In their 2000 paper, Wayne Rossman, Edward Thayer and Meinhard Wohlgemuth began a systematic study of embedded doubly periodic minimal surfaces with parallel ends of higher genus, adding handles to the KMR surfaces. Their simplest discovery is a surface of genus 2 they dub M1+, different from Fusheng Wei’s genus 2 surface (M1- in their notation).
Like Wei’s surface, M1+ has reflectional symmetries at planes parallel to the coordinate planes and it comes in a 1-parameter family.
One extreme case suggests an end-to-end gluing of doubly periodic Scherk surfaces using catenoidal necks to attach the ends, the other an end to end gluing of 8-ended singly periodic Scherk surfaces. Neither of these have been investigated in the context of doubly periodic surfaces. The second limit on the right suggests a close connection the Alan Schoen’s triply periodic I-WP surface.
Another open question is whether M1+ can be deformed into M1- through embedded, doubly periodic minimal surfaces.
W. Rossman, E.C. Thayer, M. Wohlgemuth: Embedded, Doubly Periodic Minimal Surfaces, Experimental Mathematics, Vol. 9 (2000), 197-219