graph in $\C^2$ of $f=u+iv:\C\to\C$ is a complex surface if and only if $u+iv$ is holomorphic, which implies that $u,v$ are harmonic. We know that special Lagrangians in $\C^2$ are complex surfaces for a different complex structure, so this is expected.

If $n=3$, $$I+i\text{Hess}\, F=\left(\begin{array}{ccc} 1+iF_{xx} & i F_{xy} & iF_{xz} \\ iF_{yx} & 1+iF_{yy} & iF_{yz} \\ iF_{zx} & iF_{zy} & 1+iF_{zz} \end{array}\right).$$ Hence, $$\begin{aligned}
\operatorname{Im}\operatorname{det}_{\C}(I+i\operatorname{Hess}F)=&\;F_{xx}+F_{yy}+F_{zz}\\
&-F_{xx}(F_{yy}F_{zz}-F_{yz}^2)-F_{xy}(F_{yz}F_{zx}-F_{xy}F_{zz})-F_{zx}(F_{xy}F_{yz}-F_{yy}F_{zx}).\end{aligned}$$ Therefore, $N$ is special Lagrangian if and only if $$\begin{aligned}
-\Delta F&=F_{xx}+F_{yy}+F_{zz}\\
&=F_{xx}(F_{yy}F_{zz}-F_{yz}^2)-F_{xy}(F_{xy}F_{zz}-F_{yz}F_{zx})+F_{zx}(F_{xy}F_{yz}-F_{yy}F_{zx})\\
&=\operatorname{det}\operatorname{Hess}F.\end{aligned}$$

We now wish to describe some very important examples of special Lagrangians, which are asymptotic to pairs of planes.

$\operatorname{SU}(n)$ acts transitively on the space of special Lagrangian planes with isotropy $\operatorname{SO}(n)$. So any special Lagrangian plane is given by $A\cdot\R^n$ for $A\in\operatorname{SU}(n)$ where $\R^n$ is the standard real $\R^n$ in $\C^n$.

Given $\theta=(\theta_1,\ldots,\theta_n)$ we can define a plane $P(\theta)=\{(e^{i\theta_1}x_1,\ldots,e^{i\theta_n}x_n)\in\C^n\,:\,(x_1,\ldots,x_n)\in\R^n\}$ (where we can swap orientation). We see that $P(\theta)$ is special Lagrangian if and only if $\operatorname{Re}\Upsilon|_P=\pm\cos(\theta_1+\ldots+\theta_n)=1$ so that $\theta_1+\ldots+\theta_n\in\pi\Z$. Given any $\theta_1,\ldots,\theta_n\in(0,\pi)$ with $\theta_1+\ldots+\theta_n=\pi$, there exists a special Lagrangian $N$ (called a *Lawlor neck*) asymptotic to $P(0)\cup P(\theta)$: see, for example, [@Joycebook2 Example 8.3.15] or $\S$\[s:angle\] for details. It is diffeomorphic to $\mathcal{S}^{n-1}\times\R$. By rotating coordinates we 