= Solution
The map $\phi$ is an <immersion> when $d\phi$ has rank two, equivalently $\phi_u$ and $\phi_v$ are linearly independent. It is <isothermal coordinates>[isothermal] when its first fundamental form has
$$
E=G=\lambda^2>0,
\qquad F=0.
$$
In general the <mean curvature> is
$$
H=\frac{eG-2fF+gE}{2(EG-F^2)}.
$$
The surface is <minimal surface>[minimal] when $H=0$. In isothermal coordinates the <Laplace-Beltrami operator> identity is
$$
\phi_{uu}+\phi_{vv}=2\lambda^2H N.
$$
Thus $H=0$ exactly when each coordinate function $x,y,z$ is <harmonic function>[harmonic].
For
$$
\phi(u,v)=(f(u)\cos v,f(u)\sin v,g(u)),
$$
the isothermal conditions are
$$
f'^2+g'^2=f^2,
$$
and harmonicity gives
$$
f''-f=0,
\qquad g''=0.
$$
Consequently
$$
f=Ae^u+Be^{-u},
\qquad g=cu+d,
$$
and the isothermal equation reduces to
$$
c^2=4AB.
$$
These are all the solutions for which $f$ never vanishes. When $AB>0$ they parameterize a <catenoid>, up to translations, reflection and scaling. The limiting cases $AB=0$, $c=0$, with exactly one of $A,B$ nonzero, parameterize a punctured plane.
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