In 2011, Kelly Roberta Mazzutti Lübeck and Valério Ramos Batista proved the existence of a 1-parameter family of new doubly periodic minimal surfaces.

In the quotient, it has four annular ends and genus 3. The layers are connected by Costa saddles, maintaining the straight lines of the Costa surface. In addition, the surface has straight lines between the Costa saddles, and reflectional symmetry planes separating the layers. One of the limits of the surface is the Callahan-Hoffman-Meeks surface, so one can view the surface as a doubly periodic version of that surface:

In the other limit, the surface likely decomposes into a KMR-surface and two doubly periodic Scherk surfaces.

The latter statement is unproven. It would be interesting to establish a general gluing construction for Scherk and KMR surfaces.

Resources

K.R.M. Lübeck & V. Batista: The Doubly Periodic Scherk-Costa Surfaces, Journal of Mathematics Research (6) 2014, 77-90.

Mathematica Notebook

PoVRay Sources