This is well known that every point of a surface admits a neighbourhood with conformal coordinates (i.e. such that the metric is written $ds^2= e^u (dx^2 + dy^2)$). My first question is can we control the diffeomorphism which pass from original coordinates to conformal coordinates? In other word, let $u : B(0,1) \rightarrow R^3$ an immersion, then there exists a diffeomorphism $\phi$ of neighbourhood of $0$ such that $u\circ \phi$ is conformal. Can we control $\Vert \phi - Id\Vert_{C^1}$ w.r.t. $\Vert \vert u_x\vert^2 -\vert u_y \vert^2 +i \langle u_x, u_y\rangle \Vert_\infty$.

And what about the global case, i.e. considering maps from $S^2$ to $R^3$. Does there exists a proof of the existence of thiese coordinates which doesn't use the uniformization theorem but which consists in "gluing" all te local diffeomorphisms.

isothermal. Isconformalthe accepted terminology now? (A more appropriate name might beconformally flatcoordinates, by analogy withflatcoordinates: those relative to which the connection one-form is identically zero.) – José Figueroa-O'Farrill Oct 3 '10 at 15:32mytime it was stilled called isothermal. But I disagree with the usage of (conformally) flat to refer to coordinates: I prefer flatness to be a condition on the intrinsic Riemannian structure, not an artefact of the coordinate system. – Willie Wong Oct 3 '10 at 17:53