Consider two smooth, compact surfaces $\mathcal{S_i} \subset \mathbb{R}^3$, with Riemannian metrics $g_i$, $i=1,2$ and a conformal mapping $f:S_1\rightarrow S_2$.
Suppose we know the conformal factor $u$, namely $g_2 = e^{2u} g_1$. Is that information enough to explicitly give the mapping of the embedding of $\mathcal{S_1}$ to corresponding points of the embedding of $\mathcal{S_2}$?