There are straightforward generalizations of Fusheng Wei’s doubly periodic surfaces that adds more handles in symmetric positions:

Above surface is of type (1,3), meaning 1 catenoidal neck pointing towards and 3 away from you in a translational fundamental piece. This description is suitable to construct the surfaces using their nodal limit and Traizet’s regeneration method, as carried out by Peter Connor and Matthias Weber. This construction produces 3-dimensional families of surfaces.

While keeping the symmetries, the surface deforms towards a limit that decomposes into two Karcher-Scherk surfaces of genus one.

Adding more handles increases the difficulty of the period problem, but Traizet’s method yields solutions for all types (1,n) of genus n.

There are sporadic further solutions, the simplest one a genus 4 surface of type (2,3), shown below in comparison with the (1,4) surface, also of genus 4.

All these surfaces appear to deform into a second limit, consisting of Karcher-Scherk surfaces of higher genus. This is not understood at all. Besides the (1,n) case, no general pattern for the positions of catenoidal necks in the Traizet limit have been found.

##### Resources

Mathematica Notebooks (1,3), (1,4), (1,6), (2,3)

PoVRay Sources (1,3), (1,4), (1,6), (2,3)

P. Connor, M. Weber: The construction of doubly periodic minimal surfaces via balance equations