In 2014, Christine Breiner and Stephen Kleene constructed minimal surfaces that are invariant under a screw motion followed by a homothety. These surfaces follow a logarithmically spiraling helix with and are embedded in an invariant tube around that helix and look like bent helicoids.
This raises the question what other translation invariant surfaces can be bent as to follow a given curve. For instance, for which embedded space curves can one minimally desingularize a pair of strips intersecting orthogonally in that curve, using bent singly periodic Scherk surfaces?
C. Breiner, S. Kleene: Logarithmically Spiraling Helicoids