Solution
= Solution
The inverse first-fundamental-form <matrix> is
$$
(I^S)^{-1}=\frac1{W^2}
\begin{pmatrix}
1+h_y^2&-h_xh_y\\
-h_xh_y&1+h_x^2
\end{pmatrix}.
$$
Thus $2H=\operatorname{tr}((I^S)^{-1}II^S)$ vanishes exactly when
$$
\boxed{
(1+h_y^2)h_{xx}-2h_xh_yh_{xy}+(1+h_x^2)h_{yy}=0.}
$$
This is the <minimal surface equation for a graph>.