I was going through this wiki page Link on the Liouville's equation. In that page there is this comment without any source 'Liouville's equation is equivalent to the Gauss–Codazzi equations for minimal immersions into the $3$-space, when the metric is written in isothermal coordinates $z$ such that the Hopf differential is $dz^2$.'
I am looking for a proof of the above statement. I am not aware whether this is a trivial/well known statement in the literature. I am asking for any source for this or a possible way to approach this proof.