= Solution
At a non-planar point of a <minimal surface>, the <shape operator> is not zero. Minimality gives nonzero opposite <principal curvatures>, so $K<0$ nearby and $dN$ is invertible. The <inverse function theorem> makes the <Gauss map> a local diffeomorphism onto a patch of the sphere; part (ii) makes it conformal.
Choose stereographic coordinates $(u,v)$ on that sphere patch. Their metric is $4(du^2+dv^2)/(1+u^2+v^2)^2$. Pulling them back by $N$, the conformal identity $N^*g_{S^2}=\lambda g_S$ gives
$$
g_S=\frac4{\lambda(1+u^2+v^2)^2}(du^2+dv^2).
$$
The positive coefficient shows that \b[these are local <isothermal coordinates>]. No general isothermal-coordinate existence theorem is needed: the nondegenerate conformal <Gauss map> supplies them explicitly.
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