I am recently studing minimal surfaces on my own. I have meet in many places the fallowing statement:
The only connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ are a plane, a helicoid, a catenoid or one of the Riemann minimal examples.
for example in arxiv. But without definition of planar domain in $\mathbb{R}^3$, may someone familiar with the topic give the definition or reference to one.
On page 3 of the pdf you linked to: