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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.

J.E.M.S
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On page 3 of the pdf you linked to:

enter image description here

Zev Chonoles
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  • Many thanks even thought i used "find" function i missed that :P, for surness embeding into plane is equivalent to having global parametrization with domain being full $\mathbb{R}^2$, right? – J.E.M.S May 22 '15 at 17:12