Parameters of Weierstrass Elliptic Function

Question

I am currently studying the Chen-Gackstatter surface, and in the link it uses Enneper-Weierstrass parameterization of the surface.

A function called Weierstrass elliptic function is used to define the parametrization, and I have seen the wiki page of the Weierstrass elliptic function, in which "periods" is used to define the function.

However, in the link above, two "parameters" were used to define the function, and I am wondering if there is any relationship between the "parameters" and the "periods".

Answer 1

There is a direct relationship between the parameters $\, g_2,\, g_3 \,$ and the periods as given in the Wikipedia article Weierstrass elliptic functions. In the particular case you linked to, the fundamental pair of periods is $\, 1,\,i \,$ and this is known as the lemniscate case in which case the points in the period lattice are exactly all of the Gaussian integers.

Answer 2

To make Somos's comments more explicit: it is not very hard to use the classical results of the "lemniscatic case" of the Weierstrass $\wp$ function to derive the required invariants $g_2,g_3$ (the proper term for what OP termed the "parameters", as they enter in the defining cubic $4u^3-g_2 u -g_3$). I gave a derivation of the parametric equations for the Chen-Gackstatter surface in this blog entry, where I started from the Enneper-Weierstrass parametrization.

As an executive summary: if you have the half-periods $\omega=1$ and $\omega^\prime=i$, then you have the corresponding invariants $g_2=\dfrac1{16\pi^2}\Gamma\left(\dfrac14\right)^8,\; g_3=0$.

Here is an plot made in Mathematica using the formulae I derived:

Similar considerations apply for the more famous Costa minimal surface, which also uses these Weierstrass invariants.

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As an additional note: because of the structure of the problem, one can in theory use the Jacobi elliptic functions instead (supplemented by the Jacobi $\varepsilon$ function, which replaces the Weierstrass $\zeta$ function) to express the parametric equations of the Chen-Gackstatter surface, but they are longer and a little more unwieldy.