Here are the simplest doubly periodic surfaces with just catenoidal ends in the quotient.

The catenoidal ends are arranged in a square lattice, and their limiting normals are perpendicular to the plane of periodicity. Because the flux is vertical, the López-Ros factor can be used to change the growth rate.

As with the translation  invariant catenoid, one can ask in what pattern the catenoids can be arranged. There likely  will be  simple answers.

Resources

Mathematica Notebook

PoVRay Sources