By Richard Schoen’s theorem, it is not possible to make a genus one catenoid, but one can make a genus one surface with Enneper ends. In the most symmetric case, one still has a 1-parameter family.

It might be amusing to study the limits for higher genus analogues. I also wonder what happens to the handle when one rotates the two Enneper ends as in the twisted version of the Double Enneper surface.

Resources

Mathematica Notebook and CDF

PoVRay Sources

Related Surfaces