Let $\Omega \subset \mathbf{R}^2$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is *not* assumed invariant in the vertical direction.) Let $u: \Omega \to \mathbf{R}$ be a $C^2$ function.

What is the expression for the quasilinear operator $F(x,z,p,X)$ expressing the mean curvature of $G$, i.e. so that $F(x,u,Du,D^2 u) = 0$ if $G$ is minimal?

*Edit. I would be happy with an expression valid for example for diagonal metrics, that is of the form $g = g_{11} \mathrm{d} x^2 + g_{22} \mathrm{d} y^2 + g_{33} \mathrm{d} z^2$.*

In the book of Colding and Minicozzi (pp. 235-236) the following formula is given: \begin{equation} F(x,z,p,X) = h^{ij}(X_{ij} + \Gamma_{ij}^3 + p_i \Gamma_{3j}^3 + p_j \Gamma_{i3}^3 + p_i p_j \Gamma_{33}^3) \\ - p_k h^{ij}(\Gamma_{ij}^k + p_i \Gamma_{3j}^k + p_j \Gamma_{i3}^k + p_i p_j \Gamma_{33}^k), \end{equation} where $h^{ij}$ is the inverse of $h_{ij} = \langle \partial_i + p_i \partial_3,\partial_j + p_j \partial_3 \rangle_g$.

Is this formula accurate? I can't follow their derivation, because it starts with an expression for the unit normal to $G$ that looks incorrect to me: \begin{equation} N = \frac{1}{W}(-u_1 \partial_1 - u_2 \partial_2 + \partial_3), \end{equation} with $W = (1 + g^{ij} u_i u_j)^{1/2}$.