= Solution
Let $\zeta=u+iv$. Direct differentiation gives
$$
\boxed{\varphi_1=1-\zeta^2,\qquad
\varphi_2=i(1+\zeta^2),\qquad
\varphi_3=2\zeta}.
$$
These are holomorphic <polynomials> and their squared sum is
$$
(1-\zeta^2)^2-(1+\zeta^2)^2+4\zeta^2=0.
$$
Moreover direct calculation of the metric gives $E=G=(1+u^2+v^2)^2>0$ and $F=0$, so the map is genuinely an <immersion> everywhere. The preceding characterizations prove that it is an \b[<isothermal parametrization> of a <minimal surface>], the <Enneper surface>.
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