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This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific.

Suppose that S is a minimal surface without any umbilical points (that is to say that $k_1 = -k_2$ everywhere). Let $barx$ be a parameterization of the unit sphere by stereograhic projection, and consider a neighborhood $V$ of a point $p$ on the surface so that the Gauss map $N: S rightarrow S^2$ restricted to the neighborhood $V$ is a diffeomorphism.

The question is to show that the parameterization $q= N^-1$ o $bar x$ is isothermal, meaning that $<q_u, q_u > = <q_v, q_v>$ and $<q_u,q_v> = 0$ everywhere on $S$.

Now when I saw this question, my first instinct was to try to verify that $q$ was isothermal directly by computing the parameterization in some kind of coordinates directly. I know that stereographic projection is given by the formulas:

$$ x= frac 4uu^2+v^2+4$$

$$ y = frac 4vu^2+v^2+4$$

$$ z = frac 2(u^2+v^2)u^2+v^2+4 $$

I figured with this information, maybe I could find a way to write down what $q$ is in terms of $u$ and $v$ and take partials. The problem is that I don't know how to write down anything about $N^-1$. Like when working with $N$, I can say that a point on a surface $S$ goes to it's normal, which is something that I can compute in terms of cross products of partial derivatives of a parameterization of the surface.

Is this method workable? I don't know if I can proceed any further this way. If not, what can I do to approach this problem?

The formulas of the stereographic projection give $langle barx_u,barx_urangle = langle barx_v,barx_vrangle$ and $langle barx_u,barx_vrangle = 0$, and the first part of this exercise says $langle mathrmdN(q_u),mathrmdN(q_u)rangle = lambda_p langle q_u,q_urangle$, hence $$beginalign*
langle q_u,q_urangle &= lambda_p^-1langle mathrmdN(q_u),mathrmdN(q_u)rangle \
&= lambda_p^-1 langle (Ncirc q)_u,(Ncirc q)_urangle\
&= lambda_p^-1 langle barx_u,barx_urangle \
&= lambda_p^-1 langle barx_v,barx_vrangle \
&= langle q_v,q_vrangle
endalign*$$
Similarly $$beginalign*
langle q_u,q_vrangle &= lambda_p^-1langle mathrmdN(q_u),mathrmdN(q_v)rangle \
&= lambda_p^-1 langle (Ncirc q)_u,(Ncirc q)_vrangle\
&= lambda_p^-1 langle barx_u,barx_vrangle \
&= 0
endalign*$$

Stereographic projection is conformal and you can also show the Gauss map is conformal for a minimal surface.

This is not very hard; simply assume $langle dN_p(t_1),dN_p(t_2)rangle=lambda(p)langle t_1,t_2rangle forall t_1,t_2 in T_pS$ and then take the basis of $T_pS$ consisting of the principal directions of the gauss map. A quick bit of algebra will show that the principal curvatures satisfy $k_1=-k_2$.

Hence the parameterisation you mention is a conformal mapping from the plane, hence isothermal.