occur), later work introduced the “total weight of totally ramified values” ν₍g₎, which captures not only omitted values but also the multiplicities of values that are totally ramified. Historically, only two examples were known with ν₍g₎ > 2: a three‑punctured sphere (the Miyaoka–Sato surface) and a four‑punctured sphere (the Kawakami–Watanabe surface), both having ν₍g₎ = 2.5.

The author’s main contribution is a systematic method for constructing meromorphic functions on punctured Riemann spheres that achieve ν₍g₎ > 2, and a complete classification of all such minimal surfaces within the topological types of the known examples. The paper proceeds as follows:

1. Preliminaries (Section 2). The Weierstrass representation is recalled, linking a minimal immersion f to a pair (g, ω) where g is the complex Gauss map and ω a holomorphic 1‑form. The total curvature satisfies C(Σ) = −4π deg(g). Known bounds D₍g₎ ≤ ν₍g₎ ≤ 2 + 2R < 4 (with R depending on genus and degree) are presented, together with the sharper inequality D₍g₎ ≤ ν₍g₎ < 3 when the underlying compact surface is the sphere (genus 0).

2. Value‑distribution estimates (Section 3). Proposition 3.1 gives a universal relation between the degree d = deg(g), the number of punctures n, and the sum of ramification orders of totally ramified values. From this, Corollaries 3.5 and 3.6 derive explicit upper bounds for ν₍g₎ in terms of d, n, and the branching data. In particular, when d = 2 and n = 3 (the three‑punctured sphere case), the bound forces ν₍g₎ ≤ 2.5, and equality can only be achieved by a unique meromorphic function, proving the uniqueness of the Miyaoka–Sato surface (Corollary 3.7).

3. Classification for four punctures (Section 4). For n = 4 and d = 4, the author enumerates all possible ramification patterns that yield ν₍g₎ = 2.5. By solving the associated algebraic constraints on the coefficients of g, a whole family of meromorphic functions is obtained. The corresponding Weierstrass data produce a complete list of minimal immersions with D₍g₎ = 2, ν₍g₎ = 2.5, and total curvature −16π. This family contains the Kawakami–Watanabe example as a special case but also many previously unknown surfaces. Moreover, by allowing one of the omitted values to disappear (i.e., D₍g₎ = 1), the author constructs a new surface on the four‑punctured sphere with ν₍g₎ = 2.5, demonstrating that the condition D₍g₎ = 2 is not necessary for ν₍g₎ > 2.

4. Open problems (Section 4.10). The paper concludes with several questions: (a) Does any complete minimal surface with finite total curvature satisfy ν₍g₎ > 2.5? (b) What are the possible ν₍g₎ values for higher genus (γ ≥ 1) surfaces? (c) Can the method be extended to construct examples with prescribed ν₍g₎ in the interval (2, 3)?

Overall, the work achieves three major results: (i) it proves the uniqueness of the three‑punctured sphere example, (ii) it completely determines all four‑punctured sphere examples with ν₍g₎ = 2.5, and (iii) it introduces a new four‑punctured sphere example with a single omitted value. The systematic construction based on ramification data provides a powerful tool for future investigations into Osserman’s problem and the finer invariants of minimal surfaces.