Let $\mathbb{D}$ denote the unit disc in the complex plane centered at origin. Let us define the set $M=\{(z_1, z_2) \in \mathbb{D}^2 : |z_1|+|z_2|=1\}$. Also, let $\Omega=\mathbb{D}^2\setminus \{(0,0)\}$. Note that $\Omega$ is an open set in $\mathbb{C}^2$ containing $M$. We want to show that $M$ is a Levi-flat smooth hypersurface in $\mathbb{C}^2$.
$M$ is a smooth hypersurface with the locally defining function $r(z_1, z_2)=|z_1|+|z_2|-1$. Indeed, for every $z=(z_1, z_2) \in M$, take the open neighbourhood $U_z=\Omega$ of $z$. Then $r(z_1, z_2)$ is a smooth function on $\Omega$ and gradient of $r$ is non-vanishing on $\Omega$. It also follows that $M \cap U_z=\{w \in U_z : r(w)=0 \}$. Thus, $M$ becomes a smooth hypersurface in $\mathbb{C}^2$.
I want to prove that $M$ is Levi-flat, i.e. the Levi form of $M$ vanishes. Mathematically, the following holds: (Levi-flat condition) $\overset{2}{\underset{i, j=1}{\sum}}\bar{a}_ia_j\frac{\partial ^2 r}{\partial \bar{z}_i\partial z_j}=0$ whenever $\overset{2}{\underset{i=1}{\sum}}a_i\frac{\partial r}{\partial z_i}=0$ on $M$.
I am new to several complex variables and I do not know how to prove this. Apologies in advance.