Conformal coordinates on surface

Question

Suppose $\Omega$ is a domain in $R^2$, let $F: \Omega\rightarrow R^3$ be a smooth map such that $F(\Omega)$ is smooth surface. The metric on surface is $$g=F_x\cdot F_x dx^2+2F_x\cdot F_y dxdy+F_y\cdot F_y dy^2$$ Do we have another coordinate system such that the metric is $g=\rho(u,v)(du^2+dv^2)$? I have found something about isothermal coordinates online. From my understanding, those results are local. Can we find it in whole $\Omega$? I am not sure if the topology (simply-connected, etc) affects the result.

Answer 1

Let's assume that $\partial\Omega$ is smooth. Then as $\Omega\subset \mathbb{R}^2$, we know that after 'filling in the holes' we obtain a genus $0$ surface and this answer applies and guarantees the existence of a harmonic map $u:\Omega\rightarrow\mathbb{R}$ without critical points. Then $*du$ is a closed one-form (where $*$ is the Hodge star for the pull-back metric $g =F^*\langle\cdot,\cdot\rangle_{\mathbb{R}^3}$).

If $*du$ happens to be exact, then we have $*du=dv$ for a smooth function $v:\Omega\rightarrow \mathbb{R}$. Exactness of $*du$ follows for example if $\Omega$ is simply connected (or more generally if $H^1(\Omega,\mathbb{R})=0$). Then $(u,v):(\Omega,g)\rightarrow \mathbb{R}^2$ will give you global isothermal coordinates, which is exactly what you want.