Let $\Sigma$ be a Riemann surface with complex structure $j$ and a volume form $dvol_\Sigma$. I read somewhere that one can take the so-called 'conformal coordinates' $z=s+it$ so that $j\partial_s = \partial_t$ and $dvol_\Sigma = f^2(ds\wedge dt)$ for some function $f$.

Question: When can we find a conformal coordinate so that locally $dvol_\Sigma =ds\wedge dt$, equivalently, the induced metric is locally $ds^2+dt^2$? For example, can we simply use a new coordinate $s'=f(s,t) s$ and $t' =f(s,t) t$?

I guess it would be sufficient if $\Sigma$ is flat but I am not sure. Can someone tell me the whole picture of this question? Thanks!


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