These surfaces can be interpreted as symmetrizations of the Double Enneper surface, or as a variation of the Jorge-Meeks k-Noids with the catenoidal ends being replaced by Enneper ends.

These surface probably don’t appear in the literature. They are not embedded, have high degree of the Gauß map, and their construction doesn’t pose any interesting challenges. The Enneper-Weierstraß representation is a straightforward modification of the one for the Jorge Meeks k-Noids:

G(z)= z^{k-1}\frac{z^k-r^k}{1-r^k z^k}\qquad\text{and}\qquad dh =\frac{1-\frac{z^{-k}+z^k}{r^{-k}+r^k}}{z \left(z^{-k}+z^k-2\right)^2}\, dz

There even is an additional parameter r that squeezes the surface. I have, unfortunately, a weakness for the baroque ornaments these Enneper ends provide.


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