It's a well-known but difficult theorem that any $C^2$ surface in $R^3$ with parameter domain the unit disk can be put into isothermal parameters. The Wikipedia article on isothermal coordinates references several proofs (none of which I can access immediately). Suppose given a family of surfaces $X^t$ such that the map $t \mapsto X^t$ is continuous from some interval of $t$ values to $C^n(D,R^3)$. Then can we find surfaces $Y^t$ such that $Y^t$ gives isothermal parameters for $X^t$ and the map $t \mapsto Y^t$ is continuous into $C^n(D,R^3)$?

It looks like this might follow if we know (a) the isothermal parameters are solutions of some differential equation and (b) solutions of that differential equation depend continuously in the $C^n$ metric on parameters. That proof promises to involve checking a lot of details in two complicated proofs and will look like hand-waving. So perhaps this is stated somewhere in the literature?