Is there a "distinguished" parametrization of a surface in $\mathbb{R}^3$ Let $\Gamma \subset \mathbb{R}^n$ be a smooth surface diffeomorphic to $\mathbb{R}^{n-1}$. Does there exist a smooth parametrization, $f:\mathbb{R}^{n-1} \to \Gamma$, $f$ is a diffeomorphism, such that f is "canonical" or "distinguished" is some sense?
For $n=2$, I know of at least two such distinguished parametrizations. One is given by the arc length parametrization (modulo translations). The other one is given by the boundary value of the Riemann map from the upper half plane to the region "above" $\Gamma$ (modulo Mobius transformations). 
Clearly the obvious generalization of the arc length parametrization to higher dimensions namely a parametrization which is an isometry does not exist (due to curvature). But is there a generalization coming from Riemann map? (even though there is no Riemann Map in higher dimensions). I am interested in any sort of parametrization which always exist for a smooth surface and is unique (upto small group of diffeomorphism). If there are no such results for general dimensions, is there a result of this form for $n=3$?
 A: In the case when your surface is 2-dimensional you can use the the uniformization theorem. The caveat, however, is that the domain of your map will be, most of the time, the open unit disk $B\subset R^2$ rather $R^2$. Here is relevant the statement:
Suppose that $S$ is a smooth surface in $R^m$ diffeomorphic to $R^2$ equipped with the induced Riemannian metric. Then one of the following holds:


*

*There exists a conformal or anticonformal diffeomorphism  $f: R^2\to S$. It is unique up to precomposition with the group of Euclidean similarities. 


2.There exists a conformal or anticonformal diffeomorphism  $f: B\to S$. It is unique up to precomposition with the group of isometries of the Poincare metric on $B$. 
In the case $m=3$, you can do a bit better since $S$ has the canonical orientation induced from $R^3$. Then $f$ can be taken to be conformal and, hence, unique up to precomposition with conformal automorphisms of its domain. 
When the domain has dimension $>2$, I am sure that there is no canonical parameterization in any sense. 
