The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that $\Psi^* g_{U}=e^{u}g_{D}$ where $u\in C^\infty(D)$ and $g$ denotes the Euklidean metric on $U$ respectively $D$.

My question is now whether the "inverse" is true, i.e.: given a function $u\in C^\infty(D)$, is there a domaine $U$ and a diffeomorphism $\Psi: D \to U$ such that $\Psi^* g_{U}=e^{u}g_{D}$ holds?

If it doesn't hold in general: Under what conditions on $u$ does it hold?