For Costa type surfaces (i.e. surfaces of finite total curvature with catenoidal and planar ends), adding more handles without increasing the dihedral symmetry is a delicate affair. For Costa’s surface attempt, adding such a handle leads the Horgan surface, which does not exist. For Wohlgemuth’s surface, adding a single handle is possible, but then increasing the dihedral symmetry to k=3 fails.
Above is the Weber-Wolf surface with one added handle. This surface also exists with 3-fold dihedral symmetry.
However, closing the periods for 4-fold symmetry and larger mysteriously appears to fail.
Resources
Mathematica Notebook (by Ramazan Yol for higher dihedral symmetry)
Related Surfaces
- Costa Surface and Horgan surface
- Wohlgemuth’s First and Second Surface
- Weber-Wolf Surface
Minimal Surfaces 