Let $D:=\{x\in\mathbb{R}^2:\ |x|\leq 1\}$ and $f:D\rightarrow \mathbb{R}^3$ a smooth (analytic if it helps) Immersion with boundary, such that the boundary curve $c\subset\mathbb{R}^3$ is smooth and regular and $f|_{\partial D}\rightarrow c$ is a diffeomorphism.
The Korn-Lichtenstein theorem provides us with local isotherm coordinates, hence $f(\dot{D})$ is a Riemann surface. The uniformization theorem now tells us, that $f(\dot{D})$ is biholormophic to $\Omega$ and $\Omega$ is either the Riemann Sphere, the complex plane or the complex disc. Hence we found global isotherm coordinates $F:\Omega\rightarrow f(\dot{D})$.
Now I have two questions:
1) Can we extend these isotherm coordinates $F$ to the boundary $f(\partial D)$?
2) If the answer to the first question is yes, can we assume $\Omega$ to be the complex disc $D$?
Maybe you can even provide me with a good reference which answers one of my questions. In any case, thank you for your time.
The answer to both questions seems to be yes. Mazzeo and Taylor sketched a proof in the end of section 2 in their paper 'Curvature and Uniformization', see also https://arxiv.org/abs/math/0105016.