Embedded and Non-Parametric Surface definition What does it mean for a minimal surface to be embedded? For example the Scherk surfaces?
How would I define what 'an embedded surface' is? And also what does it mean for a surface to be 'non-parametric'?
I am a third year maths undergrad, but I'm not familiar with topology per say, though I've taken a module on the geometry of surfaces
Thank you
 A: Note:  If you mention the level that you would like this question answered at (i.e. undergraduate differential geometry, beginning graduate level differential geometry, etc.) then the following response can be adjusted accordingly.
A surface $S$ is said to be embedded in $\mathbb{R}^3$ if there is a one-to-one immersion $f : S \to \mathbb{R}^3R$ such that $S$ is homeomorhpic to the image $f(S)$ (where $f(S)$ has the subspace topology from $\mathbb{R}^3$). Note that an immersion simply requires the push-forward (or differential) $f_{*} : T_{p}S \to T_{f(p)}\mathbb{R}^3$ be one-to-one for all $p \in S$.  The global requirement that the mapping $f$ be one-to-one prevents self-intersections.
As far as `non-parametric' surfaces are concerned, my guess is that what is intended is for one to view a surface $S$ more abstractly (i.e. without an embedding).  The two-sphere $S^2$ is a surface in its own right and does not need to be embedded in $\mathbb{R}^{3}$ to be realized as a surface (although this is what we traditionally do).  Further, the two-sphere can't be covered with a single parametrization or coordinate chart, which helps establish the idea that abstract surfaces are more than just simply "parametrized surfaces" in $\mathbb{R}^3$. 
