conformal structure of a disc

Question

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$ is the flat metric on $D^2$? I guess this may follows from the fact that the conformal structure on the unit sphere $S^2$ is unique.

Answer 1

The uniformization theorem says that there are exactly two conformal structures on the open unit disk $D^2$ (up to conformal mapping): The standard one and the obtained via pull-back of the conformal structure on ${\mathbb C}$ via a diffeomorphism $D^2\to {\mathbb C}$. These two structures are not conformally equivalent. However, in general, if you put some Riemannian metric on $D^2$ it is very hard (actually, usually impossible) to construct explicitly a conformal mapping to either one of the standard models. Even figuring out which one to use is not that easy. For instance, the one you mentioned, $g=dx^2 + 2dy^2$ is conformal to the flat metric on an ellipse, but I do not think there is a nice formula for the Riemann mapping from an ellipse to the unit disk, see here and here.