A uniqueness theorem for the singly periodic genus-one helicoid
  The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic genus-one helicoid provided the existence of one symmetry. 
A. Alarcon, L. Ferrer and F. Martin