In the theory of minimal surfaces, we could construct various types of minimal surface based on the way we choose a pair of holomorphic & meromorphic fucntion. To be more specific, let $f(\zeta)$ be a holomorphic function on an open set $U$ in the complex plane, not identically zero while $g(\zeta)$ be a meromorphic function on the same domain such that if $\zeta_0$ is a pole of order $m \geq 1$ of $g$ then it is a zero of order $\geq 2m$ of $f$ then we have a parametrization of a minimal surface $$\sigma(u,v) = \mathfrak{Re} \int \varphi(\zeta)d\zeta, \zeta = u + iv$$ in which $\sigma$ is a surface patch and $\varphi = \frac{1}{2}(f(1-g^2),if(1+g^2),2fg)$. For example when $(f,g) = (2,\zeta)$ we have the classical Enneper surface $$\sigma(u,v) = (u - \frac{u^3}{3} + uv^2, -v + \frac{v^3}{3} - u^2v, u^2 - v^2)$$ In this wikipedia, I know that the classical Enneper surface is algebraic which means that it can be defined as the zero locus of a polynomial of degree $9$. Naturally, I can generalize this example by choosing $(f,g)=(2,\zeta^n)$ which gives us $$\sigma(u,v) = (x(u,v),y(u,v),z(u,v)$$ in which $$x(u,v) = u - \frac{1}{2n+1}\sum_{k=0}^n \binom{2n+1}{2k+1}u^{2k+1}(iv)^{2(n-k)}$$ $$y(u,v) = -v - \frac{1}{2n+1}\sum_{k=0}^{n}\binom{2n+1}{2k}u^{2k}i^{2(n+1)-2k}v^{2n+1-2k}$$ $$z(u,v) = 2\sum_{2 \mid k-n-1}\binom{n+1}{k}u^k i^{n+1-k}v^{n+1-k} \ (0 \leq k \leq n+1)$$ and hope these general Enneper surfaces are algebraic. A quick search convinces me that this class of Enneper surfaces are algebraic so I ask for books providing a criterion for determining whether a given minimal surface is algebraic or not, at least in case the pair $(f,g)$ are both polynomials in $\zeta$. Moreover, if it is algebraic then is there an algorithm to derive its defining polynomial-equation?
Other questions, let $\mathcal{S}$ be a algebraic minimal surface (obtained by a pair $(f,g)$ as above) we define $\mathrm{deg}(\mathcal{S})$ to be the least degree of a polynomial (in three variables) defining $\mathcal{S}$.
- Is there a lower bound for $\mathrm{deg}$?
- For a given $n \in \mathbb{N}$, how can we know whether there exists a surface $\mathcal{S}$ with $\mathrm{deg}(\mathcal{S})=n$?
you can see the following 2 papers: https://www.mdpi.com/2227-7390/6/12/281 Güler E. Family of Enneper Minimal Surfaces. Mathematics. 2018; 6(12):281. https://doi.org/10.3390/math6120281 and https://www.mdpi.com/2075-1680/11/1/4 Güler E. The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space. Axioms. 2022; 11(1):4. https://doi.org/10.3390/axioms11010004 Best, Erhan Güler