The prototype for this symmetrization is the Enneper surface with two catenoids attached. To make it symmetric and avoid distortions of the Enneper surface, we choose the limiting normals of the catenoidal ends perpendicular to the limiting normal of the Enneper end.

There are (at least) two versions of this, depending which way the catenoids point. I call these the extrovert and introvert versions.

These surfaces come in 1-parameter families, changing the growth rates of catenoids for taste. Because they are all spheres, the Enneper-Weierstrass representation is by rational functions:

G(z)=\rho  z^{n-1}\frac{ z^n-b^n}{z^n-a^n}\qquad\text{and}\qquad dh =\frac {z^{n - 1} \left (z^n - a^n \right) \left (z^n - b^n \right)} {\left (z^n - 1 \right)^2}

Here, 𝝆 and b need to be adjusted to make the punctures at the roots of unity into horizontal catenoids.

Besides their esthetic appeal, there is a question attached to relatively complex surfaces like these: What are the simplest minimal surfaces who have a connected moduli space with non-trivial topology?


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