I'm trying to prove the Theorem of Catalan of minimal ruled surfaces, which states that the only minimal ruled surfaces are planes and helicoids.
According to Wolfgang Kühnel's Differential Geometry exercise 12, ch. 3, one should first verify that the Gaussian curvature $K$ and the mean curvature $H$ of a ruled surface
$$ f(t,s) = c(t) + sX(t) $$
given by standard parameters (i.e. such that $\lVert X \rVert = \lVert X' \rVert = 1$ and $\langle c', X' \rangle = 0$), are given by
$$ K = \frac{-\lambda}{(\lambda ^2 + s^2)^2} \quad \quad H = -\frac{1}{2 (\lambda^2 + s^2)^\frac{3}{2}} (Js^2 + \lambda' s + \lambda(\lambda J + F))$$
where
$$ F := \langle c', X \rangle $$ $$ \lambda := \langle c' \times X, X' \rangle = \det (c', X, X') $$ $$ J := \langle X'', X \times X' \rangle = \det (X, X', X'') $$
each of which is a function only of $t$.
With this proved, one should be able to determine all minimal ruled surfaces "with ease".
I have proved the expressions for $K$ and $H$ after somewhat lengthy calculations, but I don't know how to use them in order to prove that the plane and the helicoid are the only minimal ruled surfaces. I know that for a surface to be minimal, $H = 0$.
$H = 0$ tells you that $Js^2 + \lambda' s + \lambda(\lambda J + F) \equiv 0$. Since this is a quadratic polynomial in $s$ whose coefficients (which depend only on $t$) must vanish, we get $J = 0, \lambda' = 0$, which means $\lambda$ is constant, and $J = \lambda F = 0$. Therefore $X''$ is a linear combination of $X$ and $X'$, but since $\langle X'', X' \rangle = 0$ (because $||X'|| \equiv 1$) and $\langle X'', X \rangle \equiv - 1$ (because $$||X|| \equiv 1 \implies \langle X, X' \rangle = 0\implies \langle X'', X \rangle = -||X'||^2 = - 1$$), it follows that $X'' = -X$, i.e, $X$ defines a circle. If $\lambda = 0$ then we get $K = H = 0$ and the surface must be a plane. If $\lambda \neq 0$ then we must have $F = 0$, which implies $c' = \lambda X \wedge X'$ and differentiating we get $X'' = 0$, therefore $X$ must define a line perpendicular to the circle defined by $X$. Without loss of generality we can suppose that $c$ and $X$ are given by $c(t) = (0, 0, ct)$ for some $c \neq 0$ and $X(t) = (\cos(t), \sin(t), 0)$. It's easy to see that the correspending parametrization defines a helicoid, which proves Catalan's theorem.