Free boundary minimal surfaces of genus one

Two parallel discs which are very close to the equatorial disc can be connected by a sufficiently large number of half-necks along the equator and by one catenoidal neck in the center and then deformed to a free boundary minimal surface in the unit ball which has genus one and a large number of boundary components. However, it remains an open question whether similar surfaces with a small number of boundary components exist, as conjecturally visualised below.

The extended family of free boundary minimal disc doublings is also conjectured to contain less symmetric examples with genus one which only have two planes of symmetry.